On the Thom-Boardman symbols for polynomial multiplication maps (Q1932240)

The multiplication of monic polynomials of given respective degrees \(n\) and \(r\) defines in a natural way a map \( \mu_{n,r}: {\mathbb C}^n\times {\mathbb C}^r\to {\mathbb C}^{n+r}\) expressing the coefficients of the product as a function of the coefficients of the factors. According to a conjecture of Varley, the Thom-Boardman symbol of the ideal generated by the components of \(\mu_{n,r}\) in the local ring of differentiable functions at the origin can be expressed in terms of successive quotients and remainders for the Euclidean algorithm applied to \(n\) and \(r\). In the present paper, the authors prove Varley's conjecture.