On the Computation of Minimal Reduction

Let P := k [X,Y, Z] be a polynomial ring over an algebraic closed field k and (X, Y , Z, XY Z, XY Z ) · k [X,Y, Z] an (X,Y, Z)-primary ideal in P , (m, n, l, a, b, c, d, e, f are integers). The ideal Q=(X, Y , Z, XY Z, XY Z ) ·R is (X,Y, Z) · R-primary ideal in the local ring R = k [X,Y, Z](x,y,z). In this short note we give a formula for the calculation of Samuel multiplicity e0(Q,R) of the ideal Q in R. Remark, that the multiplicity e0(Q,R) is the leading coefficient in the Hilbert-Samuel polynomial P (n) = l(R/Q), where l(R/Q) is the length of the R-module R/Q. We use the notion of a reduction of ideal for the proof of a main theorem. We say, that the ideal q is a reduction of the m-primary ideal q in the local ring (A,m), if q ⊂ q and for same integer n ∈ N it holds q·qn=qn+1. If q is the reduction of the ideal q in A then we know that e0(q, A) = (q, A) [5, Theorem 1]. Let’s formulate the first statement of this note. For the monomial ideal Q = (X, Y , Z, XY Z, XY Z ) ·R we set