The v-function of powers of sums of ideals

Abstract

Let K be a field, I a graded ideal of R=K[x_1,...,x_n] and J a graded ideal of T=K[y_1,...,y_m]. Set S = R \otimes_K T and let L = IS + JS. The behaviour of the v-function v(L^k) in terms of the v-functions v(I^k) and v(J^k) is investigated. When I and J are monomial ideals, we describe v(L^k), giving an explicit formula involving the local v-numbers v_p(I^k) and v_q(J^k).