Motivated by questions concerning the multilinear and homogeneous complexity of the elementary symmetric polynomials, we prove the following results:

We first show that by making small modifications to the nonzero coefficients of the degree-$K$, $N$-variate elementary symmetric polynomial $\sigma_{N,K}$, one obtains a polynomial that can be computed by a monotone ... more >>>

In this paper, we initiate the study of deterministic PIT for $\Sigma^{[k]}\Pi\Sigma\Pi^{[\delta]}$ circuits over fields of any characteristic, where $k$ and $\delta$ are bounded. Our main result is a deterministic polynomial-time black-box PIT algorithm for $\Sigma^{[3]}\Pi\Sigma\Pi^{[\delta]}$ circuits, under the additional condition that one of the summands at the top $\Sigma$ ... more >>>

We prove a non-linear Edelstein-Kelly theorem for polynomials of constant degree, fully settling a stronger form of Conjecture 30 in Gupta (2014), and generalizing the main result of Peleg and Shpilka (STOC 2021) from quadratic polynomials to polynomials of any constant degree.

As a consequence of our result, we obtain ... more >>>