Repr\'esentations d\'eterminantales effectives des polyn\^omes univari\'es par les matrices fl\`eches

Abstract

We first show the existence of an effective determinantal representation for any univariate polynomial with real coefficients. Then, we more precisely establish that any univariate polynomial with real coefficients has an effective determinantal representation with signature (r+s,s) if and only if it has at least r real roots with multiplicity. The effective determinantal representations we construct used arrow matrices.