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Deciding lower-boundedness of polynomials

Nguyen Hong Duc, Vu Trung Hieu

TL;DR

The paper tackles the challenge of deciding lower-boundedness for real multivariate polynomials by introducing the n-th non-critical tangency value polynomial $\varphi_{p,a}$ in the Puiseux field and the notion of T-good points. It reduces the decision to verifying that the Sturm-sequence-based invariant $v(p,a)$ vanishes for a generic T-good point $a$, enabling a probabilistic, algebraic-geometry–driven algorithm that avoids full quantifier elimination. The authors prove genericity of T-goodness, provide practical sufficient conditions and algorithms for certification, and demonstrate efficiency gains over cylindrical algebraic decomposition in small dense cases, with applications to non-negativity and convexity. The work offers a new avenue for polynomial optimization by marrying Gröbner basis computations with Sturm’s theorem in a Puiseux-analytic framework, while acknowledging potential complexity and scalability considerations for larger problems.

Abstract

This paper addresses the problem of deciding the lower-boundedness of an arbitrary real polynomial p in n variables.

Deciding lower-boundedness of polynomials

TL;DR

The paper tackles the challenge of deciding lower-boundedness for real multivariate polynomials by introducing the n-th non-critical tangency value polynomial in the Puiseux field and the notion of T-good points. It reduces the decision to verifying that the Sturm-sequence-based invariant vanishes for a generic T-good point , enabling a probabilistic, algebraic-geometry–driven algorithm that avoids full quantifier elimination. The authors prove genericity of T-goodness, provide practical sufficient conditions and algorithms for certification, and demonstrate efficiency gains over cylindrical algebraic decomposition in small dense cases, with applications to non-negativity and convexity. The work offers a new avenue for polynomial optimization by marrying Gröbner basis computations with Sturm’s theorem in a Puiseux-analytic framework, while acknowledging potential complexity and scalability considerations for larger problems.

Abstract

This paper addresses the problem of deciding the lower-boundedness of an arbitrary real polynomial p in n variables.

Paper Structure

This paper contains 19 sections, 14 theorems, 70 equations, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background and notations
  3. Asymptotic tangency values
  4. Ideals and Gröbner bases
  5. The field of Puiseux series over $\mathop{\mathrm{\mathbb{R}}}\nolimits$
  6. Sturm's theorem
  7. Properties of non-critical tangency value polynomials
  8. Tangency points and values over $\mathop{\mathrm{\overline{\mathbb{F}}}}\nolimits$
  9. Non-critical tangency value polynomials
  10. Computing $\varphi_{p,a}$
  11. The property of being T-good
  12. Definition and properties
  13. Genericity of the T-goodness
  14. A sufficient condition
  15. Deciding lower-boundedness
  16. ...and 4 more sections

Key Result

Theorem 2.1

Let $a\in\mathop{\mathrm{\mathbb{R}}}\nolimits^n$ be given. Then, the set $T_{\infty}(p,a)$ is finite. Moreover, the polynomial $p$ is lower-bounded if and only if

Theorems & Definitions (40)

  • Theorem 2.1: vui2008lojasiewicz, and pham2023tangencies
  • Theorem 2.2: Sturm's theorem
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Example 3.4
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • ...and 30 more