Spectrum of non-degenerate functions with simplicial Newton polytopes

TL;DR

This paper provides a detailed proof of Steenbrink's formula for the spectrum of non-degenerate functions with simplicial Newton polytopes, introduces the $\Gamma$-spectrum, and explores spectral properties of certain singularities.

Contribution

It offers a new proof of Steenbrink's formula, introduces the $\Gamma$-spectrum for simplicial functions, and extends spectral analysis to specific non-isolated surface singularities.

Findings

Proof of Steenbrink's formula for simplicial cases

Introduction of the $\Gamma$-spectrum as an approximation

Identification of cases where the Yomdin-Steenbrink formula fails

Abstract

We show a precise proof of Steenbrink's formula for the spectrum of convenient Newton non-degenerate functions, and prove the symmetry of combinatorial polynomials in the simplicial case. Combined with the modified Steenbrink conjecture for spectral pairs (that is, weighted spectrum) which is recently proved in that case, this simplifies quite a lot of their calculations in such a case. We also introduce the $\Gamma$-spectrum of simplicial convenient non-degenerate functions as a first approximation of the spectrum, generalizing Arnold's picture in the 2 variable case. Analyzing their difference, we can find simple formulas for weighted spectrum in the 3 or 4 variable case. This is proved by using the symmetry of combinatorial polynomials, and fails in the non-simplicial case. Combining these with the Yomdin-Steenbrink formula for the spectrum, we can prove a formula for the spectrum of certain non-isolated surface singularities with simplicial non-degenerate Newton boundaries. As a byproduct of these arguments, we find an example where the Yomdin-Steenbrink formula for spectral pairs does not hold because of fusions of compact faces under projections.