The v-function of powers of sums of ideals

Let K be a field, I⊂ R=K[x₁,,xₙ] and J⊂ T=K[y₁,,yₘ] be graded ideals. Set S=R⊗KT and let $L=IS+JS$. The behaviour of the v-function v(Lᵏ) in terms of the v-functions v(Iᵏ) and v(Jᵏ) is investigated. When I and J are monomial ideals, we describe v(Lᵏ), giving an explicit formula involving v(Iᵏ) and v(Jᵏ).