The Effective Lasserre's Perturbative Positivstellensatz
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arXiv
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| Format: | Preprint |
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2026
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| author | Klep, Igor Magron, Victor Schötz, Matthias |
| author_facet | Klep, Igor Magron, Victor Schötz, Matthias |
| contents | We study sum-of-squares (SOS) certificates for nonnegative polynomials $p$ on $\mathbb{R}^d$ and their implications for polynomial optimization over unbounded domains. Building on Lasserre's perturbation approach, we consider SOS representations of $p$ augmented by weighted polynomial tails of the form $\sum_{n=0}^N (x\cdot x)^n/(n!)^t$ for $0 < t < 1$. Our main result provides an explicit quantitative bound on the truncation order $N$ required to achieve an $\varepsilon$-accurate certificate. Using positivity properties of the Mehler kernel and techniques inspired by polynomial kernel methods, we show that $N$ grows polynomially in $1/\varepsilon$, with rate $N = O((\|p\|/\varepsilon)^{1/(1-t)})$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_13954 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Effective Lasserre's Perturbative Positivstellensatz Klep, Igor Magron, Victor Schötz, Matthias Optimization and Control Commutative Algebra Algebraic Geometry 90C26, 13J30, 90C22, 42C05, 14P10, 33C45, 41A25 We study sum-of-squares (SOS) certificates for nonnegative polynomials $p$ on $\mathbb{R}^d$ and their implications for polynomial optimization over unbounded domains. Building on Lasserre's perturbation approach, we consider SOS representations of $p$ augmented by weighted polynomial tails of the form $\sum_{n=0}^N (x\cdot x)^n/(n!)^t$ for $0 < t < 1$. Our main result provides an explicit quantitative bound on the truncation order $N$ required to achieve an $\varepsilon$-accurate certificate. Using positivity properties of the Mehler kernel and techniques inspired by polynomial kernel methods, we show that $N$ grows polynomially in $1/\varepsilon$, with rate $N = O((\|p\|/\varepsilon)^{1/(1-t)})$. |
| title | The Effective Lasserre's Perturbative Positivstellensatz |
| topic | Optimization and Control Commutative Algebra Algebraic Geometry 90C26, 13J30, 90C22, 42C05, 14P10, 33C45, 41A25 |
| url | https://arxiv.org/abs/2603.13954 |