The spectral mapping theorem, norms on rings, and resultants.
From the paper: It is a well-known and useful result in spectral theory of complex finite dimensional vector spaces that if the characteristic polynomial of an \(n\times n\)-matrix \(M\) splits as \(P_M (t)=\det(tI_n-M)=\prod^n_{i=1}(t-\lambda_i)\) then, for any polynomial \(F(x)\), we have that \(\det(tI_n-F(M))=\prod^n_{i=1}(t-F(\lambda_i))\). We call this result the Spectral Mapping Theorem, because it is similar to the Spectral Mapping Theorem for Banach algebras. We give here a short, simple and self-contained proof of the Spectral Mapping Theorem for matrices with entries in an arbitrary commutative ring. The result is placed in the wider framework of norms on algebras. It is shown that the Spectral Mapping Theorem follows from a uniqueness result for norms on polynomial rings in one variable. The results are used to generalize classical formulas for the resultant of polynomials.
scientific article; zbMATH DE number 1736564