Powers of sums and their associated primes

Let $A, B$ be polynomial rings over a field $k$, and $I\subseteq A, J\subseteq B$ proper homogeneous ideals. We analyze the associated primes of powers of $I+J\subseteq A\otimes_k B$ given the data on the summands. The associated primes of large enough powers of $I+J$ are determined. We then answer positively a question about the persistence property of $I+J$ in many new cases.

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