On Newton polytopes and growth properties of multivariate polynomials

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

Atypical points at infinity and algorithmic detection of the bifurcation locus of real polynomials

We show that the variation of the topology at infinity of a two-variable polynomial function is localisable at a finite number of "atypical points" at infinity. We construct an effective algorithm with low complexity in order to detect sharply the bifurcation values of the polynomial function.Comment: minor edits in this last version before print; to appear in Math.

Convergence rates of Gibbs measures with degenerate minimum

We study convergence rates for Gibbs measures, with density proportional to $e^{-f(x)/t}$, as $t \rightarrow 0$ where $f : \mathbb{R}^d \rightarrow \mathbb{R}$ admits a unique global minimum at $x^\star$. We focus on the case where the Hessian is not definite at $x^\star$. We assume instead that the minimum is strictly polynomial and give a higher order nested expansion of $f$ at $x^\star$, which depends on every coordinate. We give an algorithm yielding such a decomposition if the polynomial order of $x^\star$ is no more than $8$, in connection with Hilbert's $17^{\text{th}}$ problem. However, we prove that the case where the order is $10$ 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