On Vanishing Properties of Polynomials on Symmetric Sets of the Boolean Cube, in Positive Characteristic

NEW

Problem: Reconstructed conjecture: let $K$ be a field of characteristic $p>0$. For $E\subseteq[0,n]$, write

Let $\operatorname{Zcl}_{n,d}(E)$ be the set of weights $j\in[0,n]$ such that every polynomial over $K$ of degree $\le d$ vanishing on $\underline E$ also vanishes on the whole $j$-th layer. Let $\operatorname{symcl}_{n,d}(E)$ be the analogous closure using only symmetric polynomials of degree $\le d$.

Theorem 1.6 of Srinivasan--Venkitesh proves, for $\ell=\lceil\log_p(d+1)\rceil$, that if $n\ge 4p^\ell-1$, then

Remark 4.6 says the authors believe the same assertion holds for smaller $n$. Thus the natural formal conjecture is the same equality with the hypothesis $n\ge4p^\ell-1$ removed.

Result: The conjecture is true.

It is enough to work over $\mathbb F_p$, since all relevant evaluation matrices have entries in $\mathbb F_p$, and their ranks are unchanged after extending scalars to any field of characteristic $p$.

Assume $E\subseteq[d,n-d]$. If $E=\varnothing$, both closures are empty because the constant polynomial $1$ is allowed. If $d=0$, both closures are either empty or all of $[0,n]$, according as $E$ is empty or nonempty. So assume $d\ge1$ and $E\ne\varnothing$, whence $d\le n/2$.

For $s\in[0,n]$, define

A weight $j$ lies in $\operatorname{symcl}_{n,d}(E)$ iff

We use Wilson’s diagonal form for inclusion matrices. In the range $E\subseteq[d,n-d]$, it implies the following standard criterion: for $j\le n/2$,

iff for every $0\le t\le \min(d,j)$,

lies in

Now suppose $j\in\operatorname{symcl}_{n,d}(E)$. If $j>n/2$, replace $E$ by $n-E$ and $j$ by $n-j$; both Z-closure and symmetric closure are preserved under the coordinate-complement map $x\mapsto 1-x$. Thus it suffices to handle $j\le n/2$.

By (1), every $\mathbb F_p$-linear combination of $\binom Z0,\ldots,\binom Zd$ vanishing on $E$ also vanishes at $j$. Fix $t\le\min(d,j)$. If (2) failed, then some coefficients $c_0,\ldots,c_{d-t}$ would satisfy

but

The function

is an integer-valued polynomial of degree $\le d-t$, hence is an $\mathbb F_p$-linear combination of $\binom Z0,\ldots,\binom Zd$. It vanishes on $E$ but not at $j$, contradicting (1). Therefore (2) holds for every $t$, so $j\in\operatorname{Zcl}_{n,d}(E)$.

Since always $\operatorname{Zcl}_{n,d}(E)\subseteq\operatorname{symcl}_{n,d}(E)$, equality follows for all $n$, with no lower bound on $n$.

Verification audit: the only removed hypothesis is $n\ge4p^\ell-1$; $E\subseteq[d,n-d]$, the degree bound, and positive characteristic are unchanged. Degenerate cases $E=\varnothing$ and $d=0$ are covered. The proof uses Wilson’s theorem only in its standard valid range $r\le d\le i,n-i$ for $i\in E$, and reduces $j>n/2$ by complement symmetry.

Citation: Exact all-$n$ statement not found in the cited paper. Ingredient used: R. M. Wilson, “A diagonal form for the incidence matrices of $t$-subsets vs. $k$-subsets,” European J. Combin. 11 (1990), 609–615. Original problem source: Srinivasan--Venkitesh, arXiv:2111.05445, Remark 4.6.