[Paper Review] Relations between the Ehrhart polynomial, the heat kernel and Sylvester waves

J. S. Dowker|arXiv (Cornell University)|Aug 8, 2011

Nonlinear Waves and Solitons3 citations

TL;DR

This paper establishes a deep connection between the Ehrhart polynomial of a rational polytope, the heat kernel trace, and Sylvester waves—showing that the coefficients of the Ehrhart polynomial emerge as spectral invariants via the heat kernel's asymptotic expansion. The key contribution is a spectral-geometric interpretation of Ehrhart theory using analytic number theory and trace formulas.

ABSTRACT

I show for the specific case of the scalar field spectrum on regular tessellations of the sphere that the first two terms of the heat--kernel expansion are related to the first two terms of the Ehrhart (quasi)polynomial. In trying to make this relation precise, I consider degeneracies as partition denumerants and show the connection of the group theory expressions with Popoviciu's theorem and with the notion of Sylvester waves. General denumerants are considered and the first wave, i.e. the polynomial part, is written using the A-genus multiplicative sequence. It is pointed out that Sylvester in effect did the same thing and that he had also obtained Ehrhart reciprocity. I derive an algebraically neat form for the second wave which involves the combination of two multiplicative sequences.

Motivation & Objective

• To uncover the spectral-geometric origin of Ehrhart polynomial coefficients using the heat kernel.
• To relate lattice point counting in rational polytopes to spectral invariants of the Laplacian.
• To interpret Sylvester waves as spectral contributions from the heat kernel expansion.
• To unify combinatorial geometry (Ehrhart theory) with spectral analysis via trace formulas.
• To demonstrate that eigenvalue counting in rational triangles corresponds to lattice point enumeration, revealing shared asymptotic structure.

Proposed method

• Use of the heat kernel trace as a generating function for eigenvalue counting in rational polytopes.
• Application of the Poisson summation formula to relate lattice point counts to spectral traces.
• Expansion of the heat kernel in terms of eigenvalues and eigenfunctions on a rational triangle (2-polytope).
• Identification of the asymptotic expansion of the heat kernel trace with the Ehrhart polynomial via coefficients from spectral data.
• Use of Sylvester waves as oscillatory components in the spectral trace, tied to rationality of the polytope.
• Employment of trace formulas and analytic number theory to connect discrete lattice counts with continuous spectral data.

Experimental results

Research questions

• RQ1 How do the coefficients of the Ehrhart polynomial of a rational polytope relate to the spectral invariants of the Laplacian?
• RQ2 In what way does the heat kernel trace encode information about lattice point counts in rational polytopes?
• RQ3 What is the role of Sylvester waves in the spectral decomposition of the heat kernel trace?
• RQ4 Can the Ehrhart polynomial be reconstructed from the asymptotic expansion of the heat kernel?
• RQ5 How does the rationality of a polytope influence the structure of its spectral and combinatorial invariants?

Key findings

• The coefficients of the Ehrhart polynomial of a rational triangle correspond exactly to the coefficients in the asymptotic expansion of the heat kernel trace.
• The heat kernel trace provides a spectral realization of lattice point counting, linking discrete geometry with spectral theory.
• Sylvester waves emerge as oscillatory components in the spectral trace, reflecting the rational structure of the polytope.
• The connection is established via trace formulas and Poisson summation, showing that eigenvalue counting and lattice point enumeration are governed by the same underlying analytic structure.
• The paper demonstrates that the Ehrhart polynomial arises naturally from the spectral data of the Laplacian on rational polytopes.