Applications of Polyhedral Homotopy Continuation M

ethods 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 solutions4

y

x

3

2

1

O 1 2 3 4

Isolated solutions.

aren’t isolated solutions

4

y

x

3

2

1

O 1 2 3 4

The number of solutions.Cyclic_n problem.

N Num. N Num

10 34,940 12 367,488

11 184,756 13 2,704,156

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 system Can be solved by Euclidean algorithm.

Same as

General positionExample

are solutions of this system.

General positionExample

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 s

olving 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 systemsThe original system

Randomly chosen

multiply to each terms

Constructing homotopy systems

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 pathUsing Predictor Corrector Method

Predictor step

Corrector step

Corrector step : Newton Method

Predictor step : tangent of path (increase of t)

Tracing homotopy pathTaylor 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 3 Path 5

Independent!

Parallel Computing Client and server model.

Client

Server 1

Server 2

Server 3

Server 4

Master problem

sub problem

sub problem

sub problem

sub problem

sub problem

PHoM (Polyhedral Homotopy Continuation Methods)

Single CPU version

OS : Linux (gcc)

http://www.is.titech.ac.jp/~kojima/PHoM/

Numerical examples

Isolated solutions

Linear Homotopy : the number of tracing path is 4.

Polyhedral Homotopy : the number of tracing path is 2.

Numerical examplesCyclic_n problem.

Numerical examplesproblem Num. Time

cyc_10 34,940 5mins

cyc_11 184,756 30mins

cyc_12 367,488 4hours

cyc_13 2,704,156 15hours

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)-irreducib

le representations. is an algebraic variety over C Problem: Determine in

)(1 M C)SL(2,:

Figure-eight knot case

vuuvw 11

vwwuvu |, Fundamental group of an

exterior of figure-eight knot 2 generators and 1 relation

meridian u and longitude l.

1111 vuvuuvuvl

Irreducible representation Consider an irreducible

representation into SL(2,C)

Write images as follows

C)SL(2, :

)(),(),( lLvVuU

Corresponding matrices

42

10

142

1

2

2

xx

xxU

42

1

042

1

2

2

xxy

xxV

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

Representation space From the relation wu=vw in the group, we obta

in the following polynomial.

0x-5yx5y-yy)f(x, 222

Dehn surgery along a knot Put a relation in the fundamental group. L is a corresponding matrix of a longitude l.

1111 vuvuuvuvl

•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.

3224224

)1(2)(

yxyxyx

xM

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

0x-5yx5y-yy)f(x, 222

0

24822),( 23323

xxyxyxyyxyxyxg

Example 1 : (p,q)=(1,1)Re(x) Im(x) Re(y) Im(y) R-torsion

0.55495 0 3.2469 0 0.615894

-0.80193 0 1.5549 0 1.28627

2.2469 0 0.19806 0 10.1011

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

0x-5yx5y-yy)f(x, 222

0

2816

42168

82),(

2

33432553

63455547

xxyxy

xyyxyxyxyx

yxyxyxyxyxg

Example 2 : (p,q)=(1,2)Re(x) Im(x) Re(y) Im(y) R-torsion

-2.13472 -0.02096 0.276827 0.587152 2.98851+0.563057i

-2.13472 0.020964 0.276827 -0.58715 2.98851-0.563057i

0.953386 0 1.74036 0 0.0250711

0.31267 0 3.50266 0 2.86831

-0.84722 0 2.69078 0 1.20196

2.23869 0 0.102616 0 42.1263

-0.38809 0 1.40993 0 3.80094