Computing the Elementary Symmetric Polynomials in Positive Characteristics

We first extend the results of Chatterjee,Kumar,Shi,Volk(Computational Complexity 2022) by showing that the degree $d$ elementary symmetric polynomials in $n$ variables have formula lower bounds of $\Omega(d(n-d))$ over fields of positive characteristic. Then, we show that the results of the universality of linear projections of elementary symmetric polynomials from Shpilka(JCSS 2002) and of border fan-in two $\Sigma\Pi\Sigma$ circuits from Kumar(ACM TOCT 2020) over zero characteristic fields do not extend to fields of positive characteristic. In particular, we show that *There are polynomials that cannot be represented as linear projections of the elementary symmetric polynomials(in fact, we show linear lower bounds over the size of the sum of such linear projections) and *There are polynomials that cannot be computed by border depth-$3$ circuits of top fan-in $k$, called $\overline{\Sigma^{[k]}\Pi\Sigma}$, for $k = o(n)$. To prove the first result, we consider a geometric property of the elementary symmetric polynomials, namely, the set of all points in which the polynomial and all of its first-order partial derivatives vanish. It was previously shown that the dimension of this space was exactly $d-2$ for fields of zero characteristic. We extend this to fields of positive characteristic by showing that this dimension must be between $d-2$ and $d-1$. In fact, we provide some criterion where it is $d-2$ and others where it is $d-1$. Then, to consider the border top fan-in of linear projections of the elementary symmetric polynomials and border depth-$3$ circuits(sometimes called border affine Chow rank), we show that it is sufficient to consider the border top fan-in of the sum of such linear projections of the elementary symmetric polynomials. This is done by an explicit construction of a 'metapolynomial,' meaning that this result also applies in the border setting.