57 research outputs found

    On Newton polytopes and growth properties of multivariate polynomials

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    Many interesting properties of polynomials are closely related to the geometry of their Newton polytopes. In this dissertation thesis, we analyze the growth properties of real multivariate polynomials in terms of their so-called Newton polytopes at infinity. We show some applications of our results, relate them to the existing literature and illustrate them with several examples

    Convergence rates of Gibbs measures with degenerate minimum

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    We study convergence rates for Gibbs measures, with density proportional to e−f(x)/te^{-f(x)/t}, as t→0t \rightarrow 0 where f:Rd→Rf : \mathbb{R}^d \rightarrow \mathbb{R} admits a unique global minimum at x⋆x^\star. We focus on the case where the Hessian is not definite at x⋆x^\star. We assume instead that the minimum is strictly polynomial and give a higher order nested expansion of ff at x⋆x^\star, which depends on every coordinate. We give an algorithm yielding such a decomposition if the polynomial order of x⋆x^\star is no more than 88, in connection with Hilbert's 17th17^{\text{th}} problem. However, we prove that the case where the order is 1010 or higher is fundamentally different and that further assumptions are needed. We then give the rate of convergence of Gibbs measures using this expansion. Finally we adapt our results to the multiple well case.Comment: 25 page
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