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Rational certificates of non-negativity on semialgebraic subsets of cylinders

Gabriela Jeronimo, Daniel Perrucci

Abstract

Let $g_1,\dots, g_s \in \mathbb{R}[X_1,\dots, X_n,Y]$ and $S = \{(\bar{x},y)\in \mathbb{R}^{n+1} \mid g_1(\bar{x},y) \ge 0, \dots, g_s(\bar{x}, y) \ge 0\}$ be a non-empty, possibly unbounded, subset of a cylinder in $\mathbb{R}^{n+1}$. Let $f \in \mathbb{R}[X_1, \dots, X_n, Y]$ be a polynomial which is positive on $S$. We prove that, under certain additional assumptions, for any non-constant polynomial $q \in \mathbb{R}[Y]$ which is positive on $\mathbb{R}$, there is a certificate of the non-negativity of $f$ on $S$ given by a rational function having as numerator a polynomial in the quadratic module generated by $g_1, \dots, g_s$ and as denominator a power of $q$.

Rational certificates of non-negativity on semialgebraic subsets of cylinders

Abstract

Let and be a non-empty, possibly unbounded, subset of a cylinder in . Let be a polynomial which is positive on . We prove that, under certain additional assumptions, for any non-constant polynomial which is positive on , there is a certificate of the non-negativity of on given by a rational function having as numerator a polynomial in the quadratic module generated by and as denominator a power of .
Paper Structure (3 sections, 5 theorems, 34 equations)

This paper contains 3 sections, 5 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Assumptions and main result
  3. Proof of the main result

Key Result

Theorem 1

Let $\mathbf{g}:=g_1,\dots, g_s$ and $f$ be polynomials in $\mathbb{R}[\bar{X},Y]$ such that $f>0$ on $S$ and Assumptions assump:arq, assump:g_i and assump:f hold. Let $q \in \mathbb{R}[Y]$ be a non-constant polynomial which is positive on $\mathbb{R}$. Then, there exists $M\in \mathbb{Z}_{\ge 0}$ s

Theorems & Definitions (9)

  • Example 1
  • Theorem 1
  • Example 2
  • Lemma 2
  • Proposition 3
  • Remark 4
  • Lemma 5
  • Lemma 6
  • Remark 7