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Effective bounds for polynomial systems defined over the rationals

Teresa Krick

TL;DR

This survey collects explicit, quantitative bounds for solving affine polynomial systems over $\mathbb{Q}$ in the zero-dimensional (finite) case, emphasizing degree and height controls rather than algorithms. It develops and unifies arithmetic versions of classical results—Bézout, Nullstellensatz, shape (geometric/univariate) representations, and Perron's theorem—via Chow forms and height theory. The work provides concrete, explicit bounds on root coordinates, remainders modulo zero-dimensional ideals, and the degrees/heights of certificates, with examples illustrating tightness. The results have direct implications for effective algebraic geometry, modular methods, and certifiable computations in numerical and symbolic settings.

Abstract

This survey paper was primarily written as as the support for a course pesented at the JNCF2025: it aims to present some material that illustrates the kind of estimates one can obtain in effective algebraic geometry, for affine polynomial equation systems defined over the rational numbers Q, and focuses on the case of finite varieties.

Effective bounds for polynomial systems defined over the rationals

TL;DR

This survey collects explicit, quantitative bounds for solving affine polynomial systems over in the zero-dimensional (finite) case, emphasizing degree and height controls rather than algorithms. It develops and unifies arithmetic versions of classical results—Bézout, Nullstellensatz, shape (geometric/univariate) representations, and Perron's theorem—via Chow forms and height theory. The work provides concrete, explicit bounds on root coordinates, remainders modulo zero-dimensional ideals, and the degrees/heights of certificates, with examples illustrating tightness. The results have direct implications for effective algebraic geometry, modular methods, and certifiable computations in numerical and symbolic settings.

Abstract

This survey paper was primarily written as as the support for a course pesented at the JNCF2025: it aims to present some material that illustrates the kind of estimates one can obtain in effective algebraic geometry, for affine polynomial equation systems defined over the rational numbers Q, and focuses on the case of finite varieties.

Paper Structure

This paper contains 7 sections, 12 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Setting and notations
  3. The Bézout inequality
  4. An Arithmetic Bézout inequality
  5. An arithmetic Nullstellensatz
  6. An Arithmetic Shape Lemma
  7. An Arithmetic Perron's Theorem

Key Result

Theorem 3.1

(Bézout inequality) Let $f_1,\dots,f_s\in \mathbb{C}[{\boldsymbol{x}}]\setminus \{0\}$ define a zero-dimensional variety $V\subset \mathbb{C}^n$. Set $d_j:=\deg(f_j)$ and assume that $d:=d_1\ge d_2\ge \cdots \ge d_{s-1}$ (with no condition with respect to $d_s$). Then

Theorems & Definitions (21)

  • Theorem 3.1
  • Theorem 4.1
  • proof
  • Corollary 4.2
  • proof
  • Proposition 4.3
  • proof
  • Theorem 5.1
  • Theorem 5.2
  • Theorem 6.1
  • ...and 11 more