Let F_q be a finite field, let 3≤ k≤ q-2 (or 4≤ k≤ q-3 if q is even), and let h be a complete symmetric polynomial of positive degree m in k variables. Let N_q(h) count all zeros of h in F_q^k, and let N_q^(h) count those with pairwise distinct coordinates. Low-degree distinct-coordinate zero conjecture. If 1≤ m≤ q-k, then N_q(h)≥ N_q^(h)≥ k!. This is presented as a special case of the preceding conjecture, since the reduction condition is automatic in this degree range; the source gives no general resolution.

Low-degree distinct-coordinate zero conjecture for complete symmetric polynomials

The conjecture predicts at least $k!$ zeros with pairwise distinct coordinates for every complete symmetric polynomial of degree $m$ satisfying $1\leq m\leq q-k$ . The cited paper presents this as a special case of a broader conjecture; no proposer or date is identified in the retrieved sources.

Known results

The broader conjecture, and hence the stated low-degree case where applicable, is proved when $k\leq p$ , with $p$ the characteristic of $\mathbb{F}_q$ .
It is also proved when $k\geq \lfloor (q+1)/2\rfloor$ .
For odd $q$ , the paper gives bounds including $N_q(h)\geq 6q^{k-3}$ , but the retrieved material does not show that this settles every low-degree instance.

Current status (as of August 2026): The conjecture is settled in the cited parameter ranges $k\leq p$ or $k\geq\lfloor (q+1)/2\rfloor$ , but remains open for the rest of the allowed range; no complete proof or counterexample was found.

Low-degree distinct-coordinate zero conjecture for complete symmetric polynomials

Primary source

Progress summary

Solutions 0