Integral closures of powers of sums of ideals.

Let k be a field, let A and B be polynomial rings over k , and let S = A ⊗ k B . Let I ⊆ A and J ⊆ B be monomial ideals. We establish a binomial expansion for rational powers of I + J ⊆ S in terms of those of I and J. Particularly, for u ∈ Q + , we prove that (I + J) u = ∑ 0 ≤ ω ≤ u , ω ∈ Q I ω J u - ω , and that the sum on the right-hand side is a finite sum. This finite sum can be made more precise using jumping numbers of rational powers of I and J. We further give sufficient conditions for this formula to hold for the integral closures of powers of I + J in terms of those of I and J. Under these conditions, we provide explicit formulas for the depth and regularity of (I + J) k ¯ in terms of those of powers of I and J. [ABSTRACT FROM AUTHOR]