The Alon–Jaeger–Tarsi conjecture

Let $\mathbb{F}$ be a field, let $M$ be a nonsingular matrix over $\mathbb{F}$ , and let $x$ be a vector over $\mathbb{F}$ . Alon–Jaeger–Tarsi conjecture. For any field $\mathbb{F}$ with $|\mathbb{F}|\geq 4$ and any nonsingular matrix $M$ over $\mathbb{F}$ , there is a vector $x$ such that both $x$ and $Mx$ have only nonzero entries. The conjecture is trivial over infinite fields and is known for proper prime-power field orders and for sufficiently large primes, but the general case remains open.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Alon–Jaeger–Tarsi conjecture
Let $A$ be a nonsingular $d\times d$ matrix over a finite field $k$ of cardinality $q\geq4$ . Alon–Jaeger–Tarsi conjecture. There exists a vector $v\in k^n$ such that both $v$ and $Av$ have no zero component. This conjecture concerns simultaneous avoidance of zero coordinates by a vector and its image under an invertible matrix. The text reports proofs in several ranges of $q$ , but does not state that the full conjecture is resolved.
source: George Kirkup, “Minimal Primes Over Permanental Ideals”, arXiv:math/0510025 (2005).

Sources & referencesView supporting material

Primary source

János Nagy and Péter Pál Pach, “On a group ring identity related to the Alon-Jaeger-Tarsi conjecture”, arXiv:2604.26320 (2026).

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The Alon–Jaeger–Tarsi conjecture

Equivalent formulations 1

Alon–Jaeger–Tarsi conjecture

Primary source

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