reu on algorithmic algebraic geometry€¦ · reu on algorithmic algebraic geometry joseph rennie...

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


Outline

1 Background

2 Foundation

3 Our Goal

4 Approaches

5 Conclusion


Background

Algebraic Geometry-What

What is it?

Varieties – Zero sets of systems of polynomials

Notation/Terminology Hell...but worth it!


Background

Algebraic Geometry-Why

Pure Mathematics

Nice Problems

Connections to other areas of mathematics

Number TheoryCombinatoricsStatistics

Applied Mathematics

Physics, Mathematical Biology, Automated Geometric Reasoning,. . .


Background

(a) The ”interplanetary superhighway”

Image can be found at www.jpl.nasa.gov/images/superhighway square.jpg


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 .


Foundation

For example, let f (x1, x2) = 42 + 42x32 + 42x31 + 42x1x2 (a bivariatetetranomial) then A = {(0, 0), (0, 3), (3, 0), (1, 1)}


Foundation

A polynomial is said to be honest if its support does not lie in any(n-1)-plane.


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 .


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 .


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


Our Goal

Quadric Hypersurfaces

(a) x21 + x22 − x23 = 0

Figure 2: Degenerate Quadric Hypersurface

Image courtesy of Wikipedia


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 .


Our Goal

Isotopy




+

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 .


Our Goal

Isotopy




+ 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 .


Our Goal

Isotopy




+ 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 .


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 .


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.


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.


Conclusion

THANK YOU FOR YOUR ATTENTION!!!

:)


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.


reu on algorithmic algebraic geometry€¦ · reu on algorithmic algebraic geometry joseph rennie...

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