Making Non-Negative Polynomials into Sums of Squares
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909678441594880 |
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| author | di Dio, Philipp J. |
| author_facet | di Dio, Philipp J. |
| contents | We investigate linear operators $A:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}[x_1,\dots,x_n]$. We give explicit operators $A$ such that, for fixed $d\in\mathbb{N}_0$ and closed $K\subseteq\mathbb{R}^n$, $e^A\mathrm{Pos}(K)_{\leq 2d}\subseteq\sum\mathbb{R}[x_1,\dots,x_n]_{\leq d}^2$. We give an explicit operator $A$ such that $e^A\mathrm{Pos}(\mathbb{R}^n)\subseteq\sum\mathbb{R}[x_1,\dots,x_n]^2$. For $K\subseteq\mathbb{R}^n$, we give a condition such that $A$ exists with $e^A\mathrm{Pos}(K)\subseteq\sum\mathbb{R}[x_1,\dots,x_n]^2$. We show that, for compact $K\subseteq\mathbb{R}^n$, there is no bijective linear operator $T:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}[x_1,\dots,x_n]$ with $T\mathrm{Pos}(K)\subseteq\sum\mathbb{R}[x_1,\dots,x_n]^2$. In the framework of regular Fréchet Lie groups and Lie algebras we investigate the linear operators $A$ such that $e^{tA}:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}[x_1,\dots,x_n]$ is well-defined for all $t\in\mathbb{R}$. We give a three-line-proof of Stochel's Theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_16321 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Making Non-Negative Polynomials into Sums of Squares di Dio, Philipp J. Algebraic Geometry Functional Analysis Group Theory Operator Algebras Optimization and Control Primary 11E25, Secondary 13J30, 44A60, 47A57, 90C22 We investigate linear operators $A:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}[x_1,\dots,x_n]$. We give explicit operators $A$ such that, for fixed $d\in\mathbb{N}_0$ and closed $K\subseteq\mathbb{R}^n$, $e^A\mathrm{Pos}(K)_{\leq 2d}\subseteq\sum\mathbb{R}[x_1,\dots,x_n]_{\leq d}^2$. We give an explicit operator $A$ such that $e^A\mathrm{Pos}(\mathbb{R}^n)\subseteq\sum\mathbb{R}[x_1,\dots,x_n]^2$. For $K\subseteq\mathbb{R}^n$, we give a condition such that $A$ exists with $e^A\mathrm{Pos}(K)\subseteq\sum\mathbb{R}[x_1,\dots,x_n]^2$. We show that, for compact $K\subseteq\mathbb{R}^n$, there is no bijective linear operator $T:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}[x_1,\dots,x_n]$ with $T\mathrm{Pos}(K)\subseteq\sum\mathbb{R}[x_1,\dots,x_n]^2$. In the framework of regular Fréchet Lie groups and Lie algebras we investigate the linear operators $A$ such that $e^{tA}:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}[x_1,\dots,x_n]$ is well-defined for all $t\in\mathbb{R}$. We give a three-line-proof of Stochel's Theorem. |
| title | Making Non-Negative Polynomials into Sums of Squares |
| topic | Algebraic Geometry Functional Analysis Group Theory Operator Algebras Optimization and Control Primary 11E25, Secondary 13J30, 44A60, 47A57, 90C22 |
| url | https://arxiv.org/abs/2506.16321 |