Computing the Rational Univariate Reduction by Sparse Resultants

1. Computingthe Rational Univariate Reductionby Sparse Resultants Koji Ouchi, John Keyser, J. Maurice Rojas Department of Computer Science, Mathematics Texas A&M University ACA 2004

2. Outline • What is Rational Univariate Reduction? • Computing RUR by Sparse Resultants • Complexity Analysis • Exact Implementation

3. 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 q is a root of h then (h1(q), …, hn(q)) is a solution to the system

4. 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

5. Toric Perturbation • Toric Generalized Characteristic Polynomial Let f1*, …, fn* be n polynomials in n variables X1, …, Xn with coefficients in K and Supp(fi*) Ai =Supp(fi) , i = 1, …, n that have only finitely many solutions in (\ {0})n Define TGCP(u, Y) = Res(, A1, …, An)(aua Xa, f1 - Y f1*, …, fn - Y fn*)

6. 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]

7. Toric Perturbation • Toric Perturbation • If (1, …, n)  (\ {0})n is an isolated root of the input system f1, …, fn then aua a Pert(u) • Pert(u) completely splits into linear factors over (\ {0})n. For every irreducible component of the zero set of the input system, there is at least one factor of Pert(u)

8. Computing RUR • Step 1: • Compute Pert(u) • Use Emiris’ sparse resultant algorithm [Canny and Emiris 93, 95, 00] to construct Newton matrix whose determinant is some multiple of the resultant • Evaluate resultant with distinct u and interpolate them

9. Computing RUR • Step 2: • 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

10. Computing RUR • Step 3: • Compute h1 (T), …, hn (T) • Computation of hiinvolves - Evaluating Pert(u), - Interpolate them, and - Some univariate polynomial operations

11. Complexity Analysis • Count the number of arithmetic operations • Notation • O˜( )the polylog factor is ignored • Gaussian eliminationof • m dimensional matrix requiresO(m)

12. Complexity Analysis • Quantities • MAThe mixed volume MV(A1 , …, An) of the convex hull ofA1 , …, An • RAMV(A1, …, An) +i = 1,…,n MV(, A1, …, Ai-1, Ai+1, …, An) • The total degree of the sparse resultant • SAThe dimension of Newton matrix • Possibly exponentially bigger than RA

13. Complexity Analysis • [Emiris and Canny 95] • Evaluating Res (, A1, …, An)(aua Xa, f1, …, fn) requires O˜(n RASA) orO˜(SA1+)if char K = 0

14. Complexity Analysis • [Rojas 99] • Evaluating Pert (u) requires O˜(n RA2 SA) or O˜(SA1+)if char K = 0

15. Complexity Analysis • Computing h (T) requires O˜(n MA RA2 SA) or O˜(MA SA1+)if char K = 0

16. Complexity Analysis • Computing every hi(T) requires O˜(n MA RA2 SA) or O˜(MA SA1+)if char K = 0

17. Complexity Analysis • Computing RUR h (T),h1(T), …, hn(T) for fixed u1, …, un requires O˜(n2 MA RA2 SA) or O˜(nMA SA1+)if char K = 0

18. Complexity Analysis • Derandomize the choice of u1, …, un • Computing RUR h (T),h1(T), …, hn(T) requires O˜(n4 MA3 RA2 SA) orO˜(n3MA3 SA1+)if char K = 0

19. Complexity Analysis

20. Complexity Analysis • A great speed up is achieved if we could compute “small” Newton matrix whose determinant is the resultant  No such method is known

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

22. 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

23. ERUR • Non square system is converted to some square system • Solutions in ()n are computed by adding the origin o to supports.

24. 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.

25. 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

26. Applications by ERUR • Presented today’s last talk in Session 3 “Applying Computer Algebra Techniques for Exact Boundary Evaluation” 4:30 – 5:00 pm

27. 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

28. Conclusion • ERUR • Strong for handling degeneracies • Need more optimizations and faster algorithms

29. Future Work • RUR • Faster sparse resultant algorithms • Take advantages of sparseness of matrices [Emiris and Pan 97] • Faster univariate polynomial operations

30. 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