Demonstrates the existence of subtensors with high partition rank in order-d tensors, indicating polynomial function relationships.
We prove that for every positive integer d ≥ 2 there exist polynomial functions Fd, Gd : N → N such that for each positive integer r, every order-d tensor T over an arbitrary field and with partition rank at least Gd(r) contains a Fd(r) × · · · × Fd(r) subtensor with partition rank at least r. We then deduce analogous results on the Schmidt rank of polynomials in zero or high characteristic.
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Draisma et al. (2024) studied this question.