Critical points, critical values, and a determinant identity for complex polynomials

Given any $n$-tuple of complex numbers, one can easily define a canonical polynomial of degree $n+1$ that has the entries of this $n$-tuple as its critical points. In 2002, Beardon, Carne, and Ng studied a map $\theta \colon \mathbb {C}^n\to \mathbb {C}^n$ which outputs the critical values of the canonical polynomial constructed from the input, and they proved that this map is onto. Along the way, they showed that $\theta$ is a local homeomorphism whenever the entries of the input are distinct and nonzero, and, implicitly, they produced a polynomial expression for the Jacobian determinant of $\theta$. In this article we extend and generalize both the local homeomorphism result and the elegant determinant identity to analogous situations where the critical points occur with multiplicities. This involves stratifying $\mathbb {C}^n$ according to which coordinates are equal and generalizing $\theta$ to a similar map $\mathbb {C}^\ell \to \mathbb {C}^\ell$ where $\ell$ is the number of distinct critical points. The more complicated determinant identity that we establish is closely connected to the multinomial identity known as Dyson’s conjecture.