Semicontinuity of the spectrum at infinity

TL;DR

This paper proves that the spectrum at infinity for a family of weakly tame regular functions on an affine manifold varies semicontinuously, extending Varchenko's concept to a broader class of functions.

Contribution

It establishes the semicontinuity of the spectrum at infinity for analytic families of weakly tame regular functions, generalizing previous results.

Findings

Spectrum at infinity is semicontinuous in the family.

Applicable to weakly tame regular functions on affine manifolds.

Extends Varchenko's semicontinuity concept.

Abstract

We prove that, for an analytic family of ``weakly tame'' regular functions on an affine manifold, the spectrum at infinity of each function of the family is semicontinuous in the sense of Varchenko.