an exact toric resultant-based rur approach for solving polynomial systems
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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
Texas A&M University AMS2004 3RUR
Outline
Rational Univariate Reduction (RUR) Complexity Analysis Exact RUR Comparison with Other Work Conclusion / Future Work
Texas A&M University AMS2004 4RUR
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
Texas A&M University AMS2004 5RUR
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
Texas A&M University AMS2004 6RUR
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
*)
Texas A&M University AMS2004 7RUR
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]
Texas A&M University AMS2004 8RUR
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)
Texas A&M University AMS2004 9RUR
Computing RUR Step1: Compute Mixed Volumes Step2: Construct a Resultant Matrix Step3: Compute h Step4: Compute h1, …, hn
Texas A&M University AMS2004 10RUR
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
Texas A&M University AMS2004 11RUR
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
1. Mixed Volumes2. Resultant Matrix3. h4. h1, …, hn
Texas A&M University AMS2004 12RUR
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
1. Mixed Volumes2. Resultant Matrix3. h4. h1, …, hn
Texas A&M University AMS2004 13RUR
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
1. Mixed Volumes2. Resultant Matrix3. h4. h1, …, hn
Texas A&M University AMS2004 14RUR
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
1. Mixed Volumes2. Resultant Matrix3. h4. h1, …, hn
Texas A&M University AMS2004 15RUR
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]
1. Mixed Volumes2. Resultant Matrix3. h4. h1, …, hn
Texas A&M University AMS2004 16RUR
Computing RUR All the steps can be implemented exactly
The coefficients of h, h1, …, hn can be computedin full digits
Texas A&M University AMS2004 17RUR
Outline
Rational Univariate Reduction (RUR) Complexity Analysis Exact RUR Comparison with Other Work Conclusion / Future Work
Texas A&M University AMS2004 18RUR
Complexity Analysis
Notation O˜( ) the polylog factor is ignored
Gaussian elimination ofm dimensional matrix requiresO(m) operations
Texas A&M University AMS2004 19RUR
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)
Texas A&M University AMS2004 20RUR
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
1+)
Texas A&M University AMS2004 21RUR
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
Texas A&M University AMS2004 22RUR
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]
Texas A&M University AMS2004 23RUR
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
1+
arithmetic operations
Texas A&M University AMS2004 24RUR
Outline
Rational Univariate Reduction (RUR) Complexity Analysis Exact RUR Comparison with Other Work Conclusion / Future Work
Texas A&M University AMS2004 25RUR
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
Texas A&M University AMS2004 26RUR
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
Texas A&M University AMS2004 27RUR
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
Texas A&M University AMS2004 28RUR
Outline
Rational Univariate Reduction (RUR) Complexity Analysis Exact RUR Comparison with Other Work Conclusion / Future Work
Texas A&M University AMS2004 29RUR
The Other RUR
GB+RS [Rouillier 99, 04]
Kronecker / Newton [Giusti, Lecerf and Salvy 01] [Jeronimo, Krick, Sabia and Sombra 04]
Texas A&M University AMS2004 30RUR
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]
Texas A&M University AMS2004 31RUR
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
Texas A&M University AMS2004 32RUR
Outline
Rational Univariate Reduction (RUR) Complexity Analysis Exact RUR Comparison with Other Work Conclusion / Future Work
Texas A&M University AMS2004 33RUR
Implementation
ERUR Algorithms adapt to exact implementation naturally
Strong for handling degeneracies
Need more optimizations and faster algorithms
Texas A&M University AMS2004 34RUR
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
Texas A&M University AMS2004 35RUR
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
Texas A&M University AMS2004 36RUR
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
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