On the multiplicity of zeros of polynomials over arbitrary finite dimensional k-algebras
The author continues the study of zeros begun by \textit{H. Röhrl} [ibid. 25, 359-390 (1978; Zbl 0425.17001)]. Let \(A\subseteq B\) be m-ary algebras over a field k. Let P(x) be in the algebra of formal polynomials over A. Then P(x) has a natural image \(P_ A\) in the algebra of polynomial functions over A. The main results are: (2.2) For k algebraically closed, the number of zeros of \(P_ A\) is either infinite or at most \(n^ d\), where n is the degree of \(P_ A\) and d is the vector space dimension of A. - (4.2) For k infinite, an inequality relating \(m(P_ A:a)\), the multiplicity of a in A as a zero of \(P_ A\), to \(m(P_ B:a)\). - (5.1) Extension of the base field k leaves \(m(P_ A:a)\) unchanged.
scientific article; zbMATH DE number 3929117