America n Mathematica l Societ y

Colloquiu m Publication s Volum e 3 3

Differentia l Algebr a

Josep h Fel s Ritt

^ D E D

America n Mathematica l Societ y Providence , Rhod e Islan d

http://dx.doi.org/10.1090/coll/033

2000 Mathematics Subject Classification. P r i m a r y 12-02 ; Secondar y 12H05 .

Library o f Congres s Cataloging- in-Publ icat io n D a t a

Ritt, Josep h Fels , 1893-1951 . Differential algebra .

p. cm . — (America n Mathematica l Societ y Colloquiu m publications , ISS N 0065-925 8 ; v. 33 ) New York , America n Mathematica l Society , 1950 . Includes bibliography . ISBN 978-0-8218-4638- 4 (alk . paper ) 1. Differentia l equations . I . Title . II . Colloquiu m publication s (America n Mathematica l

Society) ; v. 33 .

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PREFACE

In 1932 , th e autho r publishe d Differential equations from the algebraic stand-point,1 a boo k dealin g wit h differentia l polynomial s an d algebrai c differentia l manifolds. I n th e sixtee n year s whic h hav e passed , th e wor k o f a numbe r o f mathematicians ha s give n fres h substanc e an d ne w colo r t o th e subject . Th e complete editio n o f th e boo k havin g bee n exhausted , i t ha s seeme d prope r t o prepare a new exposition .

The titl e Differential algebra was suggeste d b y Dr . Kolchin . Th e bod y o f algebra deal s with th e operation s o f additio n an d multiplication . W e ar e con -cerned her e wit h thre e operations—addition , multiplicatio n an d differentiation .

If I a m no t mistaken , th e genera l natur e o f th e subjec t her e treate d i s no w well enough known amon g mathematicians t o permi t m e to dispens e with a de -tailed introduction, suc h as was given in A. D. E. M y principa l task i s to sho w how muc h th e presen t boo k owe s t o m y associates . I a m referrin g t o H . W . Raudenbush, W . C . Strodt , E . R . Kolchin , Howar d Levi , El i Gouri n an d Richard M . Cohn .

Cohn's constructive proof o f the theorem o f zeros will be found i n Chapte r V . The theorem o n embedded manifold s du e to Gouri n i s contained i n Chapte r I I . Chapter V I contain s a discussio n o f Strodt' s wor k o n sequences o f manifolds .

In Chapter s I , I I I an d IX , ther e ar e presente d portion s o f Levi' s wor k o n ideals o f differentia l polynomial s an d o n th e lo w power theorem . O f Kolchin' s investigation o f exponent s o f differentia l ideals , I hav e bee n abl e t o giv e onl y a bare idea . Othe r wor k o f Kolchin , fo r instance , proof s fo r th e abstrac t cas e of results previously establishe d fo r th e analyti c case , is given in Chapte r I I . Hi s work o n th e Picard-Vessio t theory , whic h employ s th e method s o f differentia l algebra, has just appeared in the Annals of Mathematics,2 and may be permitte d to speak for itself .

The contributions o f Raudenbush ca n only be described a s fundamental. Th e basis theore m o f Chapte r I was , i n th e analyti c case , implicitl y containe d i n A. D. E. I t exist s there in two parts; the first, the theorem o n the completenes s of infinit e systems ; the second , th e theore m o f zeros . Onl y casuall y ha d I no -ticed tha t th e tw o theorem s amounte d t o a basi s theorem . I wa s acquainte d with the fac t tha t th e theorem o n the decompositio n o f manifolds amounted , i n virtue of the theorem o f zeros, to a theory o f perfect an d prime ideals of differen -tial polynomials. I n the summer o f 1933 , I suggeste d to Raudenbush the prob -lem o f constructin g a theor y o f perfec t ideal s whic h woul d b e vali d i n th e ab -stract case . Thi s h e accomplished , and , i n th e cours e o f hi s work , h e brough t the basi s theore m t o it s presen t complet e an d abstrac t form . I n th e proof o f

1 These Colloquiu m publications, vol . 14 . Calle d below A . D. E . 2 Kolchin, 14 . (Se e Bibliography , p . 180. )

iii

iv PREFACE

the basi s theorem, th e procedur e o f taking power s i s due t o Raudenbush . Th e chains, characteristic set s and methods o f reduction existe d in the olde r theore m of completeness .

Raudenbush introduce d generi c zero s o f prim e ideals . Her e h e adapte d a method o f van de r Waerden , whic h ca n be traced bac k t o Konig . Raudenbus h gave the first exampl e o f a syste m o f differentia l polynomial s with a weak basis . Systems with n o strong base s were later produce d b y Kolchin .

The problem s which thi s book treat s ar e very concret e problems . The y dea l with situation s o f th e classica l theor y o f differentia l equations . Seldo m woul d much b e lost , a s fa r a s th e result s ar e concerned , i f on e limite d onesel f t o th e material o f classical analysis . Th e abstrac t metho d whic h we generally emplo y has, however , a definit e utility . I t serve s t o separat e algebrai c method s fro m analytic methods. O n the whole, it contribute s to simplicity , althoug h a t time s an abstrac t treatmen t i s less natural than a n analytical one . Th e form i n which the result s o f differentia l algebr a ar e bein g presente d ha s thu s bee n deepl y in -fluenced b y th e teaching s o f Emmy Noether , a prime mover o f ou r period, who , in continuin g Juliu s Konig' s developmen t o f Kronecker' s ideas , brought mathe -maticians t o know algebr a a s it was never known before .

In thi s connection , I shoul d lik e to sa y somethin g concernin g basi s theorems . The basi s theorem o f Chapte r I will be see n t o play , i n th e presen t theory , th e role hel d b y Hilbert' s theore m i n th e theorie s o f polynomia l ideal s an d o f alge -braic manifolds . Whe n I bega n t o wor k o n algebrai c differentia l equations , early i n 1930 , va n de r Waerden' s excellen t Moderne Algebra ha d no t ye t ap -peared. However , Emm y Noether' s wor k o f th e twentie s wa s available , an d there was nothing t o preven t on e from learnin g i n he r paper s th e value o f basi s theorems i n decompositio n problems . Actually , I becam e acquainte d wit h th e basis theore m principl e i n th e writing s o f Jule s Drach 3 o n logica l integration , writings whic h dat e bac k t o 1898 . Ho w a basi s theore m i s employe d b y hi m will now b e described .

There ar e tw o distinc t method s fo r characterizin g a n irreducibl e algebrai c equation. O n th e on e hand , a n equatio n f(x) = 0 i s irreducible i f f(z) canno t be factored . O n th e other , ther e i s irreducibilit y i f ever y equatio n whic h i s satisfied b y a singl e solutio n o f f(x) = 0 i s satisfied b y al l suc h solutions . Th e first formulatio n o f irreducibilit y lead s t o th e notio n o f irreducibl e algebrai c manifold an d t o tha t o f irreducibl e algebrai c differentia l manifold . Th e second lead s t o th e concep t o f irreducibl e syste m o f algebrai c differentia l equations whic h wa s employe d b y Koenigsberge r an d b y Drach . A syste m o f such equations , ordinar y o r partial , i s irreducibl e i f ever y differentia l equatio n which admits a single solution o f the system admit s al l solutions. Drac h under -takes to show that, give n a system o f partial differentia l equations , the repeate d adjunction o f new equations will eventually produce an irreducible system. Fo r this he invokes a theore m o f Tresse, 4 which state s that , i n ever y infinit e syste m

3 Drach, 4 , pp. 292-296. 4 Acta Mathematica, vol , 18 (1894), p. 4.

PREFACE V

of partial differential equations , there is a finite subsystem from which the infinite system can be derived by differentiations an d eliminations. A study of Tresse's paper will quickly convince one that he claims for his work a generality which it does not have. Th e statement of his theorem, and his argument, have a definite meaning only for linear systems.

It ha s not been possible for me to present al l of the material which has been developed since the publication of A. D. E. Thus , I have had to pass by most of Kolchin' s study o f exponents and a good deal of Levi' s work on ideals. O f Strodt's paper , onl y a sketch i s given. M y ow n work o n general solution s of equations of the second order in one unknown, and of equations of the first order in two unknowns, is also omitted.

I hav e trie d t o give , t o th e presen t book , th e elementar y qualit y whic h i s possessed by A. D. E. Essentially , n o previous knowledge of abstrac t algebr a is necessary. A s in A. D. E., a treatment is given of Riquier's existence theorem for orthonomic systems of partial differential equations .

New York, N. Y. January, 1948.

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CONTENTS

CHAPTER I

DIFFERENTIAL POLYNOMIAL S AN D THEI R IDEAL S 1

Differential fields, indeterminates , differentia l polynomials , chains , characteristi c sets , re -duction, ideal s o f differentia l polynomials , bases , stron g an d wea k bases , decompositio n o f perfect ideals , relatively prim e ideals , the idea l [y p], adjunctio n o f indeterminates , field exten -sions, fields o f constants .

CHAPTER I I

ALGEBRAIC DIFFERENTIA L MANIFOLD S 2 1

Manifolds an d thei r decomposition , illustration s i n analysis , prim e ideal s an d regula r zeros , generic zero s o f a prime ideal , th e theore m o f zeros , genera l solutions , singula r zero s and solu -tions, parametri c indeterminates , th e resolvent , dimensio n o f an irreducible manifold , orde r o f the resolvent , embedde d manifolds , prim e ideal s an d field extensions , adjunction s t o fields, analogue o f Luroth' s theorem .

CHAPTER II I

STRUCTURE O F DIFFERENTIA L POLYNOMIAL S 5 7

I. Manifold of a differential polynomial. Theore m o n dimensio n o f components , arbitrar y constants, th e polygo n process , dimension s o f components , degree s o f generality . II . Low powers and singular solutions. Components , preparatio n process , th e lo w powe r theorem , sufficiency proof , necessit y proof , a n example , furthe r theorem s o n lo w powers , term s o f low -est degree , singula r solutions . III . Exponents of ideals.

CHAPTER I V

SYSTEMS O F ALGEBRAI C EQUATION S 8 1

Polynomials an d thei r ideals, algebrai c manifolds , generi c zero s o f prime polynomia l ideals , resolvents, Hilbert' s theore m o f zeros , characteristic set s of prime polynomial ideals, construc -tion o f resolvents , component s o f finite systems , a n approximation theorem , zero s and charac -teristic sets .

CHAPTER V

CONSTRUCTIVE METHOD S 10 7

Characteristic set s o f prime ideals, finite systems , tes t for a d.p. to hold a finite system , con -struction o f resolvents , constructiv e proo f o f theore m o f zeros , a second theor y o f elimination , theoretical proces s fo r decomposin g th e manifol d o f a finite syste m int o it s components .

CHAPTER V I

ANALYTICAL CONSIDERATION S 12 2

Normal zeros , adherence, th e theore m o f approximation , analytica l treatmen t o f lo w powe r theorem, differentia l polynomial s i n on e indeterminate , o f first order , sequence s o f irreducibl e manifolds, operation s upo n manifolds .

vii

viii CONTENTS

CHAPTEB VI I

INTERSECTIONS O F ALGEBRAI C DIFFERENTIA L MANIFOLD S 13 3

Dimensions o f component s o f intersections , order s o f component s o f a n intersection , inter -sections of genera l solutions, intersections of component s o f a differential polynomial , analogu e of a theore m o f Kronecker .

CHAPTER VII I

RIQXJIER'S EXISTENC E THEORE M FO R ORTHONOMI C SYSTEM S 14 7

Monomials, dissectio n o f a Taylo r series , marks , orthonomi c systems , passiv e orthonomi c systems.

CHAPTER I X

PARTIAL DIFFERENTIA L ALGEBR A 16 3

Partial differentia l polynomials , ideal s an d manifolds , component s o f a partia l differentia l polynomial, th e lo w powe r theorem , characteristi c set s o f prim e ideals , algorith m fo r decom -position, the theorem o f zeros .

APPENDIX

QUESTIONS FO R INVESTIGATIO N 17 7

BIBLIOGRAPHY 18 0

INDEX 18 3

APPENDIX. QUESTION S FO R INVESTIGATIO N

IDEALS

1. Levi' s work shows the nonexistence o f a theory o f ideals of d.p . possessin g the scop e o f th e Lasker-Noethe r theor y o f p.i . Fo r d.p. , i t wil l b e necessar y either t o us e specia l type s o f ideal s o r t o us e othe r combination s tha n intersec -tions an d products .

2. Give n a finite se t o f d.p. , F h • • • , Fry an d a d.p . (? , is it possibl e t o deter -mine whethe r G i s containe d i n [Fi, • • • , Fr]? Th e method s o f Chapte r V permit on e to decid e whethe r som e power o f G is in [Fi, • • • , Fr], I t i s thus a question o f determinin g a smalles t admissibl e exponent .

3. Kolchin' s theor y o f exponent s shoul d admi t o f extensio n i n severa l direc -tions. Th e chie f proble m examine d by Kolchin i s that o f the exponen t o f { A } relative t o [A], wher e A i s a d.p . i n y o f th e first order . I n th e theorem s ob -tained b y Kolchin , th e relativ e exponent s ar e 1 , 2 , <*> . Fo r instance , i f A = y 2 + y\ } th e exponen t i s °° . No w

[A]= fcrt-S,

with p a positiv e intege r an d 2 a n idea l whos e manifold i s the genera l solutio n of A. On e may inquir e a s t o th e exponen t o f { 2 } relative t o 2) . Tha t ex -ponent ma y easil y b e finite. Thi s proble m can , o f course , b e formulate d fo r d.p. A admittin g man y singula r zeros .

The proble m o f exponent s ma y b e examine d fo r d.p . o f orde r highe r tha n the first an d fo r p.d.p .

4. Fo r F = y p + y\, i n ${ y } , wit h q > p, wha t i s th e smalles t intege r r such tha t

y'G m 0 , [FI

where G does no t vanis h fo r y = 0 ? Thi s proble m ca n b e extende d t o genera l classes o f d.p .

5. Fo r p > 0 , i > 0 , what i s the leas t q such tha t y\ = 0 , [y p]? Fo r i = 1 , it i s not har d t o sho w tha t q = 2p — 1. I n $ { u, v } , what i s th e leas t powe r of UiVj which i s containe d i n [ uv ]?

6. Th e ideals generated b y various differentia l expression s may be examined . One ma y stud y th e wronskian , th e jacobian , th e expressio n EG — F 2 o f dif -ferential geometry , etc .

7. On e may stud y d.p . ove r a field o f characteristi c p.

THE DECOMPOSITIO N PROBLE M

8. Th e basic problem has been met i n Chapte r V . I t i s that o f determinin g the numbe r o f time s whic h th e d.p . i n a finite syste m $ mus t b e differentiate d

177

178 DIFFERENTIAL ALGEBR A

before elimination s wil l produc e finite system s whos e manifold s ar e th e com -ponents o f 3>. On e would hope to secur e a bound which depends on the numbe r of d.p. in #, thei r order s and degrees .

9. Attache d t o th e decompositio n proble m i s th e first proble m o f Laplace , mentioned i n I I I , §37 . Le t F an d A b e algebraicall y irreducibl e an d le t F hold the genera l solution o f A, I t i s required t o determine whethe r th e genera l solution o f A i s containe d i n tha t o f F. Th e autho r ha s show n ho w t o settl e this questio n fo r d.p . F o f th e secon d order, 1 Th e method s ca n perhap s b e ex -tended t o cove r th e cas e i n whic h F, i n $F { y } , i s o f orde r n , an d A o f orde r n — 2 . On e migh t perhap s undertak e t o develo p a tes t fo r th e presenc e o f y = 0 i n th e genera l solutio n o f a d.p . o f th e thir d order . Othe r problem s o f this typ e wil l readily sugges t themselves .

INTERSECTIONS

10. On e can se e from Chapte r V I I tha t i f ther e i s regularity i n th e theor y o f intersections o f algebrai c differentia l manifolds , tha t regularit y i s no t immedi -ately visible . I n VII , §1 , an anomal y i s found i n th e dimensio n o f th e inter -section o f a genera l solutio n wit h a secon d irreducibl e manifold . On e migh t try t o us e complet e manifold s o f d.p . rathe r tha n genera l solutions . Thus , le t Fij • • • , Fr b e d.p . i n ${y\ 7 • • • , yn } . Suppos e tha t r <n. I s ever y com -ponent o f th e syste m F lf • • • , Fr o f dimensio n a t leas t n — r ? Fo r r = 1 , w e see from I I I , §1 , that th e answer is affirmative .

11. On e may see k to exten d th e resul t o f VII , §6 , o n Jacobi' s boun d t o sys -tems of n d.p . in n indeterminates .

The anomal y me t i n connectio n wit h th e orde r o f a componen t o f th e inter -section o f tw o genera l solution s raise s th e followin g problem . Le t A an d B b e algebraically irreducibl e d.p . i n y an d 2 . Le t 9f t b e a componen t o f dimensio n zero i n th e intersectio n o f th e genera l solution s o f A an d B. I t i s required t o find a boun d fo r th e orde r o f 9f t i n term s o f th e order s o f A an d B i n y an d z. I t i s conceivable , o f course , tha t n o boun d exists .

12. On e may generaliz e the problem o f I I I , §1 , as follows. Le t S b e a non -trivial prim e idea l i n SF { Wi , • • • , uq; t/i , • • • , yp } wit h th e u parametri c an d with

(1) Ai, • • • , A p

a characteristi c set . Le t 2 0 b e th e prim e p.i . fo r whic h (1) , wit h th e A con -sidered a s polynomials , i s a characteristi c set . Le t 2 / b e th e syste m o f d.p . obtained fro m S 0 whe n th e polynomial s i n 2 0 ar e regarded a s d.p . Wha t ar e the dimension s o f the component s o f 2' ? Doe s th e lo w power theore m hav e a generalization fo r thi s situation ?

DIFFERENTIAL POWE R SERIE S

13. Thi s subjec t ha s bee n mentione d i n I I I , §39 . Onl y on e pape r ha s bee n 1 Ritt, 31 . I n connection with §6 5 of this paper, see the final remarks of §5 1 of Ritt, 32.

QUESTIONS FO R INVESTIGATIO N 179

written o n it . Th e entir e progra m await s development , bot h fo r ordinar y dif -ferential equation s an d fo r partial . I n th e analyti c case , th e procedur e wil l depend on whether one works in the neighborhood of a point in the space of th e independent variables or in the neighborhood of a set of functions constituting a point o f a manifold.

BlRATIONAL TRANSFORMATION S

14. Th e theory o f th e resolven t furnishe s a n instanc e o f the birationa l equiv -alence o f tw o irreducibl e manifolds . Th e genera l proble m i s tha t o f finding conditions fo r suc h equivalence . Th e result s o f algebrai c geometr y shoul d b e a guide .

In studyin g birationa l transformations , on e wil l mee t differential Cremona transformations. Fo r instance , le t

We find

-r£(f> -«|(f)-Is ther e a theore m o n th e structur e o f suc h transformation s o f y an d z simila r to M . Noether' s theore m o n ordinar y Cremon a transformations ?

The analogu e o f Luroth' s theore m presente d i n Chapte r I I ma y hav e a n ex -tension to fields formed b y the adjunctio n o f two indeterminates .

SINGULAR SOLUTION S O F PARTIA L DIFFERENTIA L EQUATION S

15. Fo r simplicity , w e use tw o independen t variables , x an d y. Le t F b e a n algebraically irreducibl e d.p . i n ${ z } , o f orde r n i n z. Le t th e component s o f F b e 2ft , 2fti , • • • , 2ft«, wit h 2f t th e genera l solution . Eac h 2ftt * i s th e genera l solution o f a d.p . F{. Suppos e that , fo r som e i, Fi i s of orde r n — 1 in z. Con -sidering Hamburger' s result s fo r ordinar y differentia l equations , on e woul d ex -pect th e function s i n 2ft * to b e envelopes , with a contac t o f som e natura l order , of function s i n 2ft . Fo r n = 1 , thi s questio n ha s bee n studie d b y th e author. 2

For n > 1 , the matte r shoul d b e more difficult , sinc e there i s no theory o f char -acteristics.

DIFFERENCE ALGEBR A

16. Thi s subjec t ha s bee n treate d i n paper s o f J . L . Doob , W . C . Strodt , F . Herzog, H . W . Raudenbush , Richar d Coh n an d th e author. 3 Th e theor y i s open fo r cultivation .

2 Ritt, 41. 3 See bibliography.

BIBLIOGRAPHY

1. COHN , R. M. On the analog for differential equations of the Hilbert-Netto theorem. Bulle-tin of the American Mathematical Society, vol. 47 (1941) , pp. 268-270.

2. Manifolds of difference polynomials. Transaction s o f th e America n Mathe -matical Society , vol. 64 (1948) , pp. 133-172 .

3. DOOB , J. L., and RITT , J. F. Systems of algebraic difference equations, American Journal of Mathematics , vol . 55 (1933) , pp. 505-514 .

4. DRACH , J . Essai sur la thSorie ginirale de VinUgration et sur la classification des tran-scendantes, Annales de l'Ecole Normal e Superieure , (3) , vol. 15 (1898), pp. 245-384.

5. GOTJRIN , E . On irreducible systems of algebraic differential equations, Bulletin o f th e American Mathematica l Society , vol . 39 (1933) , pp. 593-595.

6. HAMBURGER , M. Ueber die singuldren Losungen der algebraischen Differenzialgleichungen erster Ordnung, Journal fii r di e reine und angewandte Mathematik , vol . 11 2 (1893) , pp. 205-246. Se e also ibid., vol. 12 1 (1899), p. 265, and vol . 12 2 (1900), p. 322.

7. HERZOG , F. Systems of mixed difference equations, Transactions of the American Mathe-matical Society, vol. 37 (1935), pp. 286-300.

8. KOLCHIN , E . R., an d RITT , J . F. On certain ideals of differential polynomials, Bulletin of the American Mathematica l Society, vol. 45 (1939), pp. 895-898.

9. KOLCHIN , E . R . On the basis theorem for infinite systems of differential polynomials, Bulletin o f th e America n Mathematica l Society , vol . 45 (1939) , pp. 923-926.

10. On the exponents of differential ideals, Annals of Mathematics , vol . 42 (1941) , pp. 740-777 .

11. On the basis theorem for differential systems, Transactions o f th e America n Mathematical Society , vol. 52 (1942) , pp. 115-127 .

12. Extensions of differential fields, I, II , Annal s o f Mathematics , vol . 4 3 (1942), pp. 724-729 ; vol. 45 (1945), pp. 358-361 .

13. Extensions of differential fields, III , Bulleti n o f th e America n Mathematica l Society, vol. 53 (1947), pp. 397-401 .

14. Algebraic matric groups and the Picard-Vessiot theory of homogeneous ordinary linear differential equations, Annals of Mathematics , vol . 49 (1948) , pp. 1-42 .

15. LAGRANGE , J . L . Sur les solutions particulieres des Equations diffirentielles, Oeuvres Completes, vol. 4, pp. 5-108 .

16. LAPLACE , P. S. Mimoire sur les solutions particulieres des Equations diffSrentielles et sur les intgalites sfoulaires des planhtes, Oeuvres Completes, vol. 8, pp. 326-365.

17. LEVI , H. On the structure of differential polynomials and on their theory of ideals, Trans-actions of the American Mathematica l Society, vol. 51 (1942), pp. 532-568.

18. The low power theorem for partial differential polynomials, Annals o f Mathe -matics, vol. 46 (1945), pp. 113-119.

19. POISSON , S. D. Sur les solutions particulieres des Equations differentielles et des equations aux differences, Journal de l'Ecole Poly technique, vol. 6, no. 13 (1806), pp. 60-125.

20. RAUDENBTJSH , H . W . Differential fields and ideals of differential forms, Annal s o f Mathematics, vol. 34 (1933), pp. 509-517.

21. Ideal theory and algebraic differential equations, Transaction s o f the American Mathematical Society , vol . 36 (1934) , pp. 361-368.

22. Hypertranscendental adjunctions to partial differential fields, Bulletin o f th e American Mathematica l Society , vol . 40 (1934), pp. 714-720.

23. On the analog for differential equations of the Hilbert-Netto theorem, Bulleti n of the America n Mathematica l Society , vol . 42 (1936) , pp. 371-373.

180

BIBLIOGRAPHY 181

24. RAUDENBUSH , H . W., and RITT, J . F. Ideal theory and algebraic difference equations, Transactions of the American Mathematica l Society, vol. 46 (1939), pp. 445-453 .

25. RITT , J . F. Manifolds of functions defined by systems of algebraic differential equations, Transactions of the American Mathematica l Society , vol. 32 (1930), pp. 369-398.

26. Differential equations from the algebraic standpoint, American Mathematica l Society Colloquium Publications , vol . 14, New York, 1932.

27. Algebraic difference equations, Bulletin of the American Mathematical Society, vol. 4 0 (1934), pp. 303-308.

28. Systems of algebraic differential equations, Annals o f Mathematics , vol . 36 (1935), pp. 293-302.

29. Jacob?s problem on the order of a system of differential equations, Annal s of Mathematics, vol. 36 (1935), pp. 303-312.

30. Indeterminate expressions involving an analytic function and its derivatives, Monatshefte fu r Mathematik , vol . 43 (1936), pp. 97-104.

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INDEX

The numbers refer t o pages. Adherence adjunction o f indeterminate s adjunctions t o fields algebraic differential manifol d algebraically irreducible d.p . algebraic manifol d analytic cas e approximation theorem s

Basis , stron g , wea k

Chain characteristic se t class complete se t component

, restricte d constant

, arbitrar y

122 18

20,52 21 30 82

23, 16 6 103, 122

9, 165 11 11

3, 16 4 5 2

150 23 24

1 57

Decomposition of finite systems 109 , 118, 175 decomposition o f ideals 13 , 14, 166 decomposition o f manifold s

Field, algebraic 1 , differentia l 1 , 163

fields, extensions of 1 , 19 , 50, 52 fields of constants 2 0 finite systems 109 , 118, 175

General solution 30 , 166 , restricted 3 2

generic point 2 7 generic zero 26 , 83, 166

Ideal generated b y a system 7 ideal, nontrivial prim e 2 5 ideal of d.p . 7 ideal of polynomials 8 1 ideal, perfect 7 ideal, prime 7 ideals, decomposition o f 13 , 14, 166

, product o f 1 1 , relatively prim e 1 4

indeterminate, differentia l 2 indeterminates, adjunction o f 1 8

, parametric 34 , 84 initial 5 , 164 intersection o f genera l solutions 13 8

22, differential field differential polynomia l

over a field differential powe r series differentiation dimension dimension of intersectio n divisor

, essential prim e

Elimination theor y embedded manifold s equivalence essential prime diviso r exponents of ideal s exceptional poin t extended se t extensions o f fields

23, 117 ,

109,

118, 165 1, 163

2 2

78 1, 163 44,87

133 13,81 14,82

112, 176 49 95

14,82 78

124 150

1, 19 , 50, 52

Jacobi's bound

Kronecker's theore m

Lagrange Laplace leader low power theore m Ltiroth's theorem

Manifold, algebrai c , algebraic differentia l , irreducibl e , reducibl e , restricte d

manifolds, decompositio n of 22, 23,

, operations on , sequences of

135

146

33 77

163 64, 126 , 170

52

82 21

21,83 21,83

23

117, 118 , 165 132 131

183

184 INDEX

mark monomial multiple multiplier

Normal zero

Order of irreducible manifold of resolvent

orders of components orthonomic system

Painlev6's transformatio n parametric derivative

^determinates passive system point of contact

of manifold Poisson polynomial ideal principal derivative product of ideals

151 147 147 150

122

49 45

133 152

128 153

34,84 160 122 21 77 81

153 11

Relatively prim e ideal s remainder

, clas s resolvent

, orde r o f , constructio n o f

restricted manifol d

Separant singular zer o singular solutio n solution

, genera l

Theorem o f zero s

Zero , analyti c , generi c , norma l , regula r , singula r

zeros, theore m o f

27,

27,

28,

28,

87,

87,

111,

26

111,

14 7

26 34 ,83

45 110 23

5, 16 4 32

32 ,75 21

30, 16 6

166, 17 6

21 ,81 23

i, 83, 16 6 122 26 32

166, 17 6