  The determinantal complexity of a polynomial $P \in \mathbb{F}[x_1, \ldots,
x_n]$ over a field $\mathbb{F}$ is the dimension of the smallest matrix $M$
whose entries are affine functions in $\mathbb{F}[x_1, \ldots, x_n]$ such that
$P = Det(M)$. We prove that the determinantal complexity of the polynomial
$\sum_{i = 1}^n x_i^n$ 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.
