Jumps of Milnor numbers of Brieskorn-Pham singularities in non-degenerate families

TL;DR

This paper provides an explicit formula and an inductive algorithm to compute the minimal non-zero change in Milnor numbers for non-degenerate deformations of certain singularities, specifically Brieskorn-Pham types.

Contribution

It introduces a novel inductive algorithm based on Diophantine equations to calculate the non-degenerate jump of Milnor numbers for specific singularities.

Findings

Derived a formula for the non-degenerate jump of Milnor numbers.

Developed an inductive algorithm using Diophantine equations.

Applied the method to Brieskorn-Pham singularities.

Abstract

The jump of the Milnor number of an isolated singularity $f_{0}$ is the minimal non-zero difference between the Milnor numbers of $f_{0}$ and one of its deformation $(f_{s}).$ In the case $f_{s}$ are non-degenerate singularities we call the jump non-degenerate. We give a formula (an inductive algorithm using diophantine equations) for the non-degenerate jump of $f_{0}$ in the case $f_{0}$ is a convenient singularity with only one $(n-1)$-dimensional face of its Newton diagram which equivalently (in our problem) can be replaced by the Brieskorn-Pham singularities.