From Hodge Theory for Tame Functions to Ehrhart Theory for Polytopes

We study the interplay between Sabbah's mixed Hodge structure for tame regular functions and Ehrhart theory for polytopes. We first analyze the Poincar & eacute; polynomial of the Hodge filtration of this mixed Hodge structure (we call this Poincar & eacute; polynomial the theta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document}-vector). Using the symmetry of the Hodge numbers involved, we show that it shares many properties with the h & lowast;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h<^>*$$\end{document}-vector of a polytope. For instance, we define from the theta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document}-vector the Hodge-Ehrhart polynomial of a general tame function and we show that it satisfies a reciprocity law, analogous to the one satisfied by the Ehrhart polynomial of a polytope. We study the roots of this Hodge-Ehrhart polynomial, in particular their distribution around some critical lines. Using techniques coming from singularity theory, we also show a Thom-Sebastiani type theorem for the theta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document}-vector. Finally, we offer some linear inequalities among the coefficients of the theta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document}-vectors which could be helpful to test if a polynomial is a theta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document}-vector or not. In the very particular case of convenient and nondegenerate Laurent polynomials, we show (using the Brieskorn lattice and the V-filtration) that the previous results agree with the classical ones in combinatorics and we emphasize various combinatorial properties of Sabbah's Hodge numbers: on the way, this provides an alternative interpretation of prior results about the (limit) Hodge numbers of hypersurfaces in a torus obtained in a different framework by Danilov-Khovanskii and more recently by Katz-Stapledon.