Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Leinartas's partial fraction decomposition

Alexander Raichev

TL;DR

Addresses decomposing multivariate rational functions into a sum of fractions with controlled denominators. It combines a Nullstellensatz-based decomposition with an algebraic-dependence refinement to produce Leinartas decompositions. The paper formalizes the conditions under which the two-step process terminates and is computable, discusses uniqueness (or lack thereof) in higher dimensions, and provides Sage implementations illustrating the method and its use for computing residues. The work generalizes univariate partial fractions to the multivariate setting and supplies a concrete algorithm suitable for computer algebra systems.

Abstract

These notes describe Leinartas's algorithm for multivariate partial fraction decompositions and employ an implementation thereof in Sage.

Leinartas's partial fraction decomposition

TL;DR

Addresses decomposing multivariate rational functions into a sum of fractions with controlled denominators. It combines a Nullstellensatz-based decomposition with an algebraic-dependence refinement to produce Leinartas decompositions. The paper formalizes the conditions under which the two-step process terminates and is computable, discusses uniqueness (or lack thereof) in higher dimensions, and provides Sage implementations illustrating the method and its use for computing residues. The work generalizes univariate partial fractions to the multivariate setting and supplies a concrete algorithm suitable for computer algebra systems.

Abstract

These notes describe Leinartas's algorithm for multivariate partial fraction decompositions and employ an implementation thereof in Sage.

Paper Structure

This paper contains 2 sections, 7 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Algorithm

Key Result

Theorem 2.1

Let $f = p/q$, where $p, q \in K[X]$. Let $q = q_1^{e_1} \cdots q_m^{e_m}$ be the unique factorization of $q$ in $K[X]$, and let $V_i = \{x \in \overline{K}^d : q_i(x) = 0 \}$, the algebraic variety of $q_i$ over $\overline{K}$. The rational expression $f$ can be written in the form where the $b_i$ are positive integers (possibly greater than the $e_i$), the $p_A$ are polynomials in $K[X]$ (possi

Theorems & Definitions (16)

  • Theorem 2.1: Leınartas decompositon
  • Corollary 2.2
  • Lemma 2.3: Nullstellensatz certificate
  • Lemma 2.4: Nullstellensatz decomposition
  • proof
  • Example 2.5
  • Lemma 2.6: Algebraic dependence certificate
  • Lemma 2.7
  • proof
  • Lemma 2.8: Algebraic dependence decomposition
  • ...and 6 more