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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2508.07060 |
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| _version_ | 1866916890227507200 |
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| author | Keshari, Manoj K. Ojha, Debapriya Patra, Niladri Sekhar |
| author_facet | Keshari, Manoj K. Ojha, Debapriya Patra, Niladri Sekhar |
| contents | Let $F$ be a subfield of $\mathbb R$ and let $K$ be a basic closed semi-algebraic set in $\mathbb R$ with $\partial K\subset F$. Let $\mathcal N$ be the natural choice of generators of $K$. We show that if $f\in F[x]$ is $\geq 0$ on $K$, then $f$ can be written as $$f=\sum_{e\in\{0,1\}^s } a_eσ_e g^e $$ where $a_e\in F_{\geq 0}$, $σ_e\in \sum F[x]^2$ and $g^{e}=g_1^{e_1} \cdots g_s^{e_s}$. In other words, the preordering $T_{\mathcal N}$ of $F[x]$ is saturated. In case $F=\mathbb R$, this result is due to Kuhlmann and Marshall. As an application, we prove that if $K$ is compact, then $M_{\mathcal N}=T_{\mathcal N}=Pos(K)$. In other words, the quadratic module $M_{\mathcal N}$ of $F[x]$ is saturated. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_07060 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Descent problem for certificate of non-negativity on semi-algebraic sets Keshari, Manoj K. Ojha, Debapriya Patra, Niladri Sekhar Algebraic Geometry Commutative Algebra 14P05, 14P10, 11E25 Let $F$ be a subfield of $\mathbb R$ and let $K$ be a basic closed semi-algebraic set in $\mathbb R$ with $\partial K\subset F$. Let $\mathcal N$ be the natural choice of generators of $K$. We show that if $f\in F[x]$ is $\geq 0$ on $K$, then $f$ can be written as $$f=\sum_{e\in\{0,1\}^s } a_eσ_e g^e $$ where $a_e\in F_{\geq 0}$, $σ_e\in \sum F[x]^2$ and $g^{e}=g_1^{e_1} \cdots g_s^{e_s}$. In other words, the preordering $T_{\mathcal N}$ of $F[x]$ is saturated. In case $F=\mathbb R$, this result is due to Kuhlmann and Marshall. As an application, we prove that if $K$ is compact, then $M_{\mathcal N}=T_{\mathcal N}=Pos(K)$. In other words, the quadratic module $M_{\mathcal N}$ of $F[x]$ is saturated. |
| title | Descent problem for certificate of non-negativity on semi-algebraic sets |
| topic | Algebraic Geometry Commutative Algebra 14P05, 14P10, 11E25 |
| url | https://arxiv.org/abs/2508.07060 |