A Note on the Computation of Multiplicities

Let f1, . . . , fr denote a system of polynomials in the polynomial ring P = k[x1, . . . , xd] such that 0 = (0, . . . , 0) ∈ V (f1, . . . , fr) ⊂ AK . Suppose that fR is an xR-primary ideal, where R = P(x). Then the Hilbert-Samuel multiplicity e0(f ;R) provides a certain information about the local structure of the affine variety V = V (f1, . . . , fr) as considered in Bézout’s theorem and related problems. The goal of our interest in this note is the following question: Suppose that a d-dimensional local Noetherian ring (A,m, k) contains the residue field k. Let x = x1, . . . , xd denote a system of parameters of A. Suppose that the polynomials f satisfy the above requirements. What is the relation between the multiplicities e0(f ;R) and e0(f ;A)? Recall that the system of parameters x in A is algebraically independent (cf. [2, Corollary 11.21]), so that P = k[x1, . . . , xd] is a polynomial subring of A. ∗The authors are grateful to the DAAD and the Slovak Ministry of Education (Grant Nr. 1/7658/20) for supporting the research concerning this article.