[Paper Review] Subresultants in Recursive Polynomial Remainder Sequence

Akira Terui|arXiv (Cornell University)|Jun 3, 2008

Polynomial and algebraic computation6 references3 citations

TL;DR

This paper introduces recursive polynomial remainder sequences (recursive PRS) and recursive subresultants—extensions of classical subresultant theory for polynomials with nontrivial GCDs. By recursively applying PRS to a GCD and its derivative, the method enables stable computation of real root multiplicities via recursive Sturm sequences, with recursive subresultants providing determinant-based representations of coefficients in terms of original polynomial coefficients, enhancing numerical stability in approximate algebraic computation.

ABSTRACT

We introduce concepts of "recursive polynomial remainder sequence (PRS)" and "recursive subresultant," and investigate their properties. In calculating PRS, if there exists the GCD (greatest common divisor) of initial polynomials, we calculate "recursively" with new PRS for the GCD and its derivative, until a constant is derived. We call such a PRS a recursive PRS. We define recursive subresultants to be determinants representing the coefficients in recursive PRS by coefficients of initial polynomials. Finally, we discuss usage of recursive subresultants in approximate algebraic computation, which motivates the present work.

Motivation & Objective

• To extend classical subresultant theory to handle recursive polynomial remainder sequences (recursive PRS), particularly for polynomials with nontrivial GCDs.
• To define recursive subresultants as determinants that represent coefficients in recursive PRS in terms of the original polynomial coefficients.
• To improve numerical stability in approximate algebraic computation, especially for detecting small leading coefficients in recursive Sturm sequences.
• To provide a theoretical foundation for recursive subresultants analogous to the fundamental theorem of subresultants.
• To support robust zero recognition in floating-point arithmetic by linking recursive subresultants to Sylvester matrix conditions.

Proposed method

• Define recursive PRS as a sequence of PRSs computed recursively on the GCD and its derivative until a constant is reached.
• Introduce recursive subresultants as determinants formed from coefficients of initial polynomials, representing elements in the recursive PRS.
• Establish a recursive structure where each new PRS is computed from the last non-constant element and its derivative in the prior sequence.
• Use matrix transformations and column exchanges to relate recursive subresultant determinants to subresultants of subsequent PRS levels.
• Leverage Sylvester matrix conditions and normalization via leading coefficient scaling to stabilize numerical evaluation.
• Apply the recursive subresultant framework to recursive Sturm sequences for counting real roots with multiplicity.

Experimental results

Research questions

• RQ1 How can the classical subresultant theory be extended to handle recursive polynomial remainder sequences arising from repeated GCD and derivative computation?
• RQ2 What is the algebraic structure of recursive subresultants, and how do they relate to the coefficients of the polynomials in the recursive PRS?
• RQ3 Can recursive subresultants be used to improve the numerical stability of approximate algebraic computation, particularly in detecting small or near-zero leading coefficients?
• RQ4 What conditions on the Sylvester matrix ensure reliable zero recognition in recursive Sturm sequences?
• RQ5 How can recursive subresultants be used to compute the number of real roots, including multiplicities, in polynomials with multiple or close-lying roots?

Key findings

• Recursive subresultants are defined as determinants that represent coefficients in recursive PRS using the coefficients of the initial polynomials.
• The paper proves a recursive analog of the fundamental theorem of subresultants, showing that recursive subresultants determine the structure of recursive PRS.
• Recursive subresultants can be expressed as scaled products of lower-level subresultants, with scaling factors involving powers of leading coefficients from intermediate GCDs.
• The determinant of the recursive subresultant matrix is shown to be proportional to the product of subresultants from successive PRS levels, with explicit scaling constants.
• The framework enables stable detection of small leading coefficients in recursive Sturm sequences by analyzing recursive subresultant structure.
• The method supports reliable zero recognition in floating-point arithmetic by linking subresultant determinants to Sylvester matrix conditions, improving robustness in real root counting.