On Computing the Resultant of Generic Bivariate Polynomials

Abstract

An algorithm is presented for computing the resultant of two generic bivariate polynomials over a field K. For such p and q in K[x, y] both of degree d in x and n in y, the algorithm computes the resultant with respect to y using (n 2−1/ω d ) 1+o(1) arithmetic operations in K, where two n × n matrices are multiplied using O (n ω ) operations. Previous algorithms required time ( n 2 d ) 1+o(1) .The resultant is the determinant of the Sylvester matrix S (x ) of p and q, which is an n × n Toeplitz-like polynomial matrix of degree d . We use a blocking technique and exploit the structure of S (x ) for reducing the determinant computation to the computation of a matrix fraction description R(x )Q (x ) −1 of an m ×m submatrix of the inverse S (x ) −1 , where m ≪ n. We rely on fast algorithms for handling dense polynomial matrices: the fraction description is obtained from an x -adic expansion via matrix fraction reconstruction, and the resultant as the determinant of the denominator matrix.We also describe some extensions of the approach to the computation of generic Gröbner bases and of characteristic polynomials of generic structured matrices and in univariate quotient algebras.

Implementations of Efficient Univariate Polynomial Matrix Algorithms and Application to Bivariate Resultants

“…We conclude this paper with algorithms originating from Villard's recent breakthrough on computing the determinant of structured polynomial matrices [49]. Fix a field K and consider the two following questions: computing the resultant of two polynomials F , G in K[x, z] with respect to z, and computing the characteristic polynomial of an element A in K[z]/(P), for some P in K[z].…”

“…Overview of the approach. In [49], Villard designed the following algorithm to find the determinant of a matrix P over K[x]. The parameter m is chosen so as to minimize the theoretical cost.…”

“…For approximants, we take a random input in K[x] m×n of degree d −1; for interpolants, we take d random matrices in K m×n . We focus on the common case m ≃ 2n, which arises for example in kernel algorithms (Section 4.2) and in fraction reconstruction, itself used in basis reduction (Section 4.5) and in the resultant algorithm of [49] (Section 5). Concerning iterative algorithms, we observe that interpolants are slightly faster than approximants, which is explained by the cost of computing the residual R: it uses one Horner evaluation of P and one matrix product for interpolants, whereas for approximants it uses about min(k, deg(P)) matrix products at iteration k.…”

“…2.4 to 2.7]. • Coppersmith's block Wiedemann algorithm and its extensions [7,26,48] were used in a variety of contexts, from integer factorization [44] to polynomial system solving [22,49].…”

A fast randomized geometric algorithm for computing Riemann-Roch spaces

“…11.21]. The Bézout bound implies that the degree of the resultant χ(S) is at most deg(q) deg(h), hence the complexities of all the other steps We point out that the complexity of computing resultants and subresultants of bivariate polynomials have been recently improved in [26,25] under some genericity assumptions. However, since the cost in Proposition 26 will be negligible in the global complexity estimate, we make no effort to optimize it further.…”