If F is a finite field with at least 4 elements and A is an invertible n by n matrix over F, then there are vectors x, y

Definitions and mathematical statement

Let $\mathbb F_q$ be a finite field of prime-power order $q\ge4$, let $n\ge1$, and let $A\in\operatorname{GL}_n(\mathbb F_q)$ be an invertible matrix. A vector is nowhere zero if each of its coordinates belongs to $\mathbb F_q^*=\mathbb F_q\setminus\{0\}$. Jaeger's question asks whether there always exists \[ x\in(\mathbb F_q^*)^n \quad\text{with}\quad Ax\in(\mathbb F_q^*)^n. \] Equivalently, must the two sets $(\mathbb F_q^*)^n$ and $A((\mathbb F_q^*)^n)$ intersect? In row form, the problem is to choose all $x_j\ne0$ so that $\sum_{j=1}^n A_{ij}x_j\ne0$ simultaneously for all rows $i$.

All arithmetic takes place in the field, so the statement includes nonprime orders such as $q=4$. The vectors are allowed to depend on the matrix, and no lower bound on the number of eligible vectors is required. The hypotheses of invertibility and the lower bound on $q$ are part of the assertion; merely having no zero row is not the same assumption. A vector with some nonzero coordinate is insufficient: both it and its image must avoid zero in every coordinate. No requirement is placed on the individual entries of the invertible matrix itself.

Short English statement

Over every finite field with at least four elements, does every invertible matrix map some nowhere-zero vector to another nowhere-zero vector?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3100060 (AMR-030-0060), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Cooper - Combinatorial Problems I Like (2020 snapshot), Combinatorial Number Theory, source-order bullet 60. Original source indexed by AMR-030-0060. https://people.math.sc.edu/cooper/combprob.html.