Applications of Polyhedral Homotopy Continuation Methods to Topology. Takayuki Gunji (Tokyo Inst. of Tech.)

Contents  Polyhedral Homotopy Continuation Methods  Numerical examples  Applications to Topology

Introduction Polynomial systems come from various fields in science and engineering Inverse kinematics of robot manipulators. Equilibrium states. Geometric intersection problems. Formula construction. Find all isolated solutions of polynomial systems.

Introduction  Grobner Basis Using Mathematica It takes long time  Linear Homotopy  Polyhedral Homotopy PHCpack by J.Verschelde(1999) PHoM by Gunji at al.(2002)  Parallel Implementation

Isolated solutions. are isolated solutions 4 y x O1234

Isolated solutions. aren’t isolated solutions 4 y x O1234

The number of solutions. Cyclic_n problem. NNum.NNum 1034, , ,756132,704,156

Homotopy continuation method. The original system Step 2 Solving Step 1 Constructing homotopy systems such that and that can be solved easily

Homotopy continuation method. Step 3 Tracing homotopy paths. Solutions of the original system Solutions of

Linear homotopy Can be solved easily!

Polyhedral homotopy Binomial systemCan be solved by Euclidean algorithm. Same as

General position Example are solutions of this system.

General position Example When are randomly chosen, this case doesn’t happen with probability 1 (the measure of this case happening is 0) If, this system doesn’t have a continuous solution

General position Step 2: Find solutions of P(x)=0 by using this system Step 1 : P’(x)=0 solves by using polyhedral homotopy

Polyhedral Homotopy  D.N.Bernshtein “The number of roots of a system of equations”, Functional Analysis and Appl. 9 (1975)  B.Huber and B.Sturmfels “A Polyhedral method for solving sparse polynomial systems”, Mathematics of Computation 64 (1995)  T.Y.Li “Solving polynomial systems by polyhedral homotopies”, Taiwan Journal of Mathematics 3 (1999)

Polynomial system Constructing homotopy systems Solving binomial systems Tracing homotopy paths Verifying solutions All isolated solutions

Constructing homotopy systems The original system Randomly chosen multiply to each terms

Constructing homotopy systems

Divided by

Constructing homotopy systems Ex Find all satisfying the property that. Each equation, exactly 2 of power of t are 0.

Constructing homotopy systems Find all satisfying the property that. Each equation, exactly 2 of power of t are 0.

Constructing homotopy systems All of solutions.

Tracing homotopy path Using Predictor Corrector Method Predictor step Corrector step Corrector step : Newton Method Predictor step : tangent of path (increase of t)

Tracing homotopy path Taylor series Corrector step Predictor step

Polynomial system Constructing homotopy systems Solving binomial systems Tracing homotopy paths Verifying solutions All isolated solutions

Parallel Computing Path 1 Path 2 Path 4 Path 3Path 5 Independent!

Parallel Computing  Client and server model. Client Server 1 Server 2 Server 3 Server 4 Master problem sub problem

PHoM (Polyhedral Homotopy Continuation Methods) Single CPU version OS : Linux (gcc)

Numerical examples Isolated solutions Linear Homotopy : the number of tracing path is 4. Polyhedral Homotopy : the number of tracing path is 2.

Numerical examples Cyclic_n problem.

Numerical examples problemNum.Time cyc_1034,9405mins cyc_11184,75630mins cyc_12367,4884hours cyc_132,704,15615hours The number of solutions Athlon 1200MHz 1GB(or2GB)x32CPU

Some applications to Topology  Representation space of a fundamental group in SL(2,C).  Computation of Reidemeister torsion  Joint works with Teruaki Kitano.

Representation into SL(2,C)  M: closed oriented 3-dimensional manifold  its fundamental group of M  an irreducible representation of  the set of conjugacy classes of SL(2,C)- irreducible representations.  is an algebraic variety over C  Problem: Determine in

Figure-eight knot case  Fundamental group of an exterior of figure- eight knot  2 generators and 1 relation  meridian u and longitude l.

Irreducible representation  Consider an irreducible representation into SL(2,C)  Write images as follows

Corresponding matrices  We consider conjugacy classes, then we may put U and V as follows

Representation space  From the relation wu=vw in the group, we obtain the following polynomial.

Dehn surgery along a knot  Put a relation in the fundamental group.  L is a corresponding matrix of a longitude l. The above relation gives one another polynomial g(x,y)=0 as a defining equation.

Apply the Homotopy Continuation Methods  This system of polynomial equations f=g=0 describe conjugacy classes of representations, that is, each solution is a corresponding one conjugacy class of representations.  We solve some case by using the polyhedral homotopy continuation methods.

Reidemeister torsion  Reidemeister torsion is a topological invariant of 3-manifolds with a representation parameterized by x and y.

Example 1 : (p,q)=(1,1)

Re(x)Im(x)Re(y)Im(y)R-torsion

Example 2 : (p,q)=(1,2)

Re(x)Im(x)Re(y)Im(y)R-torsion i i