A Theory and an Algorithm for Computing Sparse Multivariate Polynomial Remainder Sequence

Abstract

This paper presents an algorithm for computing the polynomial remainder sequence (PRS) and corresponding cofactor sequences of sparse multivariate polynomials over a number field. Most conventional algorithms for computing PRSs are based on the pseudo remainder (Prem), and the celebrated subresultant theory for the PRS has been constructed on the Prem. The Prem is uneconomical for computing PRSs of sparse polynomials. Hence, in this paper, the concept of sparse pseudo remainder (spsPrem) is defined. No subresultant-like theory has been developed so far for the PRS based on spsPrem. Therefore, we develop a matrix theory for spsPrem-based PRSs. The computational formula for PRS, regardless of whether it is based on Prem or spsPrem, causes a considerable intermediate expression growth. Hence, we next propose a technique to suppress the expression growth largely. The technique utilizes the power-series arithmetic but no Hensel lifting. Simple experiments show that our technique suppresses the intermediate expression growth fairly well, if the sub-variable ordering is set suitably.