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Computing greatest common divisor of several parametric univariate polynomials via generalized subresultant polynomials

Hoon Hong, Jing Yang

TL;DR

The paper tackles the problem of computing the gcd of several univariate polynomials with parametric coefficients by partitioning the parameter space into cells where the gcd expression is uniform. It introduces a natural extension of subresultants to multiple polynomials and develops generalized Sylvester's and Habicht's theories, yielding determinant-based formulas that remain non-nested and easier to analyze. The authors propose two core algorithms, PGCD and EPGCD, which compute parametric gcds by leveraging generalized Sylvester and Habicht relations to produce explicit gcd expressions $G_i$ on cells $C_i$ with conditions $C_i$. Performance assessments show substantial improvements in output size and running time compared with recursive and matrix-based approaches, making the method scalable to larger families of polynomials. The work provides a concrete, efficient framework for parametric gcds with multiple polynomials, enabling structural analysis and robust symbolic computation in parameterized settings.

Abstract

In this paper, we tackle the following problem: compute the gcd for several univariate polynomials with parametric coefficients. It amounts to partitioning the parameter space into ``cells'' so that the gcd has a uniform expression over each cell and constructing a uniform expression of gcd in each cell. We tackle the problem as follows. We begin by making a natural and obvious extension of subresultant polynomials of two polynomials to several polynomials. Then we develop the following structural theories about them. 1. We generalize Sylvester's theory to several polynomials, in order to obtain an elegant relationship between generalized subresultant polynomials and the gcd of several polynomials, yielding an elegant algorithm. 2. We generalize Habicht's theory to several polynomials, in order to obtain a systematic relationship between generalized subresultant polynomials and pseudo-remainders, yielding an efficient algorithm. Using the generalized theories, we present a simple (structurally elegant) algorithm which is significantly more efficient (both in the output size and computing time) than algorithms based on previous approaches.

Computing greatest common divisor of several parametric univariate polynomials via generalized subresultant polynomials

TL;DR

The paper tackles the problem of computing the gcd of several univariate polynomials with parametric coefficients by partitioning the parameter space into cells where the gcd expression is uniform. It introduces a natural extension of subresultants to multiple polynomials and develops generalized Sylvester's and Habicht's theories, yielding determinant-based formulas that remain non-nested and easier to analyze. The authors propose two core algorithms, PGCD and EPGCD, which compute parametric gcds by leveraging generalized Sylvester and Habicht relations to produce explicit gcd expressions on cells with conditions . Performance assessments show substantial improvements in output size and running time compared with recursive and matrix-based approaches, making the method scalable to larger families of polynomials. The work provides a concrete, efficient framework for parametric gcds with multiple polynomials, enabling structural analysis and robust symbolic computation in parameterized settings.

Abstract

In this paper, we tackle the following problem: compute the gcd for several univariate polynomials with parametric coefficients. It amounts to partitioning the parameter space into ``cells'' so that the gcd has a uniform expression over each cell and constructing a uniform expression of gcd in each cell. We tackle the problem as follows. We begin by making a natural and obvious extension of subresultant polynomials of two polynomials to several polynomials. Then we develop the following structural theories about them. 1. We generalize Sylvester's theory to several polynomials, in order to obtain an elegant relationship between generalized subresultant polynomials and the gcd of several polynomials, yielding an elegant algorithm. 2. We generalize Habicht's theory to several polynomials, in order to obtain a systematic relationship between generalized subresultant polynomials and pseudo-remainders, yielding an efficient algorithm. Using the generalized theories, we present a simple (structurally elegant) algorithm which is significantly more efficient (both in the output size and computing time) than algorithms based on previous approaches.
Paper Structure (20 sections, 14 theorems, 62 equations, 1 figure, 2 tables, 4 algorithms)

This paper contains 20 sections, 14 theorems, 62 equations, 1 figure, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Review on subresultant theory of two polynomials
  4. Subresultants for two polynomials
  5. Sylvester's theory for two polynomials
  6. Habicht's theory for two polynomials
  7. Generalization of subresultants to several polynomials
  8. Generalization of Sylvester's theory to several polynomials
  9. Main result
  10. Proof of the generalized Sylvester's theorem (Theorem \ref{['thm:gsylvester']})
  11. An algorithm for computing parametric gcd with the generalized Sylvester's theorem
  12. Generalization of Habicht's theory to several polynomials
  13. Main result
  14. Proof of the generalized Habicht's theorem (Theorem \ref{['thm:ghabicht']})
  15. An algorithm for computing parametric gcd with the generalized Habicht's theorem
  16. ...and 5 more sections

Key Result

Theorem 14

Given $F_{0},F_{1}\in\mathcal{Z}[x]$ with $\deg (F_{i})=d_{i}$. Then the following claims are equivalent: Furthermore, we have: $\deg\gcd(F_{0},F_{1})=i\ \ \ \Longleftrightarrow\ \ \ \gcd(F_{0},F_{1})=R_{d_{0}-i}(F_{0},F_{1}).$

Figures (1)

  • Figure 1: Ratios of determinants required for partitioning the parameter space.

Theorems & Definitions (49)

  • Remark 2
  • Example 4
  • Definition 5: Determinant polynomial of matrix
  • Example 6
  • Definition 7: Coefficient matrix of a list of polynomials
  • Example 8
  • Example 10
  • Definition 11
  • Example 12
  • Remark 13
  • ...and 39 more