On the structure of Thom polynomials of singularities

TL;DR

This paper introduces a product rule for Thom polynomials of singularities, allowing their calculation via associated Thom series linked to local algebras, thus advancing the understanding of their structure.

Contribution

It establishes a product rule for Thom polynomials and defines Thom series that encode all singularities with a given local algebra, enabling systematic computation.

Findings

Product rule for Thom polynomials derived.

Thom series encode all singularities with a given local algebra.

Method simplifies calculation of Thom polynomials for complex singularities.

Abstract

Thom polynomials of singularities express the cohomology classes dual to singularity submanifolds. A stabilization property of Thom polynomials is known classically, namely that trivial unfolding does not change the Thom polynomial. In this paper we show that this is a special case of a product rule. The product rule enables us to calculate the Thom polynomials of singularities if we know the Thom polynomial of the product singularity. As a special case of the product rule we define a formal power series (Thom series, Ts_Q) associated with a commutative, complex, finite dimensional local algebra Q, such that the Thom polynomial of {\em every} singularity with local algebra Q can be recovered from Ts_Q.