A Lower Bound on Determinantal Complexity
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The determinantal complexity of a polynomial P ∈𝔽[x_1, …, x_n] over a field 𝔽 is the dimension of the smallest matrix M whose entries are affine functions in 𝔽[x_1, …, x_n] such that P = Det(M). We prove that the determinantal complexity of the polynomial ∑_i = 1^n x_i^n is at least 1.5n - 3. For every n-variate polynomial of degree d, the determinantal complexity is trivially at least d, and it is a long standing open problem to prove a lower bound which is super linear in max{n,d}. Our result is the first lower bound for any explicit polynomial which is bigger by a constant factor than max{n,d}, and improves upon the prior best bound of n + 1, proved by Alper, Bogart and Velasco [ABV17] for the same polynomial.
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