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A Kronecker algorithm for locally closed sets over a perfect field

Nardo Giménez, Joos Heintz, Guillermo Matera, Luis Miguel Pardo, Mariana Pérez, Melina Privitelli

TL;DR

This work advances a probabilistic Kronecker-type framework for solving zero-dimensional sections of algebraic varieties over perfect fields by integrating simultaneous Noether normalization, lifting fibers, lifting curves, and Newton-Hensel lifting within the TERA-Kronecker suite. It replaces costly naive elimination with a homotopic, curve-based deformation strategy that yields a Kronecker representation with soft-quadratic dependence on input degrees/heights, and provides sharp complexity bounds for arbitrary perfect fields, finite fields, and the rational numbers. Key contributions include a robust construction of lifting curves, a discriminant-driven intersection step, and a modular/p-adic reconstruction pipeline that enables solving over Q with probabilistic success guarantees. The resulting algorithms offer practical, scalable symbolic solving for multivariate polynomial systems outside prescribed hypersurfaces, with explicit bounds on field size, primes, and bit complexity, making them suitable for use in computer algebra systems and related first-pass context tasks.

Abstract

We develop a probabilistic algorithm of Kronecker type for computing a Kronecker representation of a zero-dimensional linear section of an algebraic variety $V$ defined over a perfect field $k$. The variety $V$ is the Zariski closure of the set of common zeros $\{F_1=0,\ldots,F_r=0,G\not=0\}$ of multivariate polynomials $F_1,\ldots,F_r\in k[X_1,\ldots,X_n]$ outside a prescribed hypersurface $\{G=0\}$. We assume that $F_1,\ldots,F_r$ satisfy natural geometric conditions, such as regularity and radicality, in the local ring $k[X_1,\ldots,X_n]_G$. Our approach combines homotopic deformation techniques with symbolic Newton-Hensel lifting and elimination. We discuss the concept of lifting curves as intermediate geometric objects that enable efficient computation. The complexity of the algorithm is expressed in terms of the degrees and arithmetic size of the input and achieves soft-quadratic complexity in these parameters. We provide detailed complexity analyses for arbitrary perfect fields, as well as for two important cases in computer algebra: finite fields and the field of rational numbers. For each case, we obtain sharp bounds on the size of the base field or required primes.

A Kronecker algorithm for locally closed sets over a perfect field

TL;DR

This work advances a probabilistic Kronecker-type framework for solving zero-dimensional sections of algebraic varieties over perfect fields by integrating simultaneous Noether normalization, lifting fibers, lifting curves, and Newton-Hensel lifting within the TERA-Kronecker suite. It replaces costly naive elimination with a homotopic, curve-based deformation strategy that yields a Kronecker representation with soft-quadratic dependence on input degrees/heights, and provides sharp complexity bounds for arbitrary perfect fields, finite fields, and the rational numbers. Key contributions include a robust construction of lifting curves, a discriminant-driven intersection step, and a modular/p-adic reconstruction pipeline that enables solving over Q with probabilistic success guarantees. The resulting algorithms offer practical, scalable symbolic solving for multivariate polynomial systems outside prescribed hypersurfaces, with explicit bounds on field size, primes, and bit complexity, making them suitable for use in computer algebra systems and related first-pass context tasks.

Abstract

We develop a probabilistic algorithm of Kronecker type for computing a Kronecker representation of a zero-dimensional linear section of an algebraic variety defined over a perfect field . The variety is the Zariski closure of the set of common zeros of multivariate polynomials outside a prescribed hypersurface . We assume that satisfy natural geometric conditions, such as regularity and radicality, in the local ring . Our approach combines homotopic deformation techniques with symbolic Newton-Hensel lifting and elimination. We discuss the concept of lifting curves as intermediate geometric objects that enable efficient computation. The complexity of the algorithm is expressed in terms of the degrees and arithmetic size of the input and achieves soft-quadratic complexity in these parameters. We provide detailed complexity analyses for arbitrary perfect fields, as well as for two important cases in computer algebra: finite fields and the field of rational numbers. For each case, we obtain sharp bounds on the size of the base field or required primes.

Paper Structure

This paper contains 30 sections, 51 theorems, 238 equations, 12 algorithms.

Table of Contents

  1. Introduction
  2. Outline of the TERA-Kronecker suite
  3. The solution variety
  4. The left action of $GL(n,\overline{k})$ on the solution variety
  5. First approach: a procedure by jumping through solution varieties
  6. Second approach: deformations through zero-dimensional smooth varieties
  7. Third approach: homotopic deformations by smooth curves
  8. Main results and outline of the paper
  9. Notions, notations and preliminary results
  10. Kronecker representations
  11. Computational model and basic operations
  12. Basic operations
  13. Probabilistic aspects
  14. Noether normalization, primitive elements and lifting fibers
  15. Simultaneous Noether normalization and a primitive element
  16. ...and 15 more sections

Key Result

Theorem 10

Let $\boldsymbol \Lambda:=(\Lambda_{ij})_{1\le i,j\le n}$ be a matrix of indeterminates over $\overline{k}$, and let $\widetilde{\boldsymbol Y}:=\boldsymbol \Lambda \cdot \boldsymbol X$. There exists a nonzero polynomial $B\in\overline{k}[\boldsymbol \Lambda, \widetilde{Y}_1 ,,$,…,$\widetilde{Y}_{n-

Theorems & Definitions (107)

  • Remark 1
  • Remark 6
  • Remark 7
  • Remark 8
  • Remark 9
  • Theorem 10
  • Theorem 11
  • Theorem 12
  • Theorem 13
  • Definition 15
  • ...and 97 more