A Lower Bound on Determinantal Complexity

The determinantal complexity of a polynomial P∈ F[x₁, … , x_n] over a field F is the dimension of the smallest matrix M whose entries are affine functions in F[x₁, … , x_n] such that P= Det (M). We prove that the determinantal complexity of the polynomial ∑i=1nxin is at least 1.5 n- 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 et al. (2017) for the same polynomial.