On a group ring identity related to the Alon–Jaeger–Tarsi conjecture

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Source

János Nagy and Péter Pál Pach, On a group ring identity related to the Alon–Jaeger–Tarsi conjecture, arXiv:2604.26320v1 [math.CO] (29 April 2026). The retained PDF is the nine-page v1 preprint. The arXiv record (https://arxiv.org/abs/2604.26320, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Alon–Jaeger–Tarsi and the group-ring implication

Conjecture 1 (PDF p. 1) states the Alon–Jaeger–Tarsi assertion: for every field $F$ with $|F|\ge4$ and every nonsingular matrix $M$ over $F$, some vector $x$ has no zero entry while $Mx$ has no zero entry either.

For a prime $p>3$, a nonsingular $n\times n$ matrix $M$ over $\mathbb F_p$ with row vectors $a_1,\ldots,a_n$, and standard basis vectors $e_1,\ldots,e_n$ of $G=\mathbb F_p^n$, Conjecture 2 (PDF p. 1) asserts that the integer group-ring identity

$$\prod_{i=1}^n(1-g^{e_i})\prod_{i=1}^n(1-g^{a_i})=0\quad\text{in }\mathbb Z[G]$$

follows from the corresponding identity

$$\prod_{i=1}^n(1-g^{e_i})\prod_{i=1}^n(1-g^{a_i})=0\quad\text{in }\mathbb F_p[G].$$

Theorem 1 (PDF p. 2) states the exact implication proved in the note: "For every prime $p>3$ Conjecture 2 implies the Alon-Jaeger-Tarsi conjecture" (p. 2). The proof fixes a prime $p>3$ for which the conjecture fails over $\mathbb F_p$ and derives a contradiction with Conjecture 2.

Dimension reduction lemma

The paper's group-ring notation is

$$R[G]=\left\{\sum_{v\in G}r_vg^v:r_v\in R\right\},$$

with multiplication induced by addition in $G$. Lemma 1 (PDF p. 3) says that if $M$ is a minimal-dimensional counterexample to the Alon–Jaeger–Tarsi conjecture, then for every fixed $i\in[n]$,

$$\prod_{\substack{1\le j\le n\\j\ne i}}(1-g^{e_j})\prod_{j=1}^n(1-g^{a_j})=0\quad\text{in }\mathbb Z[G].$$

The proof expands each $a_j=a'_j+a_{j,i}e_i$ in the mod-$p$ identity, then uses a higher-dimensional construction and the conjecture to force the integer identity. The final proof step uses Lemma 1 and the invertibility of $M$ to remove the factor $1-g^{a_1}$, extracts the mod-$p$ identity for an $(n-1)\times(n-1)$ nonsingular submatrix, and applies Conjecture 2 to it, which yields a smaller counterexample and contradicts minimality (PDF pp. 7--8).

Version context

The official arXiv record identifies this as v1, and its comment field records text overlap with arXiv:2107.03956, which it says was split into two parts. The present digest keeps the exact v1 note and its theorem labels separate from that earlier preprint.

Proof scope

This digest records the source-stated conjectures, implication theorem and reduction lemma with PDF page locators. The proof is summarized only at method level; no independent proof reconstruction, independent proof review, or full-proof credit is claimed.