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Solving parameter-dependent semi-algebraic systems

Louis Gaillard, Mohab Safey El Din

TL;DR

This work tackles the problem of classifying real roots of parameter-dependent polynomial systems by combining Hermite quadratic forms with parametric Hermite matrices. The authors develop an algorithm that encodes TaQ-based real-root information into determinantal expressions and uses sample-point techniques to describe parameter regions where the real root count remains constant, without requiring radical equidimensionality. Under mild genericity assumptions, the method achieves a singly exponential complexity in key parameters and yields output formulas with explicit degree bounds, enabling practical computation. Empirical results demonstrate substantial performance gains over existing tools and solve previously open instances, including the Perspective-Three-Point problem in certain regimes, highlighting the approach’s applicability to real algebraic geometry problems with inequalities.

Abstract

We consider systems of polynomial equations and inequalities in $\mathbb{Q}[\boldsymbol{y}][\boldsymbol{x}]$ where $\boldsymbol{x} = (x_1, \ldots, x_n)$ and $\boldsymbol{y} = (y_1, \ldots,y_t)$. The $\boldsymbol{y}$ indeterminates are considered as parameters and we assume that when specialising them generically, the set of common complex solutions, to the obtained equations, is finite. We consider the problem of real root classification for such parameter-dependent problems, i.e. identifying the possible number of real solutions depending on the values of the parameters and computing a description of the regions of the space of parameters over which the number of real roots remains invariant. We design an algorithm for solving this problem. The formulas it outputs enjoy a determinantal structure. Under genericity assumptions, we show that its arithmetic complexity is polynomial in both the maximum degree $d$ and the number $s$ of the input inequalities and exponential in $nt+t^2$. The output formulas consist of polynomials of degree bounded by $(2s+n)d^{n+1}$. This is the first algorithm with such a singly exponential complexity. We report on practical experiments showing that a first implementation of this algorithm can tackle examples which were previously out of reach.

Solving parameter-dependent semi-algebraic systems

TL;DR

This work tackles the problem of classifying real roots of parameter-dependent polynomial systems by combining Hermite quadratic forms with parametric Hermite matrices. The authors develop an algorithm that encodes TaQ-based real-root information into determinantal expressions and uses sample-point techniques to describe parameter regions where the real root count remains constant, without requiring radical equidimensionality. Under mild genericity assumptions, the method achieves a singly exponential complexity in key parameters and yields output formulas with explicit degree bounds, enabling practical computation. Empirical results demonstrate substantial performance gains over existing tools and solve previously open instances, including the Perspective-Three-Point problem in certain regimes, highlighting the approach’s applicability to real algebraic geometry problems with inequalities.

Abstract

We consider systems of polynomial equations and inequalities in where and . The indeterminates are considered as parameters and we assume that when specialising them generically, the set of common complex solutions, to the obtained equations, is finite. We consider the problem of real root classification for such parameter-dependent problems, i.e. identifying the possible number of real solutions depending on the values of the parameters and computing a description of the regions of the space of parameters over which the number of real roots remains invariant. We design an algorithm for solving this problem. The formulas it outputs enjoy a determinantal structure. Under genericity assumptions, we show that its arithmetic complexity is polynomial in both the maximum degree and the number of the input inequalities and exponential in . The output formulas consist of polynomials of degree bounded by . This is the first algorithm with such a singly exponential complexity. We report on practical experiments showing that a first implementation of this algorithm can tackle examples which were previously out of reach.
Paper Structure (13 sections, 20 theorems, 20 equations, 1 table, 2 algorithms)

This paper contains 13 sections, 20 theorems, 20 equations, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Hermite's quadratic form
  3. Definition
  4. Real root counting
  5. Parametric Hermite matrices
  6. Basic construction and properties
  7. Algorithm for computing Hermite matrices
  8. Degree bounds
  9. Algorithm
  10. Sign determination on a finite set of points
  11. General sign determination
  12. Complexity analysis
  13. Practical experiments

Key Result

Theorem 1.1

Let $\boldsymbol{f} \subset \mathbb{Q}[\boldsymbol{y}][\boldsymbol{x}]$ be a regular sequence such that $\boldsymbol{f}$ satisfies both Assumptions assump:finite_nb_sol_generically and assump:generic_degree_grobner_basis. Let $\mathfrak{D} \coloneqq (2sd + n(d-1))d^{n}$. There exists an algorithm wh arithmetic operations in $\mathbb{Q}$ and outputs at most $(sdn)^{\mathcal{O}(t^2 + nt)}$ formulas

Theorems & Definitions (25)

  • Theorem 1.1
  • Definition 2.1: Hermite's quadratic form
  • Proposition 2.2
  • Definition 2.3: Tarski-query
  • Theorem 2.4: roy2006algo_real_alg_geom
  • Theorem 2.5: ghys2016signatures
  • Proposition 3.1
  • Corollary 3.2
  • Definition 3.3: Regular sequence
  • Lemma 3.4
  • ...and 15 more