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
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