Let w=(w_0,w_1,ots,w_m)∈N_≥ 1^m+1, let P(w) be the weighted projective space over F_q, and let e_q(d;w_0,w_1,…,w_m) denote the maximum number of zeros on P(w)(F_q) of a weighted-homogeneous polynomial of weighted degree d. Write w'=(w_1,…,w_m) and p_r=(q^r+1-1)/(q-1). Aubry–Castryck–Ghorpade–Lachaud–O'Sullivan–Ram conjecture. If w_0=1 and operatornamelcm(w')| d, then e_q(d;1,w_1,…,w_m)=minp_m,fracdw_1q^m-1+p_m-2. The paper proves this conjecture, establishing the weighted analogue of the Serre bound with equality under the stated divisibility condition; the equal-weight case was previously conjectured by Tsfasman and proved by Serre and Sørensen.

Aubry–Castryck–Ghorpade–Lachaud–O'Sullivan–Ram conjecture for weighted projective spaces

Let $w=(w_0,w_1,ots,w_m)\in\mathbb{N}_{\geq 1}^{m+1}$ , let $\mathbb{P}(w)$ be the weighted projective space over $\mathbb{F}_q$ , and let $e_q(d;w_0,w_1,\dots,w_m)$ denote the maximum number of zeros on $\mathbb{P}(w)(\mathbb{F}_q)$ of a weighted-homogeneous polynomial of weighted degree $d$ . Write $w'=(w_1,\dots,w_m)$ and $p_r=(q^{r+1}-1)/(q-1)$ . Aubry–Castryck–Ghorpade–Lachaud–O'Sullivan–Ram conjecture. If $w_0=1$ and $\operatorname{lcm}(w')\mid d$ , then

e_q(d;1,w_1,\dots,w_m)=\min\left\{p_m,\frac{d}{w_1}q^{m-1}+p_{m-2}\right\}.

The paper proves this conjecture, establishing the weighted analogue of the Serre bound with equality under the stated divisibility condition; the equal-weight case was previously conjectured by Tsfasman and proved by Serre and Sørensen.

Sources & referencesView supporting material

Primary source

Jade Nardi and Rodrigo San-José, “Maximum number of zeroes of polynomials on weighted projective spaces over a finite field”, arXiv:2507.22597 (2026).

1 week ago

Progress summary

Refreshed 4 days ago

Claimed solved

A 2025 preprint claims to prove the conjecture, but independent confirmation of the proof has not been found.

The conjecture predicts the exact maximum number of finite-field zeros for weighted hypersurfaces when the first weight is $1$ and the degree satisfies the stated divisibility condition. It was formulated by Aubry, Castryck, Ghorpade, Lachaud, O'Sullivan, and Ram in 2017.

Known results

Aubry, Castryck, Ghorpade, Lachaud, O'Sullivan, and Ram (2017) established the upper bound and conjectured sharpness under $w_0=1$ and $\operatorname{lcm}(w')\mid d$ .
A 2024 paper proved important special cases, including arbitrary degree when $w_1=1$ , and the full formula when $w_0=w_1=1$ .

July 2025 claimed proof

Nardi and San-José state that their preprint proves the conjectured equality for $w_0=1$ , removes the divisibility restriction on $d$ , and uses footprint methods, Delorme's reduction, and Serre's bound. It also reports failures of equality for some $w_0\geq 2$ . The claim remains unverified by an independent exposition or referee report.

Current status (as of August 2026): The conjectured case $w_0=1$ is claimed proved by the 2025 preprint, but absent independent verification it remains a claimed solution rather than a settled theorem.

Sources

No solutions have been posted yet.