Algorithms for Determining the Topologyof Positive Zero Sets
REU on Algorithmic Algebraic Geometry
Joseph Rennie
Texas A&M
07/24/2013
Rennie (Texas A&M) Mini Symp Talk 1 07/24/2013 1 / 19
Acknowledgements
Funded by the National Science Foundation and Texas A&MUniversity
Special thanks to Svetlana Kaynova for her contributions to thisproject
Thank you also to our mentors Dr. J. Maurice Rojas and KaitlynPhillipson
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Outline
1 Background
2 Foundation
3 Our Goal
4 Approaches
5 Conclusion
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Background
Algebraic Geometry-What
What is it?
Varieties – Zero sets of systems of polynomials
Notation/Terminology Hell...but worth it!
Rennie (Texas A&M) Mini Symp Talk 1 07/24/2013 4 / 19
Background
Algebraic Geometry-What
What is it?
Varieties – Zero sets of systems of polynomials
Notation/Terminology Hell...but worth it!
Rennie (Texas A&M) Mini Symp Talk 1 07/24/2013 4 / 19
Background
Algebraic Geometry-Why
Pure Mathematics
Nice Problems
Connections to other areas of mathematics
Number TheoryCombinatoricsStatistics
Applied Mathematics
Physics, Mathematical Biology, Automated Geometric Reasoning,. . .
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Background
(a) The ”interplanetary superhighway”
Image can be found at www.jpl.nasa.gov/images/superhighway square.jpg
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Foundation
Terms: at the gates
The support A of an n-variate t-nomial f , where
f (x1, . . . , xn) = c1xa1 + · · ·+ ctx
at
is given by A = {a1, . . . , at} where each ai ∈ Rn and wherexai = xai11 . . . xainn .
Rennie (Texas A&M) Mini Symp Talk 1 07/24/2013 7 / 19
Foundation
For example, let f (x1, x2) = 42 + 42x32 + 42x31 + 42x1x2 (a bivariatetetranomial) then A = {(0, 0), (0, 3), (3, 0), (1, 1)}
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Foundation
A polynomial is said to be honest if its support does not lie in any(n-1)-plane.
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Foundation
Notation
Z+(f ) is the set of roots of f in the positive orthant Rn+.
ZR(f ) is the set of real roots of f .
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Our Goal
Conjecture
Given f ∈ R[x1, . . . , xn] an honest (n + 2)-nomial, Z+(f ) has topologyisotopic to a quadric hypersurface of the form
x21 + · · ·+ x2j − (x2j+1 + · · ·+ x2n ) = ε
where j and the sign of ε are computable in polynomial time (for fixed n)from the support A and coefficients of f .
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Our Goal
Quadric Hypersurfaces
(b) x21 + x22 + x23 = 1 (c) x21 + x22 − x23 = 1 (d) x21 + x22 − x23 = −1
Figure 1: Nondegenerate Quadric Hypersurfaces
Images courtesy of Wikipedia
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Our Goal
Quadric Hypersurfaces
(a) x21 + x22 − x23 = 0
Figure 2: Degenerate Quadric Hypersurface
Image courtesy of Wikipedia
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Our Goal
Isotopy
A homemorphism (resp. diffeomorphism) is a continuous (resp.differentiable) bijection with a continuous (resp. differentiable)inverse.
Given any subsets X ,Y ⊆ Rn+,we say that they are isotopic (resp.
diffeotopic) iff there is a continuous (resp. differentiable) functionH : [0, 1]× X −→ Rn
+ such that H(t, ·) is a homeomorphism (resp.diffeomorphism) for all t ∈ [0, 1], H(0, ·) is the identity on X , andH(1,X ) = Y .
Rennie (Texas A&M) Mini Symp Talk 1 07/24/2013 14 / 19
Our Goal
Isotopy
A homemorphism (resp. diffeomorphism) is a continuous (resp.differentiable) bijection with a continuous (resp. differentiable)inverse.
Given any subsets X ,Y ⊆ Rn+,we say that they are isotopic (resp.
diffeotopic) iff there is a continuous (resp. differentiable) functionH : [0, 1]× X −→ Rn
+
such that H(t, ·) is a homeomorphism (resp.diffeomorphism) for all t ∈ [0, 1], H(0, ·) is the identity on X , andH(1,X ) = Y .
Rennie (Texas A&M) Mini Symp Talk 1 07/24/2013 14 / 19
Our Goal
Isotopy
A homemorphism (resp. diffeomorphism) is a continuous (resp.differentiable) bijection with a continuous (resp. differentiable)inverse.
Given any subsets X ,Y ⊆ Rn+,we say that they are isotopic (resp.
diffeotopic) iff there is a continuous (resp. differentiable) functionH : [0, 1]× X −→ Rn
+ such that H(t, ·) is a homeomorphism (resp.diffeomorphism) for all t ∈ [0, 1],
H(0, ·) is the identity on X , andH(1,X ) = Y .
Rennie (Texas A&M) Mini Symp Talk 1 07/24/2013 14 / 19
Our Goal
Isotopy
A homemorphism (resp. diffeomorphism) is a continuous (resp.differentiable) bijection with a continuous (resp. differentiable)inverse.
Given any subsets X ,Y ⊆ Rn+,we say that they are isotopic (resp.
diffeotopic) iff there is a continuous (resp. differentiable) functionH : [0, 1]× X −→ Rn
+ such that H(t, ·) is a homeomorphism (resp.diffeomorphism) for all t ∈ [0, 1], H(0, ·) is the identity on X , andH(1,X ) = Y .
Rennie (Texas A&M) Mini Symp Talk 1 07/24/2013 14 / 19
Our Goal
Conjecture
Given f ∈ R[x1, . . . , xn] an honest (n + 2)-nomial, Z+(f ) has topologyisotopic to a quadric hypersurface of the form
x21 + · · ·+ x2j − (x2j+1 + · · ·+ x2n ) = ε
where j and the sign of ε are computable in polynomial time (for fixed n)from the support A and coefficients of f .
Rennie (Texas A&M) Mini Symp Talk 1 07/24/2013 15 / 19
Approaches
Discriminant Varieties
Given any A ∈ Zn, we define the A-discriminant variety, written ∇A, tobe the topological closure of
{[c1 : · · · : cT ] ∈ PT−1C |c1xa1 + · · ·+ cT x
aT
has a degenerate root in (C∗)n}
The real part of ∇A determines where in coefficient space the real zero setof a polynomial (with support A) changes topology.
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Conclusion
Consequences of the conjecture
Tells us about the topology of positive zero sets of honest n-variate(n + 2)-nomials of arbitrary degree
Results
We currently have a bound on the number of connected components ofn-variate (n + 2)-nomials.
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Conclusion
THANK YOU FOR YOUR ATTENTION!!!
:)
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Conclusion
Bibliography
I.M. Gelfand, M.M. Kapranov, and A.V. Zelevinsky Discriminants,Resultants, and Multidimensional Determinants 2008: BirkhauserBoston, N.Y.
“Faster Real Feasibility via Circuit Discriminants,” (by Frederic Bihan,J. Maurice Rojas, and Casey Stella), proceedings of ISSAC 2009 (July28-31, Seoul, Korea), pp. 39-46, ACM Press, 2009.
“Faster Topology Computation for Positive Zero Sets of CertainSparse Polynomials,” F. Bihan, E. Refsland, J. R. Rennie, and J. M.Rojas, in progress.
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