High-order lifting for polynomial Sylvester matrices

Abstract

A new algorithm is presented for computing the resultant of two generic bivariate polynomials over an arbitrary field. For $p,q$ in $K[x,y]$ of degree d in x and n in y, the resultant with respect to y is computed using $O\left(n^{1.458}d\right)$ arithmetic operations if $d=O\left(n^{1/3}\right)$. For $d=1$, the complexity estimate is therefore reconciled with the estimates of Neiger et al. 2021 for the related problems of modular composition and characteristic polynomial in a univariate quotient algebra. The 3/2 barrier in the exponent of n is crossed for the first time for the resultant. The problem is related to that of computing determinants of structured polynomial matrices. We identify new advanced aspects of structure for a polynomial Sylvester matrix. This enables to compute the determinant by mixing the baby steps/giant steps approach of Kaltofen and Villard 2005, until then restricted to the case $d=1$ for characteristic polynomials, and the high-order lifting strategy of Storjohann 2003 usually reserved for dense polynomial matrices.