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Applications of Polyhedral Homotopy Continuation M
ethods to Topology.
Takayuki Gunji (Tokyo Inst. of Tech.)
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Contents
Polyhedral Homotopy Continuation Methods
Numerical examples Applications to Topology
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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.
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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
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Isolated solutions.
are isolated solutions4
y
x
3
2
1
O 1 2 3 4
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Isolated solutions.
aren’t isolated solutions
4
y
x
3
2
1
O 1 2 3 4
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The number of solutions.Cyclic_n problem.
N Num. N Num
10 34,940 12 367,488
11 184,756 13 2,704,156
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Homotopy continuation method.The original system
Step 2 Solving
Step 1 Constructing homotopy systems such that
and that can be solved easily
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Homotopy continuation method.Step 3 Tracing homotopy paths.
Solutions of the original system
Solutions of
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Linear homotopy
Can be solved easily!
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Polyhedral homotopy
Binomial system Can be solved by Euclidean algorithm.
Same as
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General positionExample
are solutions of this system.
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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
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General position
Step 2: Find solutions of P(x)=0 by using this system
Step 1 : P’(x)=0 solves by using polyhedral homotopy
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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)
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Polynomial system
Constructing homotopy systems
Solving binomial systems
Tracing homotopy paths
Verifying solutions
All isolated solutions
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Constructing homotopy systemsThe original system
Randomly chosen
multiply to each terms
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Constructing homotopy systems
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Constructing homotopy systems
Divided by
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Constructing homotopy systems
Ex
Find all satisfying the property that.Each equation, exactly 2 of power of t are 0.
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Constructing homotopy systems
Find all satisfying the property that.Each equation, exactly 2 of power of t are 0.
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Constructing homotopy systems
All of solutions.
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Tracing homotopy pathUsing Predictor Corrector Method
Predictor step
Corrector step
Corrector step : Newton Method
Predictor step : tangent of path (increase of t)
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Tracing homotopy pathTaylor series
Corrector step
Predictor step
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Polynomial system
Constructing homotopy systems
Solving binomial systems
Tracing homotopy paths
Verifying solutions
All isolated solutions
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Parallel Computing
Path 1 Path 2
Path 4
Path 3 Path 5
Independent!
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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
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PHoM (Polyhedral Homotopy Continuation Methods)
Single CPU version
OS : Linux (gcc)
http://www.is.titech.ac.jp/~kojima/PHoM/
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Numerical examples
Isolated solutions
Linear Homotopy : the number of tracing path is 4.
Polyhedral Homotopy : the number of tracing path is 2.
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Numerical examplesCyclic_n problem.
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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
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Some applications to Topology Representation space of a fundamental group in SL(2,
C). Computation of Reidemeister torsion
Joint works with Teruaki Kitano.
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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,:
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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
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Irreducible representation Consider an irreducible
representation into SL(2,C)
Write images as follows
C)SL(2, :
)(),(),( lLvVuU
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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
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Representation space From the relation wu=vw in the group, we obta
in the following polynomial.
0x-5yx5y-yy)f(x, 222
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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.
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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.
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Reidemeister torsion Reidemeister torsion is a topological invariant of
3-manifolds with a representation parameterized by x and y.
3224224
)1(2)(
yxyxyx
xM
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Example 1 : (p,q)=(1,1)
0x-5yx5y-yy)f(x, 222
0
24822),( 23323
xxyxyxyyxyxyxg
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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
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Example 2 : (p,q)=(1,2)
0x-5yx5y-yy)f(x, 222
0
2816
42168
82),(
2
33432553
63455547
xxyxy
xyyxyxyxyx
yxyxyxyxyxg
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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