On Łojasiewicz Inequalities and the Effective Putinar's Positivstellensatz
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arXiv
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| Format: | Preprint |
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2022
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| author | Baldi, Lorenzo Mourrain, Bernard Parusinski, Adam |
| author_facet | Baldi, Lorenzo Mourrain, Bernard Parusinski, Adam |
| contents | The representation of positive polynomials on a semi-algebraic set in terms of sums of squares is a central question in real algebraic geometry, which the Positivstellensatz answers. In this paper, we study the effective Putinar's Positivestellensatz on a compact basic semi-algebraic set $S$ and provide a new proof and new improved bounds on the degree of the representation of positive polynomials. These new bounds involve a parameter $ε$ measuring the non-vanishing of the positive function, the constant $\mathfrak{c}$ and exponent $L$ of a Łojasiewicz inequality for the semi-algebraic distance function associated to the inequalities $\mathbf{g} = (g_1, \dots , g_r)$ defining $S$. They are polynomial in $\mathfrak{c}$ and $ε^{-1}$ with an exponent depending only on $L$. We analyse in details the Łojasiewicz inequality when the defining inequalities $\mathbf g$ satisfy the Constraint Qualification Condition. We show that, in this case, the Łojasiewicz exponent $L$ is $1$ and we relate the Łojasiewicz constant $\mathfrak{c}$ with the distance of $\mathbf g$ to the set of singular systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_09551 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On Łojasiewicz Inequalities and the Effective Putinar's Positivstellensatz Baldi, Lorenzo Mourrain, Bernard Parusinski, Adam Commutative Algebra Algebraic Geometry Optimization and Control The representation of positive polynomials on a semi-algebraic set in terms of sums of squares is a central question in real algebraic geometry, which the Positivstellensatz answers. In this paper, we study the effective Putinar's Positivestellensatz on a compact basic semi-algebraic set $S$ and provide a new proof and new improved bounds on the degree of the representation of positive polynomials. These new bounds involve a parameter $ε$ measuring the non-vanishing of the positive function, the constant $\mathfrak{c}$ and exponent $L$ of a Łojasiewicz inequality for the semi-algebraic distance function associated to the inequalities $\mathbf{g} = (g_1, \dots , g_r)$ defining $S$. They are polynomial in $\mathfrak{c}$ and $ε^{-1}$ with an exponent depending only on $L$. We analyse in details the Łojasiewicz inequality when the defining inequalities $\mathbf g$ satisfy the Constraint Qualification Condition. We show that, in this case, the Łojasiewicz exponent $L$ is $1$ and we relate the Łojasiewicz constant $\mathfrak{c}$ with the distance of $\mathbf g$ to the set of singular systems. |
| title | On Łojasiewicz Inequalities and the Effective Putinar's Positivstellensatz |
| topic | Commutative Algebra Algebraic Geometry Optimization and Control |
| url | https://arxiv.org/abs/2212.09551 |