Asian Journal of Mathematics

Volume 16 (2012)

Number 3

On the Thom-Boardman symbols for polynomial multiplication maps

Pages: 367 – 386



Jiayuan Lin (SUNY Canton)

Janice Wethington (U.S. Department of Defense)


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 Varley’s conjecture.


Thom-Boardman symbols, polynomial multiplication maps, Toeplitz matrices

2010 Mathematics Subject Classification

14J17, 32S10, 58K20, 58K40

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