Solving parameter-dependent semi-algebraic systems

2024-02-12

International Symposium on Symbolic and Algebraic Computation

AI Summary

This paper addresses the classification problem of the number of real solutions to parametric semi-algebraic systems as parameters vary, aiming to partition the parameter space into cells where the real solution count remains invariant. For systems of high degree (maximum degree $d$), multiple inequalities ($s$), and high dimension ($n$ variables), we propose the first single-exponential complexity algorithm. Our method integrates symbolic computation, polynomial ideal theory, and real algebraic geometry, combining parametric elimination with critical point techniques to construct a discriminant formula endowed with a determinant structure. The formula’s degree is bounded above by $(2s+n)d^{n+1}$, and its arithmetic complexity—under genericity assumptions—is polynomial in $d$ and $s$, and exponential in $nt + t^2$, where $t$ denotes the number of distinct monomials. Experiments demonstrate that our approach successfully solves large-scale instances previously intractable, significantly extending the scope of solvable problems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We consider systems of polynomial equations and inequalities in $mathbb{Q}[oldsymbol{y}][oldsymbol{x}]$ where $oldsymbol{x} = (x_1, ldots, x_n)$ and $oldsymbol{y} = (y_1, ldots,y_t)$. The $oldsymbol{y}$ 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.

Problem

• Parameterized Semi-algebraic Systems
• Real Solution Enumeration
• High-degree Polynomials

Innovation

• Parameterized Semi-Algebraic Systems
• High-Degree Equations
• Unique Mathematical Structure