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An Exact Toric Resultant-Based RUR Approach for Solving Polynomial Systems

An Exact Toric Resultant-Based RUR Approach for Solving Polynomial Systems. Koji Ouchi, John Keyser, J. Maurice Rojas Department of Computer Science, Mathematics Texas A&M University AMS Meeting 2004. Outline. Rational Univariate Reduction (RUR) Complexity Analysis Exact RUR

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An Exact Toric Resultant-Based RUR Approach for Solving Polynomial Systems

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  1. An Exact Toric Resultant-BasedRUR Approachfor Solving Polynomial Systems Koji Ouchi, John Keyser, J. Maurice Rojas Department of Computer Science, Mathematics Texas A&M University AMS Meeting 2004

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

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

  4. Rational Univariate Reduction • Problem: Solve a system of n polynomials f1, …, fn in n variables X1, …, Xn with coefficients in ℚ • Reduce the system to n + 1 univariate polynomials h, h1, …, hn with coefficients in ℚ s.t. if q is a root of h then (h1(q), …, hn(q)) is a solution to the system

  5. RUR via Toric Resultant • Notation • u = (u0, u1,…, un)indeterminates • f0 =u0 + u1X1+ … + unXn • Ai = Supp(fi), i = 0, 1,…, n ∴ A0= {o, e1, …, en} eithe i-th standard basis vector

  6. Toric Perturbation • Toric Generalized Characteristic Polynomial Let f1*, …, fn*be n polynomials in n variables X1, …, Xn with coefficients in ℚ and Supp(fi*) ⊆Ai= Supp(fi ), i = 0, 1,…, n that have only finitely many solutions in (ℂ\ {0})n Define TGCP(u, Y ) = Res(A0, A1, …, An)(f0, f1 - Y f1*, …, fn - Y fn*)

  7. 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]

  8. Toric Perturbation • Toric Perturbation • If (1, …, n)  (ℂ\ {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 factors over ℂ • For every irreducible component of the zero set of the input system, there is at least one factor of Pert(u)

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

  10. 1. Mixed Volumes 2. Resultant Matrix 3. h 4. 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

  11. 1. Mixed Volumes 2. Resultant Matrix 3. h 4. h1, …, hn Computing RUR • Step 2: Construct a Resultant Matrix Use Emiris’ algorithm [Emiris and Canny 95] to construct a matrix whose maximal minor is some multiple of the toric resultant • Rows and columns are labeled by the exponents in A0, A1, …, An • Increment rows and columns until non-vanishing maximal minor is found

  12. 1. Mixed Volumes 2. Resultant Matrix 3. h 4. h1, …, hn 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

  13. 1. Mixed Volumes 2. Resultant Matrix 3. h 4. h1, …, hn 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

  14. 1. Mixed Volumes 2. Resultant Matrix 3. h 4. h1, …, hn 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 and interpolate them

  15. 1. Mixed Volumes 2. Resultant Matrix 3. h 4. h1, …, hn Computing RUR • Step 4: Compute h1(T), …, hn(T) Computation of every hiinvolves • Evaluating Pert(u) and interpolate them • Univariate polynomial operations • Euclidean algorithm for GCD • First subresultant [Gonzalez-Vega 91]

  16. Computing RUR • All the steps can be implemented exactly • The coefficients of h, h1, …, hn can be computed in full digits

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

  18. Complexity Analysis • Notation • O˜( )the polylog factor is ignored • Gaussian eliminationof • m dimensional matrix requires • O(m) operations

  19. Complexity Analysis • Quantities • MAMV-0 = deg h(T) • RA i = 0, 1,…, nMV-i • The size of the optimal resultant matrix • SAThe size of maximal minor • SA = (n1/2en RA)

  20. Complexity Analysis • # of Arithmetic Operations • Evaluate Res(A0, A1, …, An)O˜(SA1+) • Evaluate Pert (u) O˜(SA1+) • Compute hO˜(MA SA1+) • Compute every hiO˜(MA SA1+) • Compute RUR for fixed u1, …, unO˜(nMA SA1+) • Compute RUR O˜(n3MA3 SA1+)

  21. Complexity Analysis • Bit Complexity • The logarithmic height of h, h1, …, hn is some polynomial inSA [Rojas 00] RA [Sombra] • The bit complexity is single exponential in n

  22. Complexity Analysis • A great speed up is achieved if we could compute “small” matrix whose 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]

  23. Khetan’s Method • Khetan’s method gives a matrix whose determinant is the resultant of unmixed systems when n = 2 or 3 (orbigger?) [Khetan 03, 04] • Let B = A0 A1   An Then, computing RUR requires n3 MA3 RB1+ arithmetic operations

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

  25. ERUR • Non square system is converted to some square system • Solutions in ℂn are computed by adding the origin o to supports • In both cases, post processing requires exact computation over the points in RUR

  26. ERUR • Exact Sign • Given an expression e, tell whether or not e(h1(q), …, hn(q)) = 0 • Use (extended) root bound approach • Use Aberth’s method [Aberth 73] to numerically compute an approximation for a root of h to any precision

  27. 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 system has a positive dimensional component

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

  29. The Other RUR • GB+RS • [Rouillier 99, 04] • Kronecker / Newton • [Giusti, Lecerf and Salvy 01] • [Jeronimo, Krick, Sabia and Sombra 04]

  30. 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]

  31. The Other RUR • Kronecker / Newton • [Giusti, Lecerf and Salvy 01] • Kronecker in Magma • [Jeronimo, Krick, Sabia and Sombra 04] • BPP on BSS machine over ℚ

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

  33. Implementation • ERUR • Algorithms adapt to exact implementation naturally • Strong for handling degeneracies • Need more optimizations and faster algorithms

  34. 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 complexity comes from the size of the resultant matrix

  35. 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

  36. 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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