Leading Coefficients and the Multiplicity of Known Roots

TL;DR

This paper presents an explicit algorithm to determine the multiplicities of known roots of a monic polynomial over fields of characteristic zero, with applications to hypergraph characteristic polynomials and numerical stability.

Contribution

It introduces a novel algorithm that uses leading coefficients to compute root multiplicities, extending to non-algebraically closed fields and ensuring numerical stability over complex numbers.

Findings

Algorithm accurately computes root multiplicities from leading coefficients.

Extension to non-algebraically closed fields using minimal polynomials.

Application to deriving characteristic polynomials of hypergraphs.

Abstract

We show that a monic univariate polynomial over a field of characteristic zero, with $k$ distinct non-zero known roots, is determined by its $k$ proper leading coefficients by providing an explicit algorithm for computing the multiplicities of each root. We provide a version of the result and accompanying algorithm when the field is not algebraically closed by considering the minimal polynomials of the roots. Furthermore, we show how to perform the aforementioned algorithm in a numerically stable manner over $\mathbb{C}$, and then apply it to obtain new characteristic polynomials of hypergraphs.