Ruppert matrix as subresultant mapping
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1 Ruppert matrix as subresultant mapping Kosaku Nagasaka Kobe University JAPAN This presentation is powered by Mathematica : 29 : Ruppert matrix as subresultant mapping Prev Next
2 2 CASC2007slideshow.nb Outline of this talk How to compute polynomial GCDs Sylvester matrix and Subresultant mapping Ruppert matrix and factoring polynomials Ruppert matrix for GCD of two polynomials Ruppert matrix for GCD of several polynomials Conclusion 09 : 29 : Ruppert matrix as subresultant mapping Prev Next
3 CASC2007slideshow.nb 3 How to compute polynomial GCDs Euclidean Algorithm Eculidean algorithm for polynomials f x and gx over reals. function GCD(f(x), g(x)) if g(x) = 0 return f(x) else return GCD( g(x), remainder of f(x) by g(x) ) QR factoring of Sylvester matrix The usual subresultant mapping is important. See the next slide... Lots of other powerful approximate algorithms 09 : 29 : Ruppert matrix as subresultant mapping Prev Next
4 4 CASC2007slideshow.nb QR factoring of Sylvester matrix Sylvester matrix of the following f x and gx : f x f n x n f n1 x n1 f 1 x f 0, gx g m x m g m1 x m1 g 1 x g 0. f n f n1 f 1 f f n f n1 f 1 f 0 0 Syl f, g 0 0 f n f n1 f 1 f 0 g m g m1 g 1 g g m g m1 g 1 g g m g m1 g 1 g 0 Well known relation between GCD and QR factoring of Sylvester matrix. The last non-zero row vector of R where Syl f, g Q R, is the coefficient vector of GCD of f x and gx. (See: Laidacker, M.A., 1969 and others) 09 : 29 : Ruppert matrix as subresultant mapping Prev Next.
5 CASC2007slideshow.nb 5 Subresultant mapping Subresultant mapping S r of f 0 x and f 1 x S r : n 1 r1 n0 r1 n0 n 1 r1 u 0, u 1 u 1 f 0 u 0 f 1 where f 0 x f 0,n0 x n 0 f0,1 x f 0,0 f 1 x f 1,n1 x n 1 f1,1 x f 1,0 Sylvester subresultant matrix S r f 0, f 1 S r f 0, f 1 C n0 r f 1 C n1 r f 0 where C k p is a k-th convolution matrix of px and S 0 f 0, f 1 is the transpose of Sylvester matrix of f 0 x and f 1 x. 09 : 29 : Ruppert matrix as subresultant mapping Prev Next
6 6 CASC2007slideshow.nb Convolution matrix k-th convolution matrix of polynomial px. p n p n1 p n p n1 0 C k p p 0 p n 0 0 p 0 p n1 p n 0 p n1 p p 0 where px p n x n p 1 x p 0. Subresultant mapping and Sylvester matrix n1 r1 n0 r1 n0 n 1 r1 S r : u 0, u 1 u 1 f 0 u 0 f 1 S r f 0, f 1 C n0 r f 1 C n1 r f 0, S 0 f 0, f 1 t Syl f 0, f 1 09 : 29 : Ruppert matrix as subresultant mapping Prev Next
7 CASC2007slideshow.nb 7 Subresultant mapping and GCD Well known fact (see Rupprecht, D. (1999) and so on) (1) If r is the largest integer that the subresultant mapping S r is not injective, we can compute the GCD of f 0 x and f 1 x from the right null vector of S r f 0, f 1. (2) The dimension of the null space is the degree of the polynomial GCD. 09 : 29 : Ruppert matrix as subresultant mapping Prev Next
8 8 CASC2007slideshow.nb Ruppert criterion Reducibility of bivariate polynomials (Ruppert, W.M., 1999) f x, y is absolutely irreducible the following differential equation does not have any non-trivial solutions gx, y and hx, y. g f f h f y g y h x f x 0, g, h x, y with degree constraints: deg x g deg x f 1, deg y g deg y f, deg x h deg x f, deg y h deg y f 2 Newton polytope ver. by Gao, S. and Rodrigues, V.M. (2003) Multivariate version by May, J.P. (2005) 09 : 29 : Ruppert matrix as subresultant mapping Prev Next
9 CASC2007slideshow.nb 9 Ruppert matrix Rf The differential equation w.r.t. gx, y and hx, y a linear equation w.r.t. unknown coefficients. g f y f g y f h x h f x 0, g, h x, y, deg x g deg x f 1, deg y g deg y f, deg x h deg x f, deg y h deg y f 2 R f t coefficient vectors of g, h SVD of Ruppert matrix: irreducibility radius. Ruppert matrix depends term orders: lexicographic in this talk. 09 : 29 : Ruppert matrix as subresultant mapping Prev Next
10 10 CASC2007slideshow.nb Factoring with Ruppert and Gao s matrix Known Algorithms using Corollary 1 Corollary 1 For a given f x, y x, y that is square-free over y, the dimension (over ) of the null space of R f is equal to "(the number of absolutely irreducible factors of f x, y over ) 1". Correction: Corollary 1 from John May s thesis. and others by Kaltofen, Gao and so on. Kaltofen, E., May, J., Yang, Z., and Zhi, L. Approximate factorization of multivariate polynomials using singular value decomposition. Manuscript, 22 pages. Submitted, Jan : 29 : Ruppert matrix as subresultant mapping Prev Next
11 CASC2007slideshow.nb 11 Purpose of this talk GCD and Factorization Primitive operations in Symbolic (and Numeric) computations. Common Properties Can be computed by some matrix decompositions. Sylvester and Ruppert matices Is there any relation between them? 09 : 29 : Ruppert matrix as subresultant mapping Prev Next
12 12 CASC2007slideshow.nb How to link them f x, y f 0 x f 1 xy f x, y is reducible if f 0 x and f 1 x have a non-trivial GCD. f x, y is irreducible if they don t have any non-trivial GCD. Coprimeness of f 0 x and f 1 x can be tested by a rank deficiency of their Sylvester matrix. Irreducibility of f x, y f 0 x f 1 xy can be tested by a rank deficiency of its Ruppert matrix. 09 : 29 : Ruppert matrix as subresultant mapping Prev Next
13 CASC2007slideshow.nb 13 Simple Result Lemma 1 For any polynomials f 0 x and f 1 x, the Sylvester matrix of f 0 x and f 1 x and the Ruppert matrix of f x, y f 0 x f 1 xy have the same information for computing their GCD, with the Ruppert s original differential equation and degree constraints. The proof is in the proceedings. 09 : 29 : Ruppert matrix as subresultant mapping Prev Next
14 14 CASC2007slideshow.nb Simple Result (Sylvester and Ruppert) fa5 fa4 fa3 fa2 fa1 fa fa5 fa4 fa3 fa2 fa1 fa fa5 fa4 fa3 fa2 fa1 fa fa5 fa4 fa3 fa2 fa1 fa fa5 fa4 fa3 fa2 fa1 fa0 fb5 fb4 fb3 fb2 fb1 fb fb5 fb4 fb3 fb2 fb1 fb fb5 fb4 fb3 fb2 fb1 fb fb5 fb4 fb3 fb2 fb1 fb fb5 fb4 fb3 fb2 fb1 fb0 0 fa5 0 fa4 0 fa3 0 fa2 0 fa1 0 fa fb5 0 fb4 0 fb3 0 fb2 0 fb1 0 fb fa5 0 fa4 0 fa3 0 fa2 0 fa1 0 fa fb5 0 fb4 0 fb3 0 fb2 0 fb1 0 fb fa5 0 fa4 0 fa3 0 fa2 0 fa1 0 fa fb5 0 fb4 0 fb3 0 fb2 0 fb1 0 fb fa5 0 fa4 0 fa3 0 fa2 0 fa1 0 fa fb5 0 fb4 0 fb3 0 fb2 0 fb1 0 fb fa5 0 fa4 0 fa3 0 fa2 0 fa1 0 fa fb5 0 fb4 0 fb3 0 fb2 0 fb1 0 fb0 09 : 29 : Ruppert matrix as subresultant mapping Prev Next
15 CASC2007slideshow.nb 15 Alternative Result 1 Theorem 1 The polynomial GCD of f 0 x and f 1 x can be computed by Singular Value Decomposition (SVD) of Ruppert matrix of f x, y f 0 x f 1 xy with the J.P.May s differential equation and degree constraints, if f x, y is square-free over y. Difference between Ruppert and May s f g y f h x h f x 0, g deg x f 1, deg y g deg y f, h deg x f, deg y h deg y f 2 The proof is in the proceedings. 09 : 29 : Ruppert matrix as subresultant mapping Prev Next
16 16 CASC2007slideshow.nb Alternative Result 1(Ruppert with May) fb4 fb fb3 0 2 fb fb2 fb3 fb4 3 fb fb1 2 fb2 0 2 fb4 4 fb fb0 3 fb1 fb2 fb3 3 fb4 5 fb fb0 2 fb1 0 2 fb3 4 fb fb0 fb1 fb2 3 fb fb0 0 2 fb fb0 fb fa4 fa fa fb fa3 0 2 fa fa4 fa5 0 0 fb4 fb fa2 fa3 fa4 3 fa5 0 0 fa3 fa4 fa5 0 fb3 fb4 fb5 0 4 fa1 2 fa2 0 2 fa4 4 fa5 0 fa2 fa3 fa4 fa5 fb2 fb3 fb4 fb5 5 fa0 3 fa1 fa2 fa3 3 fa4 5 fa5 fa1 fa2 fa3 fa4 fb1 fb2 fb3 fb4 0 4 fa0 2 fa1 0 2 fa3 4 fa4 fa0 fa1 fa2 fa3 fb0 fb1 fb2 fb fa0 fa1 fa2 3 fa3 0 fa0 fa1 fa2 0 fb0 fb1 fb fa0 0 2 fa2 0 0 fa0 fa1 0 0 fb0 fb fa0 fa fa fb0 09 : 29 : Ruppert matrix as subresultant mapping Prev Next
17 CASC2007slideshow.nb 17 Alternative Result 2 Theorem 2 The polynomial GCD of f 0 x and f 1 x can be computed by QR factoring of the transpose of the last 3n 0 rows of their Ruppert matrix R f R f 0 x f 1 xy with J.P.May s equation. The last non-zero row vector of the triangular matrix is the coefficient vector of their polynomial GCD. The proof is in the proceedings. 09 : 29 : Ruppert matrix as subresultant mapping Prev Next
18 18 CASC2007slideshow.nb Alternative Result 2(Last 3n 0 rows) fb4 fb fb3 0 2 fb fb2 fb3 fb4 3 fb fb1 2 fb2 0 2 fb4 4 fb fb0 3 fb1 fb2 fb3 3 fb4 5 fb fb0 2 fb1 0 2 fb3 4 fb fb0 fb1 fb2 3 fb fb0 0 2 fb fb0 fb fa4 fa fa fb fa3 0 2 fa fa4 fa5 0 0 fb4 fb fa2 fa3 fa4 3 fa5 0 0 fa3 fa4 fa5 0 fb3 fb4 fb5 0 4 fa1 2 fa2 0 2 fa4 4 fa5 0 fa2 fa3 fa4 fa5 fb2 fb3 fb4 fb5 5 fa0 3 fa1 fa2 fa3 3 fa4 5 fa5 fa1 fa2 fa3 fa4 fb1 fb2 fb3 fb4 0 4 fa0 2 fa1 0 2 fa3 4 fa4 fa0 fa1 fa2 fa3 fb0 fb1 fb2 fb fa0 fa1 fa2 3 fa3 0 fa0 fa1 fa2 0 fb0 fb1 fb fa0 0 2 fa2 0 0 fa0 fa1 0 0 fb0 fb fa0 fa fa fb0 09 : 29 : Ruppert matrix as subresultant mapping Prev Next
19 CASC2007slideshow.nb 19 Generalized Sylvester matrix Generalized Subresultant mapping S r of f 0 x,, f k x. k k ni r1 n0 n i r1 i 0 i 1 S r : u 0 u 1 f 0 u 0 f 1, n i deg x f i x u k u k f 0 u 0 f k Generalized Sylvester matrix S r f 0,, f k. S r f 0,, f k C n0 r f 1 C n1 r f 0 (see Rupprecht, D. (1999) and so on) C n0 r f 2 C n2 r f 0 C n0 r f k C nk r f 0 If r is the largest integer that the subresultant mapping S r is not injective, we can compute the GCD of f 0 x,, f k x from the right null vector of S r f 0,, f k. 09 : 29 : Ruppert matrix as subresultant mapping Prev Next
20 20 CASC2007slideshow.nb How to link them f x, f 0 x f 1 xy 1 f k xy k f x, is reducible if f 0 x,, f k x have a non-trivial GCD. f x, is irreducible if they don t have any non-trivial GCD. Coprimeness of f 0 x,, f k x can be tested by a rank deficiency of their Sylvester matrix. Irreducibility of f x, f 0 x k i 1 by a rank deficiency of its Ruppert matrix. f i xy i can be tested 09 : 29 : Ruppert matrix as subresultant mapping Prev Next
21 CASC2007slideshow.nb 21 Relation for several polynomials Postponed as a future work: J.P.May s differential equation becomes f 0 g i g 0 f i Λ i f i f 0 f 0 f i 0 f 1 g i g 1 f i Λ i f i f 1 f 1 f i 0 i 1,, k f k g i g k f i Λ i f i f k f k f i 0 This means that u 0 x g 0 x Λ i f 0 u i x g i x Λ i f i i 1,, k will be a part of proof of several polynomial version of theorem. 09 : 29 : Ruppert matrix as subresultant mapping Prev Next
22 22 CASC2007slideshow.nb Conclusion Sylvester and Ruppert matrices have some relations for computing GCDs. Their relations lead to no algorithm which computes GCDs efficiently. The author hopes that the relations in this talk will make some progress. Thank you 09 : 29 : Ruppert matrix as subresultant mapping Prev Next
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