Recursive Polynomial Remainder Sequence and its Subresultants

TL;DR

This paper introduces recursive polynomial remainder sequences and subresultants, exploring their properties and providing new matrix constructions that optimize computational efficiency for recursive polynomial GCD calculations.

Contribution

It defines recursive PRS and subresultants, and presents three novel matrix constructions, including a size-reducing method using Gaussian elimination for recursive polynomial GCD computations.

Findings

Three different subresultant matrix constructions are proposed.

The last construction significantly reduces matrix size.

Recursive subresultants are expressed via determinants of these matrices.

Abstract

We introduce concepts of "recursive polynomial remainder sequence (PRS)" and "recursive subresultant," along with investigation of their properties. A recursive PRS is defined as, if there exists the GCD (greatest common divisor) of initial polynomials, a sequence of PRSs calculated "recursively" for the GCD and its derivative until a constant is derived, and recursive subresultants are defined by determinants representing the coefficients in recursive PRS as functions of coefficients of initial polynomials. We give three different constructions of subresultant matrices for recursive subresultants; while the first one is built-up just with previously defined matrices thus the size of the matrix increases fast as the recursion deepens, the last one reduces the size of the matrix drastically by the Gaussian elimination on the second one which has a "nested" expression, i.e. a Sylvester matrix whose elements are themselves determinants.