an exact toric resultant-based rur approach for solving polynomial systems

An Exact Toric Resultant-BasedRUR Approach

for Solving Polynomial Systems

Koji Ouchi, John Keyser, J. Maurice Rojas

Department of Computer Science, Mathematics

Texas A&M University

AMS Meeting 2004

Texas A&M University AMS2004 2RUR

Outline

Rational Univariate Reduction (RUR) Complexity Analysis Exact RUR Comparison with Other Work Conclusion / Future Work

Outline

Rational Univariate Reduction

Problem: Solve a system of n polynomials f1, …, fn

in n variables X1, …, Xn

with coefficients in ℚℚ

Reduce the system to

n + 1 univariate polynomials h, h1, …, hn

with coefficients in ℚℚ s.t.

if is a root of h then

(h1(), …, hn()) is a solution to the system

Notation u = (u0, u1,…, un) indeterminates f0 = u0 + u1 X1 + … + un Xn

Ai = Supp(fi), i = 0, 1,…, n ∴ A0 = {o, e1, …, en}

ei the i-th standard basis vector

RUR via Toric Resultant

Toric Perturbation

Toric Generalized Characteristic PolynomialLet f1

*, …, fn* be n polynomials

in n variables X1, …, Xn

with coefficients in ℚ ℚ andSupp(fi

*) ⊆ Ai = Supp(fi ), i = 0, 1,…, nthat have only finitely many solutions in (ℂℂ \ {0})n

DefineTGCP(u, Y ) =

Res (A0, A1, …, An) (f0, f1 - Y f1*, …, fn - Y fn

Toric Perturbation

Toric Perturbation [Rojas 99]Define Pert(u) to be

the non-zero coefficient of the lowest degree term(in Y ) of TGCP(u, Y )

Pert(u) is well-defined A version of “perturbations” [D’Andrea and Emiris

01, 03]

Toric Perturbation

Toric Perturbation If (1, …, n) (ℂℂ \ {0})n is an isolated root of

the input system f1, …, fn then u0 + u1 1 + … + un n Pert(u)

Pert(u) completely splits into linear factorsover ℂℂ

For every irreducible component of the zero setof the input system, there is at least one factor ofPert(u)

Computing RUR Step1: Compute Mixed Volumes Step2: Construct a Resultant Matrix Step3: Compute h Step4: Compute h1, …, hn

Computing RUR Step 1: Compute Mixed Volumes

Use Emiris’s algorithm [Emiris and Canny 95, 01] to compute

MV–i = MV(A0, A1, …, Ai-1, Ai+1, …, An), i = 0, 1, …, n

Use Linear Programming #P on Turing machine

1. Mixed Volumes2. Resultant Matrix3. h4. h1, …, hn

Computing RUR Step 2: Construct a Resultant Matrix

Use Emiris’ algorithm [Emiris and Canny 95]to construct a matrixwhose maximal minor is some multiple ofthe toric resultant

Rows and columns are labeledby the exponents in A0, A1, …, An

Increment rows and columns until non-vanishing maximal minor is found

Computing RUR Step 2: Construct a Resultant Matrix (Cont.)

[Pederson and Sturmfels 93]deg fi Res (A0, A1, …, An) (f0, f1, …, fn) = MV-i , i = 0, 1,…, n

Computing RUR

Step 2: Construct a Resultant Matrix (Cont.) Degeneracies have been removed by perturbation

The size of matrices must be at least Σ i = 0, 1,…, n MV-i

# of rows labeled by the exponents in Ai ≧ MV-i , i = 0, 1, …, n

# of rows labeled by the exponents in A0 = MV-0

∴ deg f0 D = MV-0

where D is the maximal minor

Computing RUR

Step 3: Compute h (T)

h(T) = Pert(T, u1, …, un) for some values of u1, …, un

Assign values to u1, …, un Evaluate Pert(u0, u1, …, un)

at deg h(T) = MV-0 distinct values of u0 andinterpolate them

Computing RUR

Step 4: Compute h1 (T), …, hn (T)

Computation of every hi involves Evaluating Pert(u) and interpolate them Univariate polynomial operations

Euclidean algorithm for GCD First subresultant [Gonzalez-Vega 91]

Computing RUR All the steps can be implemented exactly

The coefficients of h, h1, …, hn can be computedin full digits

Outline

Complexity Analysis

Notation O˜( ) the polylog factor is ignored

Gaussian elimination ofm dimensional matrix requiresO(m) operations

Complexity Analysis Quantities

MA MV-0 = deg h(T)

RA i = 0, 1,…, n MV-i

The size of the optimal resultant matrix

SA The size of maximal minor SA = (n1/2 en RA)

Complexity Analysis

# of Arithmetic Operations Evaluate Res (A0, A1, …, An) O˜(SA

1+) Evaluate Pert (u) O˜(SA

1+) Compute h O˜(MA SA

1+) Compute every hi O˜(MA SA

1+) Compute RUR

for fixed u1, …, un O˜(n MA SA1+)

Compute RUR O˜(n3 MA3 SA

Complexity Analysis

Bit Complexity The logarithmic height of h, h1, …, hn is

some polynomial in SA [Rojas 00] RA [Sombra]

The bit complexity is single exponential in n

Complexity Analysis A great speed up is achieved

if we could compute “small” matrixwhose determinant is the resultant No such method is known

Resultant matrices Sylvester-Dixon [Chtcherba and Kapur] Corner-cutting [Goldman and Zhang 00] Tate resolution [Khetan 03, 04]

Khetan’s Method Khetan’s method gives a matrix

whose determinant is the resultantof unmixed systems when n = 2 or 3 (or bigger?)[Khetan 03, 04]

Let B = A0 A1 An

Then, computing RUR requires

n3 MA3 RB

arithmetic operations

Outline

ERUR Non square system is converted to

some square system

Solutions in ℂℂn are computedby adding the origin o to supports

In both cases,post processing requires exact computationover the points in RUR

Exact Sign Given an expression e, tell whether or not

e(h1(), …, hn()) = 0

Use (extended) root bound approach Use Aberth’s method [Aberth 73] to

numerically compute an approximation fora root of hto any precision

Applications by ERUR

Real Root Given a system of polynomial equations,

list all the real roots of the system

Positive Dimensional Component Given a system of polynomial equations,

tell whether or not the zero set of the systemhas a positive dimensional component

Outline

The Other RUR

GB+RS [Rouillier 99, 04]

Kronecker / Newton [Giusti, Lecerf and Salvy 01] [Jeronimo, Krick, Sabia and Sombra 04]

The Other RUR

GB+RS [Rouillier 99, 04] Compute the exact RUR for real solutions

of a 0-dimensional system

GB computes the Gröbner basis The Gröbner basis computation is

EXPSPACE-complete (double exponential in n)on Turing machine [Mayr and Meyer 98]

The Other RUR

Kronecker / Newton [Giusti, Lecerf and Salvy 01]

Kronecker in Magma

[Jeronimo, Krick, Sabia and Sombra 04] BPP on BSS machine over ℚℚ

Outline

Implementation

ERUR Algorithms adapt to exact implementation naturally

Strong for handling degeneracies

Need more optimizations and faster algorithms

Conclusion

Deterministic algorithm

Handle degeneracies by perturbation The total degree of Pert(u) is RA

Use the incremental matrix construction algorithm Currently, the most efficient Starting at a matrix of size RA

Exponential factor appearing in the complexitycomes from the size of the resultant matrix

Future Work Faster toric resultant algorithms

Smaller resultant matrices

Take advantages of sparseness of matrices[Emiris and Pan 97]

Faster univariate polynomial operations Use rational functions for h1,…, hn

Thank you for listening!

Contact Koji Ouchi, kouchi@cs.tamu.edu John Keyser, keyser@cs.tamu.edu Maurice Rojas, rojas@math.tamu.edu

Visit Our Web http://research.cs.tamu.edu/keyser/geom/ERUR/

Thank you

an exact toric resultant-based rur approach for solving polynomial systems

hncreating use casestexas

systemcreating use casestexas

f1 y f1

fn y fn

resultant matrix3

system of n polynomials

n pertupertu

resultant matrixstep3

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