Computing the Rational Univariate Reduction by Sparse Resultants

Computingthe Rational Univariate

Reductionby Sparse Resultants

Koji Ouchi, John Keyser, J. Maurice Rojas

Department of Computer Science, Mathematics

Texas A&M University

ACA 2004

Texas A&M University ACA2004 2RUR

Outline

What is Rational Univariate Reduction? Computing RUR by Sparse Resultants Complexity Analysis Exact Implementation


Rational Univariate Reduction

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

in n variables X1, …, Xn

with coefficients in the field K

Reduce the system to

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

with coefficients in K s.t.

if is a root of h then

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


RUR via Sparse Resultant Notation

ei the i-th standard basis vector = {o, e1, …,en} u0, u1,…, un indeterminates Ai = Supp(fi) the algebraic closure of K


Toric Perturbation

Toric Generalized Characteristic PolynomialLet f1*, …, fn* be n polynomials in n variables X1, …, Xn

with coefficients in K andSupp(fi*) Ai =Supp(fi) , i = 1, …, n

that have only finitely many solutions in ( \ {0})n

DefineTGCP(u, Y) =

Res (, A1, …, An) (a ua Xa, 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 “projective operator technique”

[Rojas 98, D’Andrea and Emiris 03]


Toric Perturbation Toric Perturbation

If (1, …, n) ( \ {0})n is an isolated root ofthe input system f1, …, fn then

a ua a Pert(u)

Pert(u) completely splits into linear factorsover ( \ {0})n. For every irreducible component of the zero setof the input system, there is at least one factor ofPert(u)


Computing RUR Step 1:

Compute Pert(u) Use Emiris’ sparse resultant algorithm [Canny and

Emiris 93, 95, 00] to construct Newton matrixwhose determinant is some multiple of the resultant

Evaluate resultant with distinct u and interpolatethem



Compute h(T) Set h(T) = Pert(T, u1, …, un)

for some values of u1, …, un

Evaluate Pert(u) with distinct u0 and interpolate them



Compute h1 (T), …, hn (T) Computation of hi involves

- Evaluating Pert(u), - Interpolate them, and - Some univariate polynomial operations


Complexity Analysis

Count the number of arithmetic operations

Notation O˜( ) the polylog factor is ignored

Gaussian elimination ofm dimensional matrix requires O(m)


Complexity Analysis Quantities

MA The mixed volume MV(A1 , …, An)of the convex hull of A1 , …, An

RA MV(A1, …, An)+ i = 1,…,n MV(, A1, …, Ai-1, Ai+1, …, An)

The total degree of the sparse resultant

SA The dimension of Newton matrix Possibly exponentially bigger than RA


Complexity Analysis

[Emiris and Canny 95] Evaluating

Res (, A1, …, An) (a ua Xa, f1, …, fn)requires

O˜(n RA SA

)

or O˜(SA1+) if char K = 0


Complexity Analysis

[Rojas 99] Evaluating Pert (u) requires

O˜(n RA2 SA

)

or O˜(SA1+) if char K = 0


Complexity Analysis Computing h (T) requires

O˜(n MA RA2 SA

)

or O˜(MA SA1+) if char K = 0


Complexity Analysis Computing every hi (T) requires

O˜(n MA RA2 SA

)

or O˜(MA SA1+) if char K = 0


Complexity Analysis Computing RUR

h (T), h1 (T), …, hn (T)for fixed u1, …, un requires

O˜(n2 MA RA2 SA

)

or O˜(n MA SA1+) if char K = 0


Complexity Analysis

Derandomize the choice of u1, …, un Computing RUR

h (T), h1 (T), …, hn (T)requires

O˜(n4 MA3 RA

2 SA)

or O˜(n3 MA3 SA

1+) if char K = 0


Complexity Analysis

Emiris Division Emiris GCD

char K = 0

“Small”

Newton MatrixEvaluating Res

n RA SA SA

1+ RA

Evaluating Pert

n RA2 SA

SA1+ RA

1+

RUR

for fixed u

n2 MA RA2 SA

n MA SA1+ n MA RA

1+

RUR n4 MA3 RA

2 SA n3 MA

3 SA1+ n3 MA

3 RA1+


Complexity Analysis A great speed up is achieved

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


Khetan’s Method Khetan’s method gives Newton matrix

whose determinant is the resultantof unmixed systems when n = 2 or 3[Kehtan 03, 04]

Let B = A1 An

Then, computing RUR requires

n3 MA3 RB

1+

arithmetic operations


ERUR: Implementation

Current implementation The coefficients are rational numbers Use the sparse resultant algorithm [Emiris and

Canny 93, 95, 00] to construct Newton matrix All the coefficients of RUR h, h1,…, hn are exact


ERUR Non square system is converted to

some square system

Solutions in ( )n are computedby adding the origin o to supports.


ERUR

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.

Im1Re


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


Applications by ERUR Presented today’s last talk in Session 3

“ApplyingComputer Algebra Techniques

forExact Boundary Evaluation”

4:30 – 5:00 pm


The Other RUR

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

of a 0-dimensional system GB computes the Gröebner basis

[Giusti, Lecerf and Salvy01] An iterative method


Conclusion

ERUR Strong for handling degeneracies

Need more optimizations and faster algorithms


Future Work

RUR Faster sparse resultant algorithms Take advantages of sparseness of matrices

[Emiris and Pan 97] Faster univariate polynomial operations


Thank you for listening!

Contact Koji Ouchi, [email protected] John Keyser, [email protected] Maurice Rojas, [email protected]

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

Thank you

Computing the Rational Univariate Reduction by Sparse Resultants

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