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The Proper Basis for Polynomial Ideals

Sheng-Ming Ma

TL;DR

The paper introduces the proper basis, a new ideal basis for zero-dimensional polynomial ideals that reduces computational complexity and coefficient swell by separating the least variable $x_1$ as a parameter in $K[x_1][\tilde{\mathbf{x}}]$ and performing a two-stage elimination. The first step operates in $K[x_1][\tilde{\mathbf{x}}]$ to produce a prebasis via proper division with least multipliers; the second modular step in $R_q[\tilde{\mathbf{x}}]$ constructs partial proper bases modulo coprime factors of the eliminant and then reconstructs the full proper basis. The paper provides formal definitions, algorithmic steps, and proofs, complemented by an explicit example and extensive benchmarks showing improved performance and reduced coefficient swell compared to Buchberger’s and Möller’s approaches. This work suggests substantial opportunities to enhance state-of-the-art Buchberger-based methods by applying them to the proper basis and points toward extensions to higher-dimensional ideals.

Abstract

We define a new type of ideal basis called the proper basis that improves both Gröbner basis and Buchberger's algorithm. Let $x_1$ be the least variable of a monomial ordering in a polynomial ring $K[x_1,\dotsc,x_n]$ over a field $K$. The Gröbner basis of a zero-dimensional polynomial ideal contains a univariate polynomial in $x_1$. The proper basis is defined and computed in the variables $\tilde{\bm{x}}:=(x_2,\dotsc,x_n)$ with $x_1$ serving as a parameter in the algebra $K[x_1][\tilde{\bm{x}}]$. Its algorithm is more efficient than not only Buchberger's algorithm whose elimination of $\tilde{\bm{x}}$ unnecessarily involves the least variable $x_1$ but also Möller's algorithm due to its polynomial division mechanism. This is corroborated by a series of benchmark testings herein. The proper basis is in a modular form and neater than Gröbner basis and hence reduces its coefficient swell problem. It is expected that all the state of the art algorithms improving Buchberger's algorithm over the last decades can be further improved if we apply them to the proper basis.

The Proper Basis for Polynomial Ideals

TL;DR

The paper introduces the proper basis, a new ideal basis for zero-dimensional polynomial ideals that reduces computational complexity and coefficient swell by separating the least variable as a parameter in and performing a two-stage elimination. The first step operates in to produce a prebasis via proper division with least multipliers; the second modular step in constructs partial proper bases modulo coprime factors of the eliminant and then reconstructs the full proper basis. The paper provides formal definitions, algorithmic steps, and proofs, complemented by an explicit example and extensive benchmarks showing improved performance and reduced coefficient swell compared to Buchberger’s and Möller’s approaches. This work suggests substantial opportunities to enhance state-of-the-art Buchberger-based methods by applying them to the proper basis and points toward extensions to higher-dimensional ideals.

Abstract

We define a new type of ideal basis called the proper basis that improves both Gröbner basis and Buchberger's algorithm. Let be the least variable of a monomial ordering in a polynomial ring over a field . The Gröbner basis of a zero-dimensional polynomial ideal contains a univariate polynomial in . The proper basis is defined and computed in the variables with serving as a parameter in the algebra . Its algorithm is more efficient than not only Buchberger's algorithm whose elimination of unnecessarily involves the least variable but also Möller's algorithm due to its polynomial division mechanism. This is corroborated by a series of benchmark testings herein. The proper basis is in a modular form and neater than Gröbner basis and hence reduces its coefficient swell problem. It is expected that all the state of the art algorithms improving Buchberger's algorithm over the last decades can be further improved if we apply them to the proper basis.

Paper Structure

This paper contains 7 sections, 14 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. The Definition for Proper Basis
  3. The First Algorithmic Step in $K[x_1] [\tilde{\bm{x}}]$
  4. The Final Algorithmic Step in $R_q [\tilde{\bm{x}}]$
  5. An Example
  6. Benchmark Testings in a Cascade of Complexity
  7. Conclusion and Acknowledgement

Key Result

Theorem 3.4

Suppose that $B=\{b_j\colon 1\le j\le s\}\subset K[x_1] [\tilde{\bm{x}}]\setminus K[x_1]$ is a finite polynomial set. For every $f\in K[x_1] [\tilde{\bm{x}}]$, there exist a multiplier $\lambda\in (K[x_1])^\ast$, a remainder $r\in K[x_1] [\tilde{\bm{x}}]$ and quotients $q_j\in K[x_1] [\tilde{\bm{x}} where $r$ is properly reduced with respect to $B$, and the multiplier $\lambda$ is literally a prod

Theorems & Definitions (35)

  • Definition 2.2: Eliminant $\chi$; multiplicity $\mathrm{mult}_p (f)$
  • Definition 2.3: Proper basis
  • Definition 3.2: Proper term reduction in the algebra $K[x_1] [\tilde{\bm{x}}]$
  • Definition 3.3: Properly reduced polynomial in $K[x_1] [\tilde{\bm{x}}]$
  • Theorem 3.4: Proper division or reduction in $K[x_1] [\tilde{\bm{x}}]$
  • proof
  • Remark 3.5
  • Definition 3.6: $S$-polynomial in $K[x_1] [\tilde{\bm{x}}]$
  • Lemma 3.7
  • proof
  • ...and 25 more