On the Thom-Boardman Symbols for Polynomial Multiplication Maps

TL;DR

This paper confirms a conjecture that the Thom-Boardman symbol for polynomial multiplication maps can be computed using the Euclidean algorithm applied to the degrees of the polynomials, simplifying a complex classification task.

Contribution

The paper proves that the Thom-Boardman symbol for polynomial multiplication maps can be obtained through Euclidean algorithm steps, confirming Varley's conjecture.

Findings

Thom-Boardman symbol computation reduces to Euclidean algorithm.

Confirmation of Varley's conjecture for polynomial multiplication maps.

Simplification of singularity classification for polynomial maps.

Abstract

The Thom-Boardman symbol was first introduced by Thom in 1956 to classify singularities of differentiable maps. It was later generalized by Boardman to a more general setting. Although the Thom-Boardman symbol is realized by a sequence of non-increasing, nonnegative integers, to compute those numbers is, in general, extremely difficult. In the case of polynomial multiplication maps, Robert Varley conjectured that computing the Thom-Boardman symbol for polynomial multiplication reduces to computing the successive quotients and remainders for the Euclidean algorithm applied to the degrees of the two polynomials. In this paper, we confirm this conjecture.