Invariant equations defining coincident root loci (Q1764454)

This paper deals with a very classical and important algebraic problem for univariate polynomials: given a polynomial \(F\) with complex coefficients of degree \(n\) and a partition \((\lambda_1, \dots, \lambda_r)\) of \(n\), determine explicit algebraic conditions on the coefficients of \(F\) such that this polynomial may have roots with multiplicities \(\lambda_1\), \dots, \(\lambda_r\). The author gives an invariant theoretic solution to this problem: he exhibits a list of covariants of \(F\) whose simultaneous vanishing provides a necessary and sufficient condition. The construction of these covariants is combinatorial and involves the study of associated graphs on \(n\) vertices. The basic idea already appeared in papers of Arthur Cayley, but the present paper provides improvements on Cayley's work.