MEGA 2013E�e tive Methods in Algebrai GeometryJune 3rd to June 7th, 2013Goethe University in Frankfurt am MainAddress:Goethe UniversityCampus Bo kenheimMertonstr. 17�21Main le ture room: H IV60325 Frankfurt am MainExe utive ommittee for MEGA 2013Chair: Bernard Mourrain (INRIA Sophia Antipolis)Members: Alin Bostan (INRIA Paris-Ro quen ourt), Wolfram De ker (TU Kaiserslautern), Ali- ia Di kenstein (Univ. of Buenos Aires), Jan Draisma (TU Eindhoven), Christian Haase (GoetheUniversity, Frankfurt), Tomas Re io (U Cantabria, Santander), Thorsten Theobald (Goethe Uni-versity, Frankfurt) Lo al ommittee for MEGA 2013Chair: Thorsten Theobald (Frankfurt)Vi e- hair: Christian Haase (Frankfurt)Members: María Angéli a Cueto (Frankfurt), Jan Hofmann (Frankfurt), Mi hael Joswig (TUDarmstadt), Andreas Pa�enholz (TU Darmstadt), Christian Trabandt (Frankfurt), Annette Werner(Frankfurt), Timo de Wol� (Frankfurt)Students: Franziska Bommel (Frankfurt), Thorsten Jörgens (Frankfurt)S ienti� advisory board of the MEGA onferen e seriesIsabel Bermejo (La Laguna, Spain), Alin Bostan (Paris-Ro quen ourt, Fran e), Arjeh Cohen(Eindhoven, Netherlands), Carlos D'Andrea (Bar elona, Spain), Sandra Di Ro o (Sto kholm,Sweden), James H. Davenport (Bath, UK), Wolfram De ker (Kaiserslautern, Germany), Ali- ia Di kenstein (Buenos Aires, Argentina), Jan Draisma (Eindhoven, Netherlands), Ioannis Z.Emiris (Athens, Gree e), Jean-Charles Faugere (Paris-Ro quen ourt, Fran e), André Galligo(Ni e, Fran e), Vladimir Gerdt (Dubna, Russia), Patrizia Gianni (Pisa, Italy), Mar Giusti (Paris,Fran e), Laureano Gonzalez-Vega (Santander, Spain), Gert-Martin Greuel (Kaiserslautern, Ger-many), Dima Y. Grigoriev (Rennes, Fran e), Mark van Hoeij (Tallahassee, FL, USA), EvelyneHubert (Sophia Antipolis, Fran e), Rimvydas Krasauskas (Vilnius, Lithuania), Daniel Lazard(Paris, Fran e), Grégoire Le erf (Palaiseau, Fran e), Gunter Malle (Kaiserslautern, Germany),Bernard Mourrain (Sophia Antipolis, Fran e), Luis M. Pardo (Santander, Spain), Ragni Piene(Oslo, Norway), Tomas Re io (Santander, Spain), Marie-Françoise Roy (Rennes, Fran e), Massi-miliano Sala (Dublin, Ireland), Josef S hi ho (Linz, Austria), Mi hael Singer (Raleigh, NC, USA),Pablo Solerno (Buenos Aires, Argentina), Martin Sombra (Bar elona, Spain), Frank Sottile (TexasA&M, USA), Nobuki Takayama (Kobe, Japan), Thorsten Theobald (Frankfurt, Germany), CarloTraverso (Pisa, Italy), Franz Winkler (Linz, Austria)2

ContentsTime Table 4S hedule 4List of A epted Posters 8List of Restaurants and Cafes in Bo kenheim 9List of Restaurants in Sa hsenhausen 11Conferen e Dinner 13Re eption 13Abstra ts 15Abstra ts of Posters 41

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Time TableTime MONDAY TUESDAY WEDNESDAY THURSDAY FRIDAY8:00 - 8:50 Registration8:50 - 9:00 Wel ome9:00 - 9:50 Plenary Talk Plenary Talk Plenary Talk Plenary Talk Plenary TalkS. Sullivant L. Caporaso F. Cu ker R. Thomas B. Nill10:00 - 10:30 Co�ee Break Co�ee Break &Poster Session Co�ee Break Co�ee Break Co�ee Break10:30 - 10:55 Talk Talk (2 parallel) Talk (2 parallel) Talk (2 parallel)11:05 - 11:30 Talk Talk (2 parallel) Talk (2 parallel) Talk (2 parallel)11:40 - 12:05 Talk Talk (2 parallel) Talk (2 parallel) Talk (2 parallel) Talk (2 parallel)12:10 - 14:00 Lun h Break Lun h Break Free afternoon/Ex ursion Lun h Break Lun h Break14:00 - 14:50 Plenary Talk Plenary Talk Plenary Talk Plenary TalkM. S hweighofer G. Ottaviani B. Edixhoven F.-O. S hreyer15:05 - 15:30 Talk Talk (2 parallel) Software Presentation Talk (2 parallel)15:35 - 16:10 Co�ee Break Co�ee Break Co�ee Break Co�ee Break16:10 - 16:35 Talk Talk (2 parallel) Software Presentation Talk16:45 - 17:10 Talk Talk (2 parallel) Software Presentation Goodbye17:20 - 17:45 Talk (2 parallel) Forward Look/Dis ussion17:45 - 18:2018:30 - 19:0019:00 - Re eption at Conferen ethe Gästehaus DinnerS heduleAll plenary talks, single talks and talks announ ed as �Talk A� are lo ated in le ture room H IV.The talks announ ed as �Talk B� are lo ated in le ture room H 14.Monday 3 June8:00 - 8:50 Registration8:50 - 9:00 Wel ome / Opening Ceremony9:00 - 9:50 Plenary Talk: Seth Sullivant �Identi�able Reparametrizations of Linear Input-Output Equations�10:00 - 10:30 Co�ee Break10:30 - 10:55 Talk: J. S hi ho (with G. Hegedüs, Z. Li, H.-P. S hrö ker) �A Genus Bound forClosed 6R Linkages�11:05 - 11:30 Talk: W. Bruns (with C. Söger) �The Computation of Generalized Ehrhart Seriesin Normaliz�11:40 - 12:05 Talk: M. Rojas (with M. Avendaño, R. Kogan, M. Nisse) �Metri Estimates forAr himedean Amoebae and Tropi al Hypersurfa es�12:10 - 14:00 Lun h Break 4

14:00 - 14:50 Plenary Talk: Markus S hweighofer �Polynomial Optimization via Semide�niteProgramming�15:05 - 15:30 Talk: M. Sombra (with F. Amoroso, L. Leroux) �Overdetermined Systems ofSparse Polynomial Equations�15:35 - 16:10 Co�ee Break16:10 - 16:35 Talk: F. Bihan �Extremal Polynomial Systems Supported on Cir uits�16:45 - 17:10 Talk: L. Tabera �On Real Tropi al Bases and Real Dis riminants�19:00 - Re eption at the Gästehaus of the Goethe UniversityTuesday 4 June9:00 - 9:50 Plenary Talk: Lu ia Caporaso �Linear Series on Algebrai Curves via GraphTheory�10:00 - 11:40 Co�ee Break & Poster Session11:40 - 12:05 Talk A: J. Rauh (with T. Kahle) �Segre Produ ts, Tori Fiber Produ ts andNormality�Talk B: M. Giusti (with A. Colin) �Geometry Asso iated to a Finite Subgroupand Evaluation Te hniques�12:10 - 14:00 Lun h Break14:00 - 14:50 Plenary Talk: Giorgio Ottaviani �Complexity of Matrix Multipli ation and Ten-sor Rank�15:05 - 15:30 Talk A: P. Lella (with E. S hlesinger): �Expli it Constru tion of Degenerations ofSpa e Curves to Extremal Curves�Talk B: M. Masdeu (with X. Guitart) �Numeri al Eviden e for the Rationalityof Darmon Points�15:35 - 16:10 Co�ee Break16:10 - 16:35 Talk A: J. Hauenstein (with W. Hao, B. Hu, A. Sommese) �Solving PolynomialSystems Arising from Di�erential Equations�Talk B C. D'Andrea (with T. Cortadellas Benítez) �Minimal Generators of theDe�ning Ideal of the Rees Algebra Asso iated to a Rational Plane Parameterizationwith µ = 2�16:45 - 17:10 Talk A: N. Hein (with F. Sottile, J. Hauenstein) �Certi�able Numeri al Compu-tations in S hubert Cal ulus�Talk B: L. Busé (with N. Botbol, M. Chardin) �Fitting Ideals and Multiple-pointsof Surfa e Parametrizations�17:20 - 17:45 Talk A: M. Pellegrini (with C. Mar olla, M. Sala) �On the Hermitian Curve andits Interse tions with some Coni s�Talk B: R. Lebreton �Relaxed Hensel Lifting of Triangular Sets�Wednesday 5 June9:00 - 9:50 Plenary Talk: Felipe Cu ker �Re ent Advan es on Smale's 17th Problem�5

10:00 - 10:30 Co�ee Break10:30 - 10:55 Talk A: T. de Wol� (with S. Iliman) �Separating Inequalities for NonnegativePolynomials that Are not Sums of Squares�Talk B: S. Urbinati �Divisorial Models of Normal Varieties�11:05 - 11:30 Talk A: G. Gusev (with A. Esterov) �Systems of Equations with a Single Solution�Talk B: M. Stamps (with A. Engström) �Betti Diagrams from Graphs�11:40 - 12:05 Talk A: G. Jeronimo (with D. Perru i) �A Probabilisti Symboli Algorithm toFind the Minimum of a Polynomial Fun tion on a Basi Closed Semialgebrai Set�Talk B: A.-S. Elsenhans �Expli it Computations of Invariants of Plane Quarti Curves�12:20 - 19:00 Free afternoon / Ex ursion19:00 - Conferen e DinnerDepotThursday 6 June9:00 - 9:50 Plenary Talk: Rekha Thomas �Algebrai Geometry in Computer Vision�10:00 - 10:30 Co�ee Break10:30 - 10:55 Talk A: D. Plaumann (with A. Leykin) �Determinantal Representations of Hy-perboli Curves Via Polynomial Homotopy Continuation�Talk B: P.-V. Kosele� (with D. Pe ker) �On Alexander-Conway Polynomials ofTwo-bridge Links�11:05 - 11:30 Talk A: A. Boysal (with W. Baldoni, M. Vergne) �Multiple Bernoulli Series andVolumes of Moduli Spa es of Flat Bundles�Talk B: E. Gar ía-Llorente (with I. Bermejo) �Castelnuovo-Mumford Regularityof Proje tive Monomial Curves Asso iated to Arithmeti Sequen es�11:40 - 12:05 Talk A: T. Kri k (with C. D'Andrea, A. Szanto) �Subresultants, Sylvester Sumsand the Rational Interpolation Problem�Talk B: A. Aleksandrov: �Computing the Index of Ve tor Fields at Cohen-Ma aulay Curves�12:10 - 14:00 Lun h Break14:00 - 14:50 Plenary Talk: Bas Edixhoven �Polynomial Time Computation of Galois Repre-sentations Atta hed to Modular Forms�15:05 - 15:30 Software Presentation A: R. Krone (with C.J. Hillar, A. Leykin) �Algorithms forEquivariant Gröbner Bases�Software Presentation B: C. Söger (with B. I him, W. Bruns) �Normaliz: Algo-rithms for Rational Cones and A�ne Monoids�15:35 - 16:10 Co�ee Break16:10 - 16:35 Software Presentation A: A. Chan �Computing Tropi al Curves via Proje tions�Software Presentation B: S. Kei her (with J. Hausen) �MDS � a Pa kage for Com-putations with Mori Dream Spa es�6

16:45 - 17:10 Software Presentation A: M. Barakat, J. Boehm, Y. Ren (with W. De ker, H.S hoenemann) �Conne ting Singular, GAP, Polymake and Gfan�Software Presentation B: R. Vidunas (with M. van Hoeij) �Software for omput-ing Belyi Fun tions of Genus 0�17:20 - 18:20 Forward Look / Dis ussion SessionFriday 7 June9:00 - 9:50 Plenary Talk: Benjamin Nill �Latti e Polytopes with a View Toward Algebrai Geometry�10:00 - 10:30 Co�ee Break10:30 - 10:55 Talk A: M. Mi halek (with L. Oeding, P. Zwiernik) �Se ant Cumulants and Tori Geometry�Talk B: A. Boralevi (with E. Mezzetti, D. Faenzi) �Linear Spa es of Matri es ofConstant Rank and Instanton Bundles�11:05 - 11:30 Talk A: F. Blo k (with D. Ma lagan) �A Tropi al Approa h to Computing E�e -tive Cones�Talk B: P. Gallardo �Comparing the KSBA and the GIT Compa ti� ation of theModuli Spa e of Quinti Surfa es�11:40 - 12:05 Talk A: J. Rodriguez �Combinatorial Ex ess Interse tion�Talk B: V. Levandovskyy (with J.M. Morales) �Central Chara ters, Bernstein-Sato Polynomials and Asso iated Strati� ations�12:10 - 14:00 Lun h Break14:00 - 14:50 Plenary Talk: Frank-Olaf S hreyer �Finite Fields in Computational Algebrai Geometry�15:05 - 15:30 Talk A: S. Tavenas (with P. Koiran, N. Portier) �A Wronskian Approa h to theReal τ -Conje ture�Talk B: C. Reynoso Al ántara �Strati� ation of the Spa e of Foliations on CP2�15:35 - 16:10 Co�ee Break16:10 - 16:35 Talk: W.M. Seiler (with M. Fetzer, E. Sáenz-De-Cabezón) �On the Free Resolu-tion Indu ed by a Pommaret Basis�16:40 - 17:00 Goodbye

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List of A epted PostersThe presentation of the posters will o ur on Tuesday, June 4th, from 10:00am to 11:40am.It will take pla e in the front of the le ture hall H IV.List of a epted Posters1. P. Norén Tori Homogenous Markov Chain Models2. C. Vinzant (with D. Plau-mann, R. Sinn, D. E. Speyer) Hermitian Determinantal Representations of Hyperboli Curves3. F. Kiràly, Z. Rosen (with L.Theran) Symmetries and Finiteness for Algebrai Matroids4. C. Andradas (with T. Re io,R. Sendra, L. Tabera, C. Vil-larino) Reparametrizing Rational Revolution Surfa es over the Reals5. A. Lundman Lo al Positivity of Line Bundles on Tori Varieties and Cayley Poly-topes6. M. Compagnoni (with R.Notari, F. Antina i. A. Sarti) The Geometry of the TDOA-based Sour e Lo alization7. M. Narváez Clauss Bounds for the Height of a Parametrization of a Rational Plane Curve8. M. Roggero (with C. Ber-tone, F. Cio�) A Division Algorithm in an A�ne Framework for Flat Families Cover-ing Hilbert S hemes9. S. King An F5 Algorithm for Modules over Path Algebra Quotients and theComputation of Loewy Layers10. V. Fisikopoulos (with I.Emiris, C. Konaxis) A Software Framework for Computing Newton Polytopes of Resultantsand (Redu ed) Dis riminants11. M. Ceria JMBTest.lib and JMSConst.lib: Singular Tools for J-Marked S hemes12. B. Assarf (with M. Joswig, A.Pa�enholz) Wronski Polynomial Systems with Polymake and Singular

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List of Restaurants and Cafes in Bo kenheim1. �Albatros�, Kiesstraÿe 27, afe and restaurant, mixed uisine, 9am to 12pm,http://www. afe-albatros.de/2. �Ban Thai�, Leipziger Straÿe 26, restaurant, Thai uisine, 12am to 3pm and 6pm to 11pm (to12pm on Friday), http://www.banthairestaurants.de/3. �Bastos�, Gräfstraÿe 45, afe and restaurant, mixed uisine, 8am to 2am,http://www.bastos.de/4. �Bullys Burger�, Am Weingarten 12, burgerbar, 11.30am to 3pm and 6pm to 9.30pm, http://www.bullys-burger.de5. �Cafeteria�, Bo kenheimer Landstraÿe 133, mixed uisine, 8am to 5pm,http://www.studentenwerkfrankfurt.de/index.php?id=1416. �Crumble�, Kiesstraÿe 41, afe, breakfast & lun h spe ials, 8am to 8pm,http://www. afe rumble.de/7. �Da Cimino�, Adalbertstraÿe 29, pizzeria, 11am to 1am,http://www.pizzeria- imino.de/8. �Extrablatt�, Bo kenheimer Landstraÿe 141, afe and restaurant, mixed uisine, 8am to 1am (to2am on Friday), http://www. afe-extrablatt. om9. �farmer's market�, Bo kenheimer Warte, only on Thursday: 8am to 6pm,http://www.frankfurter-markthaendler.de/bo kenheim.html10. �Frankfurt & Friends�, Jordanstraÿe 1, afe, breakfast & restaurant, mixed uisine, 11am to12pm (to 1am on Friday), http://www.frankfurt-and-friends.de/11. �Joe Peñas�, Robert�Mayer�Straÿe 18, restaurant and bar, Spanish and Mexi an uisine, 5pm to1am (to 2am on Friday), http://www.joepenas. om/index.php12. �Kish�, Leipziger Straÿe 16a, restaurant, Persian uisine, 11.30am to 11.30pm,http://www.kish-restaurant.de/index.html13. �Mario & Lino�, Adalbertstraÿe 37 � 39, Italian uisine, 11am to 11pm,http://www.pizzeria-mario-lino.de14. �Mezzanotte�, Clemensstraÿe 6, restaurant, Italian uisine, 11.30am to 1am,http://www.restaurant-mezzanotte.de/15. �Namaste�, Jordanstraÿe 19, restaurant, Indian uisine, 11.30am to 3pm and 5.30pm to 11.30pm,http://www.namaste-frankfurt.de/16. �Peppino�, Adalbertstraÿe 13, pizzeria, 11.30am to 2.30pm and 5.30pm to 10.30pm17. �Pielok�, Jordanstraÿe 3, restaurant, home�style ooking, 11.30am to 2.30pm and 6pm to 10.30pm,http://www.restaurant-pielok.de/18. �Sausalitos�, Kiesstraÿe 36, restaurant & bar, basi ly Spanish and Mexi an uisine, 5pm to 1am(to 2am on Friday), http://www.sausalitos.de/19. �Subway�, Leipziger Str. 1, sandwi hes (fastfood), 9am to 10pm,http://www.subway-sandwi hes.de20. �Topo Gigio�, S hlossstraÿe 117, restaurant, Italian uisine, 12am to 3pm and 6pm to 11pm21. �T-Style�, Adalbertstraÿe 37, Japanese uisine, noon to 2pm, 6pm to 10pm,http://www.t-style-de. om22. �Vina Sushi�, Robert�Mayer�Straÿe 18, restaurant, Asian uisine and sushi, 11am to 3pm and5pm to 10pm, http://www.vina-sushi.de/23. �Volkswirts haft�, Jordanstraÿe 13, pub, �ngerfood, 5.30pm to 2am,http://www.vowi.net/24. �Zum Tannenbaum�, Homburger Str. 19, pub, �ngerfood, 5pm to 2am9

1 Albatros 13 Mario & Lino2 Ban Thai 14 Mezzanotte3 Bastos 15 Namaste4 Bullys Burger 16 Peppino5 Cafeteria 17 Pielok6 Crumble 18 Sausalitos7 Da Cimino 19 Subway8 Extrablatt 20 Topo Gigio9 farmer's market 21 T�Style10 Frankfurt & Friends 22 Vina Sushi11 Joe Peñas 23 Volkswirts haft12 Kish 24 Zum TannenbaumYou are here (le ture room) 10

List of Restaurants in Sa hsenhausen1. �Ambassel�, Deuts hherrenufer 28, Afri an uisine, from 5pm,http://www.ambassel-frankfurt.de2. �Apfelwein Dax�, Willemerstraÿe 11, German uisine, noon to midnight,http://www.apfelwein-dax.de3. �Bodega Los Gitanos�, Paradiesgasse 21, Spanish uisine, from 5pm ( losed on Monday)4. �Buenos Aires�, Dreiei hstraÿe 35, Argentinian and international uisine, 6pm to midnight,http://www.restaurant-buenos-aires.de5. �BurgerMeister�, Dreiei hstraÿe 20, burgerbar, noon to 11pm (to 2am on Friday),http://www.burgermeister-frankfurt.de6. �Exenberger�, Bru hstraÿe 14, German uisine, 11am to 11pm,http://www.exenberger-frankfurt.de7. �Grill-Haus�, Groÿe Rittergasse 52, German uisine, 4pm to midnight (from noon on Fri-day), http://www.grill-haus. om8. �Lokalbahnhof �, Darmstädter Landstr. 14, breakfast, German uisine, 9am to 1am,http://www.lokalbahnhof.info9. �Lorsba her Thal�, Groÿe Rittergasse 49, 4pm to midnight,http://www.lorsba her-thal.de10. �Nana�, S hweizer Straÿe 73, Vietnamese uisine, 9.30am to 6pm11. �S hreiber-Heyne Apfelweinwirts haft�, Mörfelder Landstraÿe 11, German uisine,from 5pm, http://www.s hreiber-heyne.de12. �S hreiber-Heyne Proletariat�, Abtsgäss hen 8, German uisine, from 5pm,http://www.s hreiber-heyne.de13. �Struwwelpeter�, Neuer Wall 3, German uisine, from 11am,http://www.struwwelpeter-frankfurt.de14. �Sushiko�, S hweizer Straÿe 61, sushi, Japanese uisine, noon to 2.30pm, 6pm to 10.30pm,http://www.frankfurt-sushi.de15. �Zum Ei hkatzerl�, Dreiei hstraÿe 29, German uisine, 5pm to 1am,http://www.ei hkatzerl.de16. �Zum gemalten Haus�, S hweizer Straÿe 67, German uisine, 10am to midnight ( losedon Monday), http://www.zumgemaltenhaus.de11

1 Ambassel 9 Lorsba her Thal2 Apfelwein Dax 10 Nana3 Bodega Los Gitanos 11 S hreiber-Heyne Apfelweinwirts haft4 Buenos Aires 12 S hreiber-Heyne Proletariat5 BurgerMeister 13 Struwwelpeter6 Exenberger 14 Sushiko7 Grill-Haus 15 Zum Ei hkatzerl8 Lokalbahnhof 16 Zum gemalten HausThe grey-marked area is the distri t Alt-Sa hsenhausen, whi h is famous for its ider and its pubs.Conferen e Dinner: DEPOT 1899, Textorstraÿe 33, http://www.depot1899.de12

Conferen e Dinner• The onferen e dinner takes pla e in the DEPOT 1899. The adress is Textorstraÿe 33.• The easiest way to rea h the onferen e dinner is to take the U-Bahn U1, U2, U3 or U8to Südbahnhof and walk down Brü kenstraÿe. The restaurant is on the right hand sideat the rossing Brü kenstraÿe / Textorstraÿe.• The onferen e dinner starts at 7pm.• If booked, you re eive a ti ket for the dinner with your onferen e materials.• The dinner fee in ludes water, soft drinks, jui es, beer, housewine, apple wine and hot drinks.• A vegetarian option will be available at the dinner. If you have any further wishes or needsregarding the food (e.g., vegan, kosher, halal, diet, restri tions due to an allergy), please onta t us on Monday, June 3rd.Re eptionOn Monday evening, you are ordially invited to a WELCOME RECEPTION in the beautifulguest house of Goethe University. It is lo ated some 20 minutes' walk from the onferen e site. (Seeon the map on the next page; a group led by lo als will leave for the guest house at 18:30.) Please ome and share some sna ks and drinks with us, onne t with the other parti ipants and helpthe MEGA COMMUNITY deserve its name. You re eive an entran e ti ket with your onferen ematerial.

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A: Le ture RoomB: Re eption / Gästehaus (Frauenlobstraÿe 1)

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Abstra tsIdenti�able Reparametrizations of Linear Input-Output EquationsSeth SullivantNorth Carolina State University, Raleigh, NC, USAMonday, June 3rd, 9:00 - 9:50, Room H IV, Plenary TalkSystems of ordinary di�erential equations are ubiquitous in the applied s ien es asmodels for the evolution of a system over time, o urring in mathemati al biology, hemistry, materials s ien e, physi s and numerous other areas. Su h models havemany parameters and often involve the evolution of quantities that are impossible ordi� ult to measure. A fundamental question arises: in whi h models is it possible tore over all the parameters from the measurements that are made? In models wherenot every parameter is identi�able, we an try to reparametrize the model in terms ofa smaller number of parameters to make the model identi�able.Solving these identi�ability problems usually requires tools from omputational alge-bra. First, input-output equations are al ulated using te hniques from di�erentialelimination theory. Se ond, the identi�ability of the parameter oe� ients is addressusing tools from omputational algebrai geometry. This talk will provide an overviewto the area and report on re ent theoreti al advan es for linear-input output equations.This is joint work with Ni olette Meshkat.A Genus Bound for Closed 6R LinkagesJosef S hi hoRICAM, Linz, AustriaMonday, June 3rd, 10:30 - 10:55, Room H IVA linkage is a me hanism omposed of a �nite number of rigid bodies, alled links,and onne tions between them, alled joints. The links move in three-dimensionalspa e, and when two links are onne ted by a revolute joint, then the relative motionis onstrained to a rotation around a �xed axis. A linkage onsisting of n links thatare y li ally onne ted by n revolute joints is alled a losed nR linkage.In kinemati s, one studies the set of all possible on�gurations of a linkage. If the on�guration set has positive dimension, then the linkage is mobile. This is always the ase for nR linkages when n ≥ 7. On the other hand, the lassi� ation of mobile losed6R linkages is still an open problem.Here we assume that the on�guration set of a losed 6R linkage is an irredu ible urve, and we prove that its genus is at most 5. The proof is based on the theory ofbonds, whi h has re ently been introdu ed as a method for the analysis of linkages withrevolute joints (see arxiv.org/abs/1206.4020).Coauthors: Gabor Hegedüs, Ke skemét College, Budapest, Hungary;Zijia Li, RICAM, Linz, Austria;Hans-Peter S hrö ker, University of Innsbru k, Austria.15

The Computation of Generalized Ehrhart Series in NormalizWinfried BrunsUniversität Osnabrü k, GermanyMonday, June 3rd, 11:05 - 11:30, Room H IVWe des ribe an algorithm for the omputation of generalized (or weighted) Ehrhartseries based on Stanley de ompositions as implemented in the o�spring NmzIntegrateof Normaliz. The algorithmi approa h in ludes elementary proofs of the basi results.We illustrate the omputations by examples from ombinatorial voting theory.Coauthors: Christof Söger, Universität Osnabrü k, Osnabrü k, Germany.

Metri Estimates for Ar himedean Amoebae and Tropi alHypersurfa esJ. Mauri e RojasTexas A&M University, College Station, TX, USAMonday, June 3rd, 11:40 - 12:05, Room H IVGiven any Laurent polynomial f ∈C[

x±11 , . . . , x±1

n

], we give an e� iently onstru tiblepolyhedral approximation, ArchTrop(f), of the image of any omplex algebrai hyper-surfa e under the log absolute value map: Our main result is an expli it upper bound(as a fun tion of the sparsity of f ) for the Hausdor� distan e between the two sets. Wethus obtain an Ar himedean analogue of Kapranov's Non-Ar himedean Amoeba Theo-rem, and a higher-dimensional extension of an earlier univariate estimate of Ostrowski.Our result also generalizes and re�nes an earlier estimate of Mikhalkin for the spe ial ase n=2.As appli ations, we obtain e� ient approximations for the possible norms of omplexroots of polynomial systems, and an alternative, arguably more geometri proof of aformula of Khovanski relating latti e points in polygons and urve genus.Coauthors: Martin Avendaño, University of Zaragoza, Zaragoza, Spain;Roman Kogan, Texas A&M University, College Station, TX, USA;Mounir Nisse, Texas A&M University, College Station, TX, USA.16

Polynomial Optimization via Semide�nite ProgrammingMarkus S hweighoferUniversity of Konstanz, GermanyMonday, June 3rd, 14:00 - 14:50, Room H IV, Plenary TalkPolynomial optimization is on erned with minimizing or maximizing a polynomial ob-je tive fun tion subje t to a �nite number of polynomial onstraints. The onstraintsare non-stri t real polynomial inequalities in several variables. A very su essful te h-nique to solve these problems has emerged sin e the turn of the millennium: The basi idea is to relax the problem by repla ing all nonlinear monomials by new variables. As amatter of ourse, the resulting linear program would in general arry little informationabout the original problem. To delimit the loss of information during the linearizationpro edure, one adds in�nite families of polynomial inequalities whi h are redundant inthe original polynomial optimization problem but arry important information afterbeing linearized. Ea h su h family is hosen in a way su h that it be omes a singlelinear matrix inequality when linearized. The linearized problem is no longer a linearprogram but still a semide�nite program and an therefore be solved e� iently. Thistalk is an introdu tion to the art of hoosing these families in the right way: The re-sulting semide�nite program should not be too big to solve it in pra ti e. At the sametime it should nevertheless arry enough information about the original problem. Theright hoi e is governed by results on the moment problem from fun tional analysis andon sums of squares representation of positive polynomials in real algebrai geometry.Overdetermined Systems of Sparse Polynomial EquationsMartín SombraICREA & Departament d'Àlgebra i Geometria, Universitat de Bar elona, SpainMonday, June 3rd, 15:05 - 15:30, Room H IVWe show that, for a system of univariate polynomials given in sparse odi� ation, itszero set an be des ribed by a single polynomial that an be omputed in time quasi-linear in the logarithm of the degree of the system. In parti ular, it is possible todetermine if su h a system of polynomials does have a zero, in time quasi-linear in thelogarithm of its degree. This algorithm relies on a result of Bombieri and Zannier onmultipli atively dependent points in subvarieties of an algebrai torus.We also present the following onditional partial extension to the higher dimensionalsetting: suppose that the e�e tive Zilber's onje ture holds true. Then, a system ofmultivariate polynomials given in sparse odi� ation an be redu ed, in time quasi-linear in the logarithm of its degree, to a �nite olle tion of omplete interse tionsoutside a hypersurfa e.Coauthors: Fran es o Amoroso & Louis Leroux, Laboratoire de mathématiques Ni olas Oresme,Université de Caen, Fran e.17

Extremal Polynomial Systems Supported on Cir uitsFrédéri BihanUniversité de Savoie, Chambéry, Fran eMonday, June 3rd, 16:10 - 16:35, Room H IVThe support of a real Laurent polynomial system of n equations in n variables is the setW ⊂ Zn of all the exponent ve tors of its monomials. A real Laurent polynomial systemis alled extremal if all its omplex solutions (in the n-dimensional omplex torus) arepositive solutions (lie in the positive orthant (0,+∞)n). A support W having n + 2elements is alled a ir uit. The author previously showed that the number of non-degenerate positive solutions of a system supported on a ir uit W ⊂ Zn is at mostm(W) + 1, where m(W) ≤ n is the dimension of the a�ne span of the mininal a�nelydependent subset ofW. We prove that if a ir uit W ⊂ Zn supports an extremal systemwith the maximal number m(W) + 1 of non-degenerate positive solutions, then it isunique up to the obvious a tion of the group of invertible integer a�ne transformationsof Zn. In the general ase, we prove that any any extremal system supported on a ir uit an be obtained from another one having the maximal number m(W) + 1 ofpositive solutions by means of some elementary transformations. As a onsequen e, weget for ea h n and up to the above obvious a tion a �nite list of ir uits W ⊂ Zn whi h an support extremal polynomial systems.On Real Tropi al Bases and Real Dis riminantsLuis Felipe TaberaUniversidad de Cantabria, Santander, SpainMonday, June 3rd, 16:45 - 17:10, Room H IVWe explore the on ept of real tropi al basis of an ideal in the �eld of real Puiseuxseries. We show expli it tropi al bases of zero-dimensional real radi al ideals, linearideals and hypersurfa es oming from ombinatorial pat hworking. On the other hand,a real radi al ideal without tropi al basis is presented. As an appli ation, we show howto ompute the set of singular points of a real tropi al hypersurfa e. i.e. we omputethe real part of the tropi al dis riminant.

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Linear Series on Algebrai Curves via Graph TheoryLu ia CaporasoUniversity Roma Tre, Rome, ItalyTuesday, June 4th, 09:00 - 09:50, Room H IV, Plenary TalkRe ent progress in ombinatori s and tropi al geometry have brought to light someremarkable analogies between the theory of algebrai urves and the theory of graphs;in parti ular, graph theory an be used to e�e tively ompute or bound the dimensionof linear series on algebrai urves, a well known very di� ult problem. The talk willsurvey some re ent results and onje tures within this theme.

Segre Produ ts, Tori Fiber Produ ts and NormalityJohannes RauhMax Plan k Institute for Mathemati s in the S ien es, Leipzig, GermanyTuesday, June 4th, 11:40 - 12:05, Room H IVThe tori �ber produ t is an operation that ombines two ideals that are homogeneouswith respe t to a grading by an a�ne monoid. The Segre produ t is a related onstru -tion that ombines two multigraded rings. The quotient ring by a tori �ber produ tof two ideals is a subring of the Segre produ t, but in general this in lusion is stri t.We ontrast the two onstru tions and show that any Segre produ t an be representedas a tori �ber produ t without hanging the involved quotient rings. This allows toapply previous results about tori �ber produ ts to the study of Segre produ ts.For tori ideals, we give riteria, when the Segre produ t of two a�ne tori varietiesis dense in their tori �ber produ t, and when the map from the Segre produ t to thetori �ber produ t is �nite. We give an example that shows that the quotient ring of atori �ber produ t of normal ideals need not be normal.Coauthors: Thomas Kahle, TU Mün hen, Gar hing, Germany.19

Geometry Asso iated to a Finite Subgroup and EvaluationTe hniquesMar GiustiLIX, CNRS-Polyte hnique, 91128 Palaiseau Cedex, Fran eTuesday, June 4th, 11:40 - 12:05, Room H 14Given a �nite subgroup of the general linear group, we study the geometry asso iatedto the algebra of polynomial invariants. On e hosen what are alled fundamental in-variants, the link between the dis riminant of the Noether proje tion and two othernatural dis riminants is established.Computationally speaking, the use of adequate data stru tures based on evaluationrather than writing, allows better e� ien y to �nd the multipli ation tensor.As an illustration we give a fast symboli algorithm to ompute Lagrange resolvents,either generi or spe ialized. These universal obje ts have their own interest, indepen-dently of lassi al appli ations � we don't have in mind here � to Galois theory.Eventually we show how to exhibit square-free resolvents, with a omplexity polynomialin the index of the subgroup, after a pre omputation depending only on the subgroup.Coauthor: Antoine Colin, LIX (address above) and Université de la Ro helle, Fran e.Complexity of Matrix Multipli ation and Tensor RankGiorgio OttavianiUniversity of Firenze, ItalyTuesday, June 4th, 14:00 - 14:50, Room H IV, Plenary TalkMatrix multipli ation is one of the basi operations in numeri al algorithms. Thestandard rule needs O(n3) elementary operations in order to multiply a pair of n ∗ nmatri es. Strassen showed in 1969 how to redu e this number to O(n2.81) and betterbounds are known today. Matrix multipli ation an be seen as a distinguished tensor ina spa e of dimension n6, and Strassen te hnique was based on bounding the rank of thistensor. The omputation of the rank and the border rank of the matrix multipli ationtensor remains an open and di� ult problem. We will show that the border rank(and the rank) of the matrix multipli ation tensor is bounded below by 2n2 − n, andother related results. These results are obtained, joint with J.M. Landsberg, by usingmultilinear algebra, representation theory and SL(2)-a tions.

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Expli it Constru tion of Degenerations of Spa e Curves to ExtremalCurvesPaolo LellaUniversità degli Studi di Torino, Italy.Tuesday, June 4th, 15:05 - 15:30, Room H IVWe report on re ent progress about the problem whether the Hilbert s hemes Hd,g oflo ally Cohen-Ma aulay urves in P3 are onne ted. The authors, in ollaboration withRobin Hartshorne (�Smooth urves spe ialize to extremal urves�, arXiv:1207.4588),have proven that any smooth irredu ible urve C ⊆ P3 an be spe ialized in a �atfamily to an extremal urve, in the sense of Martin-Des hamps and Perrin. It followsthat all smooth and irredu ible urves of degree d and genus g lie in a unique onne ted omponent of Hd,g. The basi tool used is Gröbner degeneration. Indeed, any extremal urve of degree d supported on a line is de�ned by an ideal homogeneous w.r.t. thegrading (d, 2, 1, 1). Hen e, it is natural to ask what are the urves whose generi initialideal w.r.t. (d, 2, 1, 1) is that of an extremal urve. In the talk, we will give algebrai andgeometri al des riptions of su h urves and we will dis uss the omputational aspe ts.We will exhibit several examples in order to show (1) how this te hnique applies to thealready-known ases and (2) what happens when this approa h fails.Coauthors: Enri o S hlesinger, Polite ni o di Milano, Italy.Numeri al Eviden e for the Rationality of Darmon PointsMar MasdeuColumbia University, New York, USATuesday, June 4th, 15:05 - 15:30, Room H 14We explain how one an extend the algorithms of [Darmon-Green℄ and [Darmon-Polla k℄ for omputing p-adi Darmon points on ellipti urves to the ase of omposite ondu tor, as well as the algorithm of [Darmon-Logan℄ for omputing ATR Darmonpoints to treat urves of nontrivial ondu tor. Both ases involve an algorithmi de- omposition into elementary matri es in ongruen e subgroups Γ1(N) for ideals N in ertain rings of S-integers. We use these extensions to provide additional eviden e insupport of the onje tures on the rationality of Darmon points.Coauthors: Xavier Guitart, Universitat Politè ni a de Catalunya, Bar elona, Spain.

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Solving Polynomial Systems Arising from Di�erential EquationsJonathan HauensteinNorth Carolina State University, Raleigh, NC, USATuesday, June 4th, 16:10 - 16:35, Room H IVSystems of nonlinear algebrai di�erential equations arise in many appli ations in awide range of subje t areas in luding biology, hemistry, e onomi s, engineering, andphysi s. Dis retization produ es a highly stru tured, large-s ale polynomial systemthat typi ally has only a few solutions of interest. This talk will explore using numeri- al methods in algebrai geometry designed for omputing su h solutions and presentexamples from appli ations where this approa h has been used.Coauthors: Wenrui Hao, University of Notre Dame, Notre Dame, IN, USA;Bei Hu, University of Notre Dame, Notre Dame, IN, USA;Andrew Sommese, University of Notre Dame, Notre Dame, IN, USA.Minimal Generators of the De�ning Ideal of the Rees AlgebraAsso iated to a Rational Plane Parametrization with µ = 2Carlos D'AndreaUniversity of Bar elona, SpainTuesday, June 4th, 16:10 - 16:35, Room H 14We exhibit a set of minimal generators of the de�ning ideal of the Rees Algebra asso- iated to the ideal of three bivariate homogeneous polynomials parametrizing a properrational urve in proje tive plane, having a minimal syzygy of degree 2.Coauthors: Teresa Cortadellas Benítez, University of Bar elona, Spain.

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Certi�able Numeri al Computations in S hubert Cal ulusNi kolas HeinTexas A&M University, College Station, TX, USATuesday, June 4th, 16:45 - 17:10, Room H IVTraditional formulations of geometri problems from the S hubert al ulus, either inPlü ker oordinates or in lo al oordinates provided by S hubert ells, yield systems ofpolynomials that are typi ally far from omplete interse tions and (in lo al oordinates)typi ally of degree ex eeding two. We present an alternative primal-dual formulationusing parametrizations of S hubert ells in the dual Grassmannians in whi h interse -tions of S hubert varieties be ome omplete interse tions of bilinear equations. Thisformulation enables the numeri al erti� ation of problems in the S hubert al ulus.Coauthors: Jonathan Hauenstein, North Carolina State University, Raleigh, NC, USA;Frank Sottile, Texas A&M University, College Station, TX, USA.Fitting Ideals and Multiple-points of Surfa e ParameterizationsLaurent BuséINRIA, Sophia Antipolis, Fran e.Tuesday, June 4th, 16:45 - 17:10, Room H 14Parameterized algebrai surfa es are ubiquitous in geometri modeling and the deter-mination of their singular lo i is an important problem. Given a parameterization φfrom P2 to P3 of a rational algebrai surfa e S, we will see in this talk that the setsof points on S whose preimage onsists in k or more points, ounting multipli ity, anbe des ribed in terms of Fitting ideals of some graded parts of the symmetri algebraasso iated to the parameterization φ. More pre isely, we will show that the drop ofrank of a ertain elimination matrix M(φ) at a given point P ∈ P3 is in relation withthe �ber of the graph of φ over P . Thus, the Fitting ideals atta hed to M(φ) provide a�ltration of the surfa e whi h is in orresponden e with the degree and the dimensionof the �bers of the graph of the parameterization φ. We will also omment on thelink with the double-point formulas that have been extensively studied in the �eld ofinterse tion theory for �nite maps.Coauthors: Ni olás Botbol, Buenos Aires University, Argentina;Mar Chardin, IMJ and Pierre et Marie Curie University, Fran e.

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On the Hermitian Curve and Its Interse tions with Some Coni sMar o PellegriniUniversità degli studi di Firenze, ItalyTuesday, June 4th, 17:20 - 17:45, Room H IVWe onsider Fq2 the �nite �eld with q2 elements, where q is a power of a prime. Forany q, the Hermitian urve H is the urve de�ned over Fq2 by the a�ne equationxq+1 = yq + y. This urve has genus g = q(q−1)

2 and has n = q3 rational a�ne points.For oding theory appli ation, it is interesting to know the number of (a�ne plane)points that lie inH and an arbitrary parabola y = ax2+bx+c, disregarding multipli ityand other similar notions. We all this number their planar interse tion. Moreover, it isessential to know pre isely the number of parabolas having a given planar interse tionwith H. We present here for the �rst time a omplete lassi� ation.The link with oding theory we studied is the following: for some high-dimensionHermitian odes, we found that a w-weight word an exist if and only if we an �nd atleast w points lying in the interse tion between H and a low-degree urve, for instan ea line or a parabola.Coauthors: Chiara Mar olla, Università degli studi di Trento, Italy;Massimiliano Sala, Università degli studi di Trento, Italy.Relaxed Hensel Lifting of Triangular SetsRomain LebretonLIRMM, Université de Montpellier II, Fran eTuesday, June 4th, 17:20 - 17:45, Room H 14In this paper, we present a new lifting algorithm for triangular sets over power series.Our ontribution is to give, for any power series triangular set, a shifted algorithm ofwhi h the triangular set is a �xed point. Then we an apply the relaxed re ursive powerseries framework and dedu e a relaxed lifting algorithm for this triangular set.We ompare our algorithm to the existing te hniques. Our algorithm always improvesthe asymptoti ost in the pre ision for the spe ial ase of univariate representations.

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Re ent Advan es on Smale's 17th ProblemFelipe Cu kerCity University of Hong Kong, Hong KongWednesday, June 5th, 09:00 - 09:50, Room H IV, Plenary TalkThe 17th of the problems proposed by Steve Smale for the 21st entury asks for theexisten e of a deterministi algorithm omputing an approximate solution of a systemof n omplex polynomials in n unknowns in time polynomial, on the average, in the sizeN of the input system. In the talk we will give a pre ise des ription of the problem andsurvey the many advan es that have been done in the last 10 years towards a de�nitesolution whi h, as of today, has not been found.

Separating Inequalities for Nonnegative Polynomials that Are notSums of SquaresTimo de WolffGoethe University, Frankfurt am Main, GermanyWednesday, June 5th, 10:30 - 10:45, Room H IVTernary sexti s and quaternary quarti s are the smallest ases where there exist nonneg-ative polynomials that are not sums of squares (SOS). A omplete lassi� ation of thedi�eren e between these ones was given by G. Blekherman via analyzing the extremerays of the orresponding dual ones. However, an exa t omputational approa h inorder to build separating extreme rays for nonnegative polynomials that are not sumsof squares is a widely open problem. We present a method substantially simplifyingthis omputation for ertain lasses of polynomials on the boundary of the PSD ones.In parti ular, our method yields separating extreme rays for every nonnegative ternarysexti with at least seven zeros. As an appli ation to further instan es, we provide arational erti� ate proving that the Motzkin polynomial is not SOS.Coauthors: Sadik Iliman, Goethe University, Frankfurt am Main, Germany.25

Divisorial Models of Normal VarietiesStefano UrbinatiUniversity of Warsaw, PolandWednesday, June 5th, 10:30 - 10:55, Room H 14We prove that the anoni al ring of a anoni al variety in the sense of de Fernexand Ha on is �nitely generated. We prove that anoni al varieties are klt if and onlyif RX(X,−KX ) is �nitely generated. We introdu e a notion of nefness for non-Q-Gorenstein varieties and study some of its properties. We then fo us on these propertiesfor non-Q-Gorenstein tori varieties.

Systems of Equations with a Single SolutionGleb GusevMos ow Institute of Physi s and Te hnology, Mos ow, Russia.Wednesday, June 5th, 11:05 - 11:30, Room H IVWe lassify general systems of polynomial equations with a single solution, or, equiva-lently, olle tions of latti e polytopes of minimal positive mixed volume. As a byprod-u t, this lassi� ation provides an algorithm to evaluate the single solution of su h asystem.Coauthors: Alexander Esterov, National Resear h University Higher S hool of E onomi s,Mos ow, Russia.

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Betti Diagrams from GraphsMatthew T. StampsAalto University, Helsinki, FinlandWednesday, June 5th, 11:05 - 11:30, Room H 14The emergen e of Boij-Söderberg theory has given rise to new onne tions between om-binatori s and ommutative algebra. Herzog, Sharifan, and Varbaro re ently showedthat every Betti diagram of an ideal with a k-linear minimal resolution arises from thatof the Stanley-Reisner ideal of a simpli ial omplex. In this talk, we will extend theirresult for the spe ial ase of 2-linear resolutions using purely ombinatorial methods.Spe i� ally, we will give bije tions between Betti diagrams of ideals with 2-linear res-olutions, threshold graphs, and anti-le ture hall ompositions. If time permits, we willshow that any Betti diagram of a module with a 2-linear resolution is realized by a di-re t sum of Stanley-Reisner rings asso iated to threshold graphs. The key observationis that these obje ts all orrespond to the latti e points of a normal re�exive latti epolytope.Coauthors: Alexander Engström, Aalto University, Helsinki, Finland.A Probabilisti Symboli Algorithm to Find the Minimum of aPolynomial Fun tion on a Basi Closed Semialgebrai SetGabriela JeronimoDepartamento de Matemáti a, FCEN, Universidad de Buenos Aires, andIMAS, CONICET�UBA, ArgentinaWednesday, June 5th, 11:40 - 12:05, Room H IVWe onsider the problem of omputing the minimum of a polynomial fun tion g on abasi losed semialgebrai set

E = {x ∈ Rn | f1(x) = · · · = fl(x) = 0, fl+1(x) ≥ 0, . . . , fm(x) ≥ 0}de�ned by polynomials f1, . . . , fm ∈ Q[x1, . . . , xn], assuming that g attains a minimumvalue over E.We will present a probabilisti symboli algorithm to �nd a �nite set of sample points ofthe subset Emin of E where the minimum of g is attained, provided that Emin satis�essome ompa tness assumption.Coauthors: Daniel Perru i, Universidad de Buenos Aires and CONICET, Argentina.27

Expli it Computations of Invariants of Plane Quarti CurvesAndreas-Stephan ElsenhansUniversity of Sydney, AustraliaWednesday, June 5th, 11:40 - 12:05, Room H 14The study of invariants is one of the oldest roots of algebrai geometry. However, anexpli it des ription of the rings of invariants is only known in a small number of ases.In this talk, we will review lassi al methods to write down and to ompute invariantsof forms. This will lead us to a short inspe tation of o- and ontravariants.In a se ond part, I will explain how to use these te hni s for an e� ient numeri al omputation of invariants.We will apply this to the ase of plane quarti urves. We get a omplete system ofinvariants. The proof of ompleteness is a ombination of several lassi al results anda magma omputation.Invariants with a known geometri interpretation are sometimes alled lassi al in-variants. I will explain how to express the lassi al invariants in terms of the knowngenerators.Finally, I will sket h some ideas how these geometri interpretations an be used tospeed up all the omputations in the proofs.Algebrai Geometry in Computer VisionRekha R. ThomasUniversity of Washington, Seattle, WA, USAThursday, June 6th, 09:00 - 09:50, Room H IV, Plenary TalkA fundamental problem in omputer vision is the re onstru tion of 3D s enes from noisyimages in a number of ameras. These problems belong to a subdis ipline alled multi-view geometry whi h is based in lassi al proje tive geometry and on erns the studyof the spa e of images in ameras. The re onstru tion problem is then an optimizationproblem over this spa e.In this talk I will explain several results on the algebrai geometry and optimizationthat arise from the situation in whi h ameras are known. The ideals that o ur arryri h ombinatori s and give rise to surprising results. The optimization problem is alsohighly stru tured and often solvable in polynomial time, and the performan e has anatural explanation from the geometry of the problem.Joint work with Chris Aholt, Sameer Agarwal and Bernd Sturmfels28

Determinantal Representations of Hyperboli Curves viaPolynomial Homotopy ContinuationDaniel PlaumannUniversity of Konstanz, GermanyThursday, June 6th, 10:30 - 10:55, Room H IVA smooth urve of degree d in the real proje tive plane is hyperboli if its ovals aremaximally nested, i.e. its real points ontain ⌊d2⌋ nested ovals. By the Helton-VinnikovTheorem, any su h urve admits a de�nite symmetri determinantal representation.But omputing su h representations expli itly is not a simple task. In this talk, wewill explain the ontext of this problem and show how to use polynomial homotopy ontinuation to �nd numeri al solutions.Coauthors: Anton Leykin, Georgia Institute of Te hnology, Atlanta, GA, USA.On Alexander-Conway Polynomials of Two-bridge LinksPierre-Vin ent KoseleffUniversité Pierre et Marie Curie (UPMC - Paris 6), ParisInstitut de Mathématiques de Jussieu (IMJ, CNRS UMR 7586)Ouragan (INRIA Paris-Ro quen ourt), Fran eThursday, June 6th, 10:30 - 10:55, Room H 14We onsider Conway polynomials of two-bridge links as Euler ontinuant polynomi-als. As a onsequen e, we obtain new and elementary proofs of lassi al Murasugi's1958 alternating theorem and Hartley's 1979 trapezoidal theorem. We give a mod-ulo 2 ongruen e for links, whi h implies the lassi al Murasugi's 1971 ongruen e forknots. We also give sharp bounds for the oe� ients of Euler ontinuants and dedu ebounds for the Alexander polynomials of two-bridge links. These bounds improve andgeneralize those of Nakanishi-Suketa'96. We easily obtain some bounds for the rootsof the Alexander polynomials of two-bridge links. This is a partial answer to Hoste's onje ture on the roots of Alexander polynomials of alternating knots.Coauthors: Daniel Pe ker, Université Pierre et Marie Curie (UPMC - Paris 6), IMJ, Paris,Fran e.

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Multiple Bernoulli Series and Volumes of Moduli Spa es of FlatBundlesArzu BoysalBo§aziçi University, Istanbul, TurkeyThursday, June 6th, 11:05 - 11:30, Room H IVUsing Szenes formula for multiple Bernoulli series, we explain how to ompute Wittenseries asso iated to lassi al Lie algebras. Parti ular instan es of these series omputevolumes of moduli spa es of �at bundles over surfa es, and some spe ial ases of multiplezeta values. We give expli it pro edures for omputations for lassi al root systems,and ompare our results with those in the urrent literature.Coauthors: Velleda Baldoni, Universita degli Studi di Roma Tor Vergata, Rome, Italy;Mi hèle Vergne, Institut de Mathématiques de Jussieu, Paris, Fran e.Castelnuovo-Mumford Regularity of Proje tive Monomial CurvesAsso iated to Arithmeti Sequen esEva Gar ía-LlorenteUniversity of La Laguna, Tenerife, SpainThursday, June 6th, 11:05 - 11:30, Room H 14Let k be an algebrai ally losed �eld and m0 < . . . < mn an arithmeti sequen e. We onsider the proje tive monomial urve C ⊂ P

n+1k parametri ally de�ned by

x0 = tm0smn−m0 , . . . , xn−1 = tmn−1smn−mn−1 , xn = tmn , xn+1 = smn .In this work, we prove that C is an arithmeti ally Cohen-Ma aulay proje tive urveand we obtain a formula for the Castelnuovo-Mumford regularity reg (C) of C in termsof the arithmeti sequen e that allows us to ompute reg (C) in an e� ient way.Coauthors: Isabel Bermejo, University of La Laguna, Tenerife, Spain.30

Subresultants, Sylvester Sums and the Rational InterpolationProblemTeresa Kri kUniversidad de Buenos Aires, ArgentinaThursday, June 6th, 11:40 - 12:05, Room H IVWe present a solution for the lassi al univariate rational interpolation problem bymeans of (univariate) subresultants. In the ase of Cau hy interpolation (interpolationwithout multipli ities), we re over expli it formulas for the solution in terms of symmet-ri fun tions of the input data, hen e generalizing the well-known formulas for Lagrangeinterpolation. In the ase of the os ulatory rational interpolation (interpolation withmultipli ities), we get determinantal expressions in terms of the input data.Coauthors: Carlos D'Andrea, Universitat de Bar elona, Spain;Agnes Szanto, North Carolina State University, Raleigh, NC, USA.Computing the Index of Ve tor Fields at Cohen-Ma aulay CurvesAlexander G. AleksandrovInstitute for Control S ien es, Russian A ademy of S ien es, Mos ow, RussiaThursday, June 6th, 11:40 - 12:05, Room H 14In 1887 H.Poin aré introdu ed the notion of topologi al index of ve tor �elds withisolated singularities given on 2-dimensional manifolds. For a long time many authorsstudied the index as a topologi al invariant in di�erent ontexts and various settings.A new algebrai on ept of the homologi al index for ve tor �elds on redu ed pure-dimensional omplex analyti spa es was originated by X.Gómez-Mont in 1998; it iseasy and well adapted for use in the theory of singular varieties.The purpose of this talk is to dis uss a new method for omputing the index of ve tor�elds at Cohen-Ma aulay urves. It is based on simple properties of regular mero-morphi di�erential forms onsidered in the framework of the theory of residues andduality. In parti ular, we show how to ompute expli itly the index of ve tor �elds givenon quasi-homogeneous Gorenstein urves, omplete interse tions, monomial urves,Cohen-Ma aulay urves of odimension two, and many others. In ontrast to pre-vious works on this subje t we do not use either te hnique of spe tral sequen es or omputer algebra systems for symboli al ulations.

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Polynomial Time Computation of Galois Representations Atta hedto Modular FormsBas EdixhovenUniversity of Leiden, The NetherlandsThursday, June 6th, 14:00 - 14:50, Room H IV, Plenary TalkModular forms give rise to number �elds with non-solvable Galois groups, a ting faith-fully on �nite subgroups of ja obian varieties of urves. In joint work with Couveignes,de Jong and Merkl, generalised by Bruin, it was shown that these number �elds an be omputed in polynomial time. The major di� ulty is that su h omputations must bedone in time polynomial in the dimension of the ja obian varieties that arise. This dif-� ulty was solved by approximate omputations and bounds that allow us to get exa tsolutions from approximate ones. The bounds will be dis ussed brie�y. We will fo us onCouveignes's algorithms for the approximate omputations. Finally, real omputationsby Bosman, Mas ot and Zeng will be presented.Algorithms for Equivariant Gröbner BasesRobert KroneGeorgia Te h, Atlanta, GA, USAThursday, June 6th, 15:05 - 15:30, Room H IVA polynomial ring over a ountably in�nite number of variables presents some obsta lesto omputation be ause it is not Noetherian. However, often ideals of interest in thissetting have ertain symmetry. Given an a tion of a monoid G on the set of variables,an ideal is G-equivariant if it is losed under the a tion of G. In parti ular we onsider ases where G is the group of all permutations, or the monoid of stri tly in reasingfun tions on the natural numbers. We des ribe an algorithm to ompute equivariantGröbner bases that may exist for su h ideals and its implementation, EquivariantGBpa kage for Ma aulay2.We give examples of omputation of the kernels of tori maps of in�nite-dimensionalrings. One reproves a result of de Loera, Sturmfels, and Thomas obtained theoreti ally.The other answers the smallest open ase of a question about the �nite-generation ofsu h kernels.Coauthors: Christopher Hillar, University of California, Berkeley, CA, USA;Anton Leykin, Georgia Te h, Atlanta, GA, USA.

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Normaliz: Algorithms for Rational Cones and A�ne MonoidsChristof SögerUniversität Osnabrü k, GermanyThursday, June 6th, 15:05 - 15:30, Room H 14Normaliz is a software tool for omputations in dis rete onvex geometry and tori algebra. Its main obje tives are the omputation of Hilbert bases and of Hilbert (orEhrhart) series of rational ones C ⊂ Rn (de�ned with respe t to a grading).Normaliz ombines several algorithms: Fourier-Motzkin elimination, (partial) lexi o-graphi triangulation in onne tion with pyramid de omposition, latti e operations,redu tion in normal monoids, rational generating fun tions, and, as a variant to trian-gulation based approa hes, Pottier's algorithm for Hilbert bases.Normaliz has been developed in the last 15 years by W. Bruns in ooperation withR. Ko h (until 2002), B. I him (sin e 2007) and C. Söger (sin e 2009). Several othermathemati ians have ontributed to it. Normaliz has interfa es to CoCoA, Singular,and Ma aulay 2, polymake and is a essible from Sage. Normaliz has been written inC++, and is highly parallelized via OpenMP.Coauthors: Winfried Bruns, Universität Osnabrü k, Germany.Bogdan I him, Institute of Mathemati s �Simion Stoilow� of the Romanian A ademy, Bu harest,Romania Computing Tropi al Curves via Proje tionsAndrew J. ChanUniversity of Warwi k, Coventry, United KingdomThursday, June 6th, 16:10 - 16:35, Room H IVTropi al geometry an be thought of as a pie ewise linear approximation to algebrai geometry where algebrai varieties are repla ed by tropi al varieties. It is useful as theyshare many of the same invariants but the tropi al variety is often easier to work withas it has a ni e polyhedral stru ture.Currently tropi al varieties an be omputed using the software gfan. This works by�rst �nding a maximal one of the tropi al variety then omputing a tropi al urvewhi h has a ray orresponding to ea h of the neighbouring maximal ones. Thus the omputation of tropi al urves is a ru ial step in the onstru tion of tropi al varietiesand so any improvement to the tropi al urve algorithm would result in improvementsto the tropi al variety algorithm.We introdu e the Ma aulay2 pa kage Tropi alCurves whi h omputes tropi al urvesby onsidering their polyhedral stru ture and re onstru ting them from their oordinateproje tions. We give examples of tropi al urves whi h annot be omputed using gfanbut whi h now an be omputed using Tropi alCurves.33

MDS � a Pa kage for Computationswith Mori Dream Spa esSimon Kei herUniversität Tübingen, GermanyThursday, June 6th, 16:10 - 16:35, Room H 14Every algebrai variety X with a �nitely generated divisor lass group Cl(X) omeswith a Cox ring Cox(X), graded by Cl(X). A Mori dream spa e is an algebrai varietyX with �nitely generated Cox ring. The lass of Mori dream spa es is a large lass omprising e.g. tori varities.Mori dream spa es admit an elementary des ription in terms of the Cox ring and further ombinatorial data extending basi prin iples of tori geometry. Using this des riptionwe developed a Maple-pa kage MDS.I will report on how our pa kage works and show by examples how one an use oursoftware to ompute geometri data, e.g. the Pi ard-group, the semi ample one, theMori hamber de omposition, Fano-tests and how to ompute the Cox ring of blowups.Coauthors: Jürgen Hausen, Universität Tübingen, Germany.

Conne ting Singular, GAP, Polymake and GfanMohamed Barakat, Janko Böhm, Yue RenTe hnis he Universität Kaiserslautern, GermanyThursday, June 6th, 16:45 - 17:10, Room H IVWe report on re ent e�orts to onne t the omputer algebra systems Singular, GAP,polymake, and Gfan on kernel level, and the orresponding extensions of the userlanguages. The goal of these e�orts is to provide powerful ross-border omputationaltools whi h an be a essed via the user's favorite environment and language. Singu-lar is a system for ommutative and non- ommutative algebra, algebrai geometry,and singularity theory. GAP o�ers a mathemati al obje t oriented language whi h wasused to develop a modern system for omputational group and representation theory.Polymake fo uses on geometri ombinatori s and onvex geometry, and Gfan on the omputation of Gröbner fans and tropi al varieties. We illustrate the new omputa-tional possibilities on various examples. These in lude a new approa h due to Barakatand Posur to onstru t ve tor bundles of low rank on proje tive spa e using tools fromrepresentation theory, ommutative and non- ommutative algebra. Furthermore, us-ing tools from ommutative algebra and onvex geometry, we dis uss Simon Kei her'sapproa h to ompute the GIT-fan in the ontext of geometri invariant theory.Coauthors: Wolfram De ker and Hans S hönemann, TU Kaiserslautern, Germany.34

Software for Computing Belyi Fun tions of Genus 0Raimundas VidunasNational Kapodistrian University of Athens, Gree eThursday, June 6th, 16:45 - 17:10, Room H 14Genus 0 Belyi fun tions ϕ(x) of degree around 60 an be omputed e� iently if theypull-ba k a hypergeometri (di�erential) equation to a Fu hsian equation with just afew singularities. The bran hing patterns of these Belyi fun tions are nearly regular:with just a few ex eptions in total, all points in the ϕ = 1 �ber have the same bran hingorder k, the points in the ϕ = 0 �ber have the same bran hing order ℓ, and the points inthe ϕ = ∞ �ber have the same bran hing order m. The developed software omputes(and attempts to simplify) the Belyi fun tions with a bran hing pattern given by k, ℓ,mand the ex eptional bran hing orders. In the ase of 4 ex eptions, the result is omputedin reasonable time for degree 60�80 Belyi fun tions, while in the ase of 5 ex eptionsthis degree bound is roughly halved.Coauthors: Mark van Hoeij, Florida State University, Tallahassee, FL, USA.Latti e Polytopes with a View Toward Algebrai GeometryBenjamin NillCase Western Reserve University, Cleveland, OH, USAFriday, June 7th, 09:00 - 09:50, Room H IV, Plenary TalkBy its very nature, tori geometry an be onsidered a beautiful example of the use ofe�e tive methods in algebrai geometry. The ombinatorial des ription of tori varietiesvia fans and polytopes allows for the expli it omputation and sharp estimation of theirinvariants su h as the Pi ard number or the degree. Moreover, using onvex geometrysoftware pa kages it is possible to get omplete lassi� ation results of interesting lassesof tori varieties in mu h higher dimensions than within rea h for non-tori varieties.On the other hand, there are still purely ombinatorial statements that wouldn't havebeen dis overed without algebro-geometri insight and annot yet be proved withoutthe power of algebrai geometry or ommutative algebra. I will provide re ent examplesof this interplay of algebrai and onvex geometry by fo using on latti e polytopes andtheir asso iated proje tive tori varieties.

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Se ant Cumulants and Tori GeometryMateusz Mi haªekPolish A ademy of S ien es, Warsaw, PolandMax Plan k Institute, Bonn, GermanyFriday, June 7th, 10:30 - 10:55, Room H IVThe study of varieties naturally embedded in a tensor produ t of ve tor spa es, likeVeronese reembeddings, Segre produ ts or Grassmannians, is a lassi al topi in math-emati s. In many appli ations, su h as Algebrai Statisti s and Computational Com-plexity, a deeper understanding of se ant varieties of these manifolds is needed.We present new results on the (�rst) se ant variety, fo using on the Segre embedding.Our approa h is based on a Cremona transformation inspired by the al ulation of umulants in statisti s. Our main result shows that the se ant variety an be overedby mu h simpler tori varieties. This enables us to obtain new insights into its geometry.In parti ular, we provide a pre ise des ription of the singular lo us and we lassify all ases when it is Gorenstein. Our te hniques an be adapted to a larger lass of se antand tangential varieties.Coauthors: Luke Oeding, University of California, Berkeley, CA, USA;Piotr Zwiernik, University of California, Berkeley, CA, USA.Linear Spa es of Matri es of Constant Rank and Instanton BundlesAda BoraleviPolish A ademy of S ien es, Warsaw, PolandFriday, June 7th, 10:30 - 10:55, Room H 14Given a omplex ve tor spa e V of dimension n, one an look at d-dimensional linearsubspa es A in Λ2V , whose elements have onstant rank r. The natural interpretationof A as a ve tor bundle map yields some restri tions on the values that r, n and d an attain. After a brief overview of the subje t and of the main te hniques used,I will on entrate on the ase r = n − 2 and d = 4. I will introdu e what used tobe the only known example, by Westwi k, and give an explanation of this example interms of instanton bundles and the derived ategory of P3. I will then present a newmethod that allows one to prove the existen e of new examples of su h spa es, showhow this method applies to instanton bundles of harge 2 and 4, and give an algorithmto onstru t expli itly a matrix of size 14 of this type.Coauthors: Daniele Faenzi, Université de Pau et des Pays de l'Adour, Pau, Fran e;Emilia Mezzetti, Università degli Studi di Trieste, Italy.36

A Tropi al Approa h to Computing E�e tive ConesFlorian Blo kUniversity of California at Berkeley, CA, USAFriday, June 7th, 11:05 - 11:30, Room H IVThe wonderful model Y of a hyperplane arrangement omplement Y is a smooth om-pa ti� ation of Y with normal rossing boundary introdu ed by De Con ini and Pro- esi. Examples in lude the blow-up of P2 at any number of points, and the modulispa e M0,n. We give an iterative method, inspired by tropi al geometry, to omputethe e�e tive one of Y . If the realization spa e R of the hyperplane arrangement isirredu ible there is a very general open set of R for whi h the e�e tive one is onstant.Coauthors: Diane Ma lagan, University of Warwi k, Coventry, United KingdomComparing the KSBA and the GIT Compa ti� ation of the ModuliSpa e of Quinti Surfa esPatri io GallardoStony Brook University, NY, USAFriday, June 7th, 11:05 - 11:30, Room H 14Geometri invariant theory allows us to onstru t an expli it ompa ti� ation of themoduli spa e of smooth hypersurfa es of a given degree in proje tive spa e. Here,we des ribe the behaviour of the log anoni al threshold, and the geometri genusasso iated to non-degenerate isolated singularities on stable surfa es. This leads us to onsider a partial ompa ti� ation by taking the lo us parametrizing surfa es with atworst minimal ellipti singularities. We des ribe some boundaries of su h lo us.

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Combinatorial Ex ess Interse tionJose RodriguezUniversity of California at Berkeley, CA, USAFriday, June 7th, 11:40 - 12:05, Room H IVWe provide formulas and algorithms for omputing the ex ess numbers of ertain ideals.The solution for monomial ideals is given by the mixed volumes of ertain polytopes.These results enable us to design spe i� homotopies for numeri al algebrai geometry.

Central Chara ters, Bernstein-Sato Polynomials and Asso iatedStrati� ationsViktor LevandovskyyRWTH Aa hen, GermanyFriday, June 7th, 11:40 - 12:05, Room H 14Let D = Dn be the n-th Weyl algebra over a �eld K with CharK = 0 and D[s] =D ⊗K K[s]. For f ∈ K[x1, . . . , xn] \ {K} we de�ne I = AnnD[s]f

s + f ⊂ D[s] andMf := D[s]/I. The global Bernstein-Sato polynomial of f is the moni generator ofthe ideal I∩K[s]. We investigate the onne tion between the Bernstein-Sato polynomialof f and entral hara ters of a holonomi module Mf .We employ the entral hara ter de omposition te hnique in order to de ompose Mfinto a �nite dire t sum of submodules Mχj

f . Ea h Mχj

f an be organized in a �niteseries of submodules and, notably, this te hnique is algorithmi .We study a strati� ation of an a�ne spa e, built a ording to the following riterion:the lo al Bernstein-Sato polynomial of f is onstant at ea h stratum (whi h is a on-stru tible set). The de omposition of Mf as above allows one to asso iate a ertainsubmodule Nk of Mf to the k-th stratum and, moreover, Mf =∑

kNk holds.We investigate questions of algorithmi omputability of mentioned de ompositions andstrati� ations and study several examples in details.Coauthors: Jorge Martín-Morales, Centro Universitario de la Defensa de Zaragoza, Spain.38

Finite Fields in Computational Algebrai GeometryFrank-Olaf S hreyerUniversität des Saarlandes, Saarbrü ken, GermanyFriday, June 7th, 14:00 - 14:50, Room H IV, Plenary TalkComputation over �nite �elds have various appli ation to the study of parameter spa esof obje ts in algebrai geometry. In the talk I report, how omputation over �nite �elds an shed light on unirationality questions of moduli spa es, and on maps between them.I will fo us on two re ent appli ation: random urves of genus 15, and, if time permits,gli i olle tions of points in P3.

A Wronskian Approa h to the Real τ -Conje tureSébastien TavenasLIP, ENS Lyon, Fran eFriday, June 7th, 15:05 - 15:30, Room H IVA ording to the real τ - onje ture, the number of real roots of a sum of produ ts ofsparse univariate polynomials should be polynomially bounded in the size of su h anexpression. It is known that this onje ture implies a superpolynomial lower bound onthe arithmeti ir uit omplexity of the permanent. In this talk, we use the Wronksiandeterminant to give an upper bound on the number of real roots of sums of produ tsof sparse polynomials of a spe ial form. We fo us on the ase where the number ofdistin t sparse polynomials is small, but ea h polynomial may be repeated severaltimes. We also give a deterministi polynomial identity testing algorithm for the same lass of polynomials. Our proof te hniques are quite versatile; they an in parti ularbe applied to some sparse geometri problems that do not originate from arithmeti ir uit omplexity.Coauthors: Pas al Koiran, LIP, ENS Lyon, Fran e;Nata ha Portier, LIP, ENS Lyon, Fran e.39

Strati� ation of the Spa e of Foliations on CP2Claudia Reynoso Al ántaraUniversidad de Guanajuato, Méxi oFriday, June 7th, 15:05 - 15:30, Room H 14In this talk we will onstru t a strati� ation of the spa e of foliations on CP2 of de-gree d with respe t to the a tion by hange of oordinates of Aut(CP2). We use theNorbert A'Campo's implementation of Popov's algorithm to obtain the indexing set ofthe strati� ation and the dimension of the strata. In some ases we hara terize thefoliations on every stratum a ording to existen e of degenerate singular points andalgebrai leaves. These strata are non-singular, lo ally- losed, algebrai varieties. Theaim of the talk is to give arguments to onvin e us of the usefulness of this strati� ationto lassify foliations with spe ial properties.On the Free Resolution Indu ed by a Pommaret BasisWerner M. SeilerUniversität Kassel, GermanyFriday, June 7th, 16:10 - 16:35, Room H IVWe relate the free resolution indu ed by the revlex Pommaret basis of a polynomialideal or module with a (slight generalisation of a) re ent onstru tion by Sköldbergusing dis rete Morse theory. This onne tion allows the expli it determination of a freeresolution with the omputation of only one Pommaret basis. For the spe ial ase ofa quasi-stable monomial ideal, we show that the indu ed resolution is a mapping oneresolution.Coauthors: Matthias Fetzer, Universität Kassel, Germany;Eduardo Sáenz-de-Cabezón, Universidad de la Rioja, Logroño, Spain.

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Abstra ts of PostersTori Homogenous Markov Chain ModelsPatrik NorénAalto University, Helsinki, FinnlandWe prove that the three-state tori homogenous Markov hain model has Markov de-gree two. In algebrai terminology this means, that a ertain lass of tori ideals aregenerated by quadrati binomials. This was onje tured by Haws, Martin del Campo,Takemura and Yoshida, who proved that they are generated by binomials of degree sixor less.

Hermitian Determinantal Representations of Hyperboli CurvesCynthia VinzantUniversity of Mi higan, Ann Arbor, MI, USABe ause all the eigenvalues of a Hermitian matrix are real, the real hypersurfa e de�nedby the determinant of a d × d Hermitian matrix of linear forms det(xI + yA + yB)is hyperboli , meaning that it has the real topology of ⌊d/2⌋ nested ovals in P2(R).In 2007, Helton and Vinnikov showed that every hyperboli plane urve has su h adeterminantal representation. Here we present an algorithm for omputing a Hermitiandeterminantal representation of a given hyperboli plane urve that uses only linearalgebra and the interse tion of plane urves. This involves showing that a matrix oflinear forms is de�nite if and only if its o-maximal minors interla e its determinantand extending a lassi al onstru tion of Dixon from 1902.Coauthors: Daniel Plaumann, Universität Konstanz, Germany;Rainer Sinn, Universität Konstanz, Germany;David E Speyer, University of Mi higan, Ann Arbor, MI, USA.41

Symmetry and Finiteness in Algebrai MatroidsFranz Király and Zvi RosenTU Berlin, Germany and University of California at Berkeley, CA, USAAlgebrai matroids apture the ombinatorial stru ture of dependen e and indepen-den e among a set of elements in a �eld extension. In parti ular, the oordinates of thefun tion �eld of a variety are naturally the ground set of an algebrai matroid. Thisperspe tive o�ers a uniform approa h to problems su h as low-rank matrix ompletionand framework rigidity, whi h are on erned with re onstru ting a point on a varietyfrom a small set of oordinate proje tions.For onstant-rank algebrai matroids in in�nitely many variables with ompa t quo-tient under an a tion by the symmetri group, we an show there are �nitely manyisomorphism lasses of ir uits.Coauthor: Louis Theran, FU Berlin, Germany.Reparametrizing Rational Revolution Surfa es over the RealsCarlos AndradasUniversidad Complutense, Madrid, SpainIn this paper we hara terize when a omplex surfa e SC parametrized in the form

P(t, s) = (φ1(t)ψ1(s), φ1(t)ψ2(s), φ2(t)) ∈ C(t, s)3, an be reparametrized over R and ifso, how to �nd su h a real reparametrization. Here (0, φ1(t), φ2(t)) and (ψ1(s), ψ2(s), 0)are omplex rational urves alled the generatrix and traje tory urves of SC, whi his obtained by gliding the generatrix along the traje tory urve. So our surfa es areinstan es of the so- alled swung surfa es used in some NURBS pa kages of CAGD.We show that SC is R-parametrizable if and only if it ontains enough real points. Al-though this is of ourse ne esary we point out that, in general, it is not known whethera real surfa e, provided with a omplex parametrization, has a real parametrization.The proof relies on the use of the parametri Weil variety asso iated to the given sur-fa e and the theory of ultraquadri s and hyper ir les spe i� ally reated by the authorsto handle over R the reparametrizing of a given omplex parametrization. The paperends with two reparametrization algorithms and a table with their performan e runningtimes on a olle tion of surfa es.Coauthors: Tomas Re io, Universidad de Cantabria, Santander, Spain;Rafael Sendra, Universidad de Al alá, Al alá de Henares, Spain;Luis-Felipe Tabera, Universidad de Cantabria, Santander, Spain;Carlos Villarino, Universidad de Al alá, Al alá de Henares, Spain.42

Lo al Positivity of Line Bundles on Smooth Tori Varieties andCayley PolytopesAnders LundmanKTH, Sto kholm, SwedenFor any non-negative integer k the k-th os ulating dimension at a given point x of avariety X embedded in proje tive spa e gives a measure of the lo al positivity of orderk at that point. In this paper we show that a smooth tori embedding having maximalk-th os ulating dimension, but not maximal (k + 1)-th os ulating dimension, at everypoint is asso iated to a Cayley polytope of order k. This result generalises an earlier hara terisation by David Perkinson. In addition we prove that the above assumpionsare equivalent to requiring that the Seshadri onstant is exa tly k at every point of X,generalising a result of Atsushi Ito.

The Geometry of the TDOA-based Sour e Lo alizationMar o CompagnoniPolite ni o di Milano, ItalyWe study the lo alization of an a ousti sour e based on time di�eren e measurements(TDOA). This is a well-established problem in the literature of spa e-time signal pro- essing, but a omplete hara terization of the statisti al TDOA model is still la king.Here, we fo us on the deterministi part of the model, onsidering every TDOA as areal number not a�e ted by error. So, we reformulate the problem in terms of theproperties of a map from the physi al spa e of sour e positions to the spa e of TDOAmeasurements. We fully explore the properties of the map in the ase of three re- eivers and a sour e lying on a plane, des ribing its image, i.e. the set of all admissiblemeasurements, and studying its invertibility. The analysis has been arried out usingdi�erent mathemati al tools, in luding lo al analyti al methods, multi�linear algebrain Minkowski spa e and algebrai geometry. In fa t, although the model is not de�nedusing rational fun tions, we des ribe the measurements set and its preimage in termsof real algebrai sets. Our work an give new insights also for the study of the GPSlo alization, in parti ular in relation with the so� alled bifur ation problem.Coauthors: R.Notari, F.Antona i, A.Sarti, Polite ni o di Milano, Italy.43

Bounds for the Height of a Parametrization of a Rational PlaneCurveMarta Narváez ClaussUniversity of Bar elona, SpainGiven a proje tive plane rational urve C of degree d with base �eld Q and ordinarysingularities only, we will study the height of an algebrai ally optimal parametrizationobtained when we apply Sendra-Winkler's Optimal-Parametrization algorithm and wewill give a bound in terms of both the height and the degree of the urve.

A Division Algorithm in an A�ne Framework for Flat FamiliesCovering Hilbert S hemesMargherita RoggeroUniversità degli Studi di Torino, ItalyLet j be a strongly stable ideal in R := K[x1, . . . , xn]. We study the family F(j) ofideals i ⊂ R whose quotients R/i share the same a�ne Hilbert polynomial p(t) and thesame monomial K-ve tor basis, the sous-es alier N (j) of j. We have already studied theanalogous problem for homogeneous ideals, but in the non-homogenous ase there aremore di� ulties that we over ome by a new division algorithm. For any �xedm ∈ Z, we onstru t suitable subsets Mf(j,m) ⊂ F(j) and prove that they are naturally endowedwith a s heme stru ture. Moreover for m≫ 0 they an be identi�ed with open subsetsof Hilbnp(t), that, up to the a tion of GL(n + 1), over it. These results allow us tomake expli it omputations on Hilbert s hemes: for example, for Hilb716, we dete tthree irredu ible omponents through a single point and we prove the smoothability ofGorenstein s hemes with Hilbert fun tion (1, 7, 7, 1). In a similar way, in Hilb512, weprove the smoothability of Gorenstein s hemes (1, 5, 5, 1).Coauthors: Cristina Bertone, Università degli Studi di Torino, Italy;Fran es a Cioffi, Università degli Studi di Napoli �Federi o II �, Italy.

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An F5 Algorithm for Modules over Path Algebra Quotients and theComputation of Loewy LayersSimon KingFriedri h S hiller University, Jena, GermanyWe provide a non- ommutative version of the F5 algorithm, for right-modules over pathalgebra quotients. Termination an generally not be guaranteed, sin e �nite standardbases do not always exist. However, if the path algebra quotient is a basi algebra,then the F5 algorithm will terminate, for every hoi e of a monomial ordering.The output of the F5 algorithm is a so- alled signed standard basis. Our main result is:If a negative degree monomial ordering is used, then the signed standard basis providesenough information to read o� the Loewy layers of the module. This information isnot provided in an unsigned standard basis. Hen e, the F5 algorithm ould not easilybe repla ed by any other algorithm for the omputation of standard bases.The �rst Loewy layer provides a minimal generating set. The best urrently imple-mented algorithm for omputing minimal generating sets, the �heady algorithm� ofDavid Green. The non- ommutative F5 algorithm yields more information than theheady algorithm and in addition is theoreti ally more e� ient.A Software Framework for Computing Newton Polytopes ofResultants and (Redu ed) Dis riminantsVissarion FisikopoulosUniversity of Athens, Gree eWe present new C++ software ResPol for omputing the Newton polytopes of sparseresultants and (redu ed) dis riminants. For the former, our software implements avertex ora le by omputing a regular triangulation of the Cayley pointset. This is usedto re onstru t the entire polytope by means of an in remental onvex hull onstru tion.Regarding the dis riminant polytope, we use two approa hes: The �rst fo uses onredu ed dis riminants and, via the Horn-Kapranov parameterization of the dis riminantpolynomial, redu es the problem to omputing a proje tion of the Newton polytope ofthe resultant of a system de�ned by the parametri polynomials. The se ond approa hto dis riminant polytopes employs vertex ora les as de�ned by Rin ón, using tropi algeometry. Our publi ly available software (http://respol.sour eforge.net) has omputedpolytopes in up to 7 dimensions. We illustrate its use through examples.Coauthors: Ioannis Z. Emiris, University of Athens, Gree e;Christos Konaxis, University of Crete, Gree e.

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JMBTest.lib and JMSConst.lib:Singular Tools for J-marked S hemes.Mi hela CeriaUniversità degli Studi di Torino, ItalyIn this poster, we examine JMBTest.lib and JMSConst.lib, two new libraries for Singu-lar, integrated in the 3-1-6 release.The �rst library, JMBTest.lib, he ks whether a J-marked set (J ⊳ k[x0, ..., xn] Borel-�xed ideal) is a J-marked basis, a ording to the de�nition given by F. Cio� and M.Roggero. The se ond library, JMSConst.lib, takes as input a Borel-�xed ideal J and omputes the equations of the asso iated J-marked s heme.Both libraries are based on a Bu hberger-type redu tion, performed on a spe ial set ofS-polynomials, alled Eliahou-Kervaire polynomials, that orrespond to a basis for the�rst syzygies of J .In the urrent release, our libraries and the underlying algorithms work on the homoge-neous ase. However, we plan to extend them to the ase of non-homogeneous markedpolynomials, also improving their performan e.The libraries are freely available athttp://www.singular.uni-kl.de/index.php/singular-download.htmlWronski Polynomial Systems with Polymake and SingularBenjamin AssarfTe hnis he Universität Darmstadt, Germanypolymake is a software system for geometri ombinatori s, in parti ular, onvex poly-topes, polyhedral fans, and similar obje ts. Singular is a software system for om-mutative algebra, in parti ular, Gröbner bases, resolution of singularities, and related on epts. Re ent progress on algorithms in algebrai geometry bene�ts from polyhedralmethods and, onversely, ombinatori s gains a lot from te hniques in ommutative al-gebra. Between these two systems this sparked a re ent �urry of intera tion, whi h isbased on a new bidire tional interfa e. Here we will report on omputational experi-ments on erning the numbers of real roots of ertain multi-variate polynomial systems.These were arried out in polymake, alling Singular through that interfa e.See http://www.polymake.org and http://www.singular.uni-kl.de/Coauthors: Mi hael Joswig, Te hnis he Universität Darmstadt, Germany;Andreas Paffenholz, Te hnis he Universität Darmstadt, Germany.

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