Rational certificates of non-negativity on semialgebraic subsets of cylinders
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| author | Jeronimo, Gabriela Perrucci, Daniel |
| author_facet | Jeronimo, Gabriela Perrucci, Daniel |
| contents | 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 $q$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_01636 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Rational certificates of non-negativity on semialgebraic subsets of cylinders Jeronimo, Gabriela Perrucci, Daniel Algebraic Geometry 12D15, 13J30, 14P10 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 $q$. |
| title | Rational certificates of non-negativity on semialgebraic subsets of cylinders |
| topic | Algebraic Geometry 12D15, 13J30, 14P10 |
| url | https://arxiv.org/abs/2305.01636 |