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Two Algorithms for Computing Rational Univariate Representations of Zero-Dimensional Ideals with Parameters

Dingkang Wang, Jingjing Wei, Fanghui Xiao, Xiaopeng Zheng

TL;DR

The paper extends Rational Univariate Representations (RUR) to zero-dimensional ideals with parameters by partitioning the parameter space into branches with fixed root structure. It leverages Comprehensive Gröbner Systems and an extended subresultant/parametric gcd framework to select separating elements branch-by-branch, producing branch-specific RUR data of the form $\left(\frac{g_{i1}(\bar{u},\beta)}{g_i(\bar{u},\beta)}, \ldots, \frac{g_{in}(\bar{u},\beta)}{g_i(\bar{u},\beta)}\right)$ where $\beta$ runs over roots of $\mathcal{X}_i(\bar{u},T)$. Two algorithms are developed: Algorithm 1 uses a four-step partitioning (Gröbner basis, number of zeros, separating elements, gcds of $\mathcal{X}_t$ and $\mathcal{X}_t'$), while Algorithm 2 directly applies parametric gcds to obtain RURs, potentially skipping explicit separating-element steps. The methods are implemented in Singular and tested on standard parametric benchmarks, showing branch- and time-based tradeoffs, with open-source code provided. These contributions enable robust, branch-aware symbolic solving of parametric polynomial systems and extend RUR techniques to the parametric setting with practical tooling.

Abstract

Based on the partition of parameter space, two algorithms for computing the rational univariate representation of zero-dimensional ideals with parameters are presented in the paper. Unlike the rational univariate representation of zero-dimensional ideals without parameters, the number of zeros of zero-dimensional ideals with parameters under various specializations is different, which leads to choosing and checking the separating element, the key to computing the rational univariate representation, is difficult. In order to pick out the separating element, we first ensure that under each branch the ideal has the same number of zeros by partitioning the parameter space. Subsequently two ideas are given to choose and check the separating element. One idea is that by extending the subresultant theorem to parametric cases, we utilize the extended subresultant theorem to choose the separating element with the further partition of parameter space and then with the help of parametric greatest common divisor theory compute rational univariate representations. Another one is that we go straight to choose and check the separating element by the computation of parametric greatest common divisors, then immediately get the rational univariate representations. Based on these, we design two different algorithms for computing rational univariate representations of zero-dimensional ideals with parameters. Furthermore, the algorithms have been implemented on Singular and the performance comparison are presented.

Two Algorithms for Computing Rational Univariate Representations of Zero-Dimensional Ideals with Parameters

TL;DR

The paper extends Rational Univariate Representations (RUR) to zero-dimensional ideals with parameters by partitioning the parameter space into branches with fixed root structure. It leverages Comprehensive Gröbner Systems and an extended subresultant/parametric gcd framework to select separating elements branch-by-branch, producing branch-specific RUR data of the form where runs over roots of . Two algorithms are developed: Algorithm 1 uses a four-step partitioning (Gröbner basis, number of zeros, separating elements, gcds of and ), while Algorithm 2 directly applies parametric gcds to obtain RURs, potentially skipping explicit separating-element steps. The methods are implemented in Singular and tested on standard parametric benchmarks, showing branch- and time-based tradeoffs, with open-source code provided. These contributions enable robust, branch-aware symbolic solving of parametric polynomial systems and extend RUR techniques to the parametric setting with practical tooling.

Abstract

Based on the partition of parameter space, two algorithms for computing the rational univariate representation of zero-dimensional ideals with parameters are presented in the paper. Unlike the rational univariate representation of zero-dimensional ideals without parameters, the number of zeros of zero-dimensional ideals with parameters under various specializations is different, which leads to choosing and checking the separating element, the key to computing the rational univariate representation, is difficult. In order to pick out the separating element, we first ensure that under each branch the ideal has the same number of zeros by partitioning the parameter space. Subsequently two ideas are given to choose and check the separating element. One idea is that by extending the subresultant theorem to parametric cases, we utilize the extended subresultant theorem to choose the separating element with the further partition of parameter space and then with the help of parametric greatest common divisor theory compute rational univariate representations. Another one is that we go straight to choose and check the separating element by the computation of parametric greatest common divisors, then immediately get the rational univariate representations. Based on these, we design two different algorithms for computing rational univariate representations of zero-dimensional ideals with parameters. Furthermore, the algorithms have been implemented on Singular and the performance comparison are presented.
Paper Structure (15 sections, 14 theorems, 73 equations, 1 table)

This paper contains 15 sections, 14 theorems, 73 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Rational univariate representation
  4. Comprehensive Gröbner System and Parametric Greatest Common Divisor
  5. Subresultant and Common Divisor
  6. RUR of zero-dimensional ideal with parameters
  7. Algorithm 1
  8. Partition according to the Gröbner basis
  9. Partition according to the number of zeros
  10. Partition according to separating elements
  11. Partitions according to GCDs of $\mathcal{X}_t$ and $\mathcal{X}_t^{\prime}$
  12. Algorithm 2
  13. Example
  14. Implementation and Comparative Performance
  15. Concluding remarks

Key Result

Theorem 1

Let $I \subset k[X]$ be a zero-dimensional ideal, $t \in k[X]$ and $m_t$ be the multiplication map. Then the eigenvalues of $M_t$ are $\{t(p):p \in {\mathbb{V}}_L(I)\}$. More specifically, the characteristic polynomial of $m_t$ is where $\mu(p)$ is the multiplicity of $p$ in ${\mathbb{V}}_L(I)$.

Theorems & Definitions (33)

  • Theorem 1: Rouillier99
  • Theorem 2: Cox06, Page 71, Theorem 5.2
  • Definition 3
  • Definition 4
  • Definition 5: Weispfenning92
  • Definition 6
  • Definition 7
  • Theorem 8: MB1993
  • Lemma 9
  • proof
  • ...and 23 more