A nowhere-zero point for several linear mappings simultaneously

Abstract

Let $q=p^k$, and let $A_1,\ldots,A_{r-1}$ be invertible $n\times n$ matrices over ${\mathbb F}_q$. We prove that, if $k\ge r$, there is a vector $x$ for which \[ x,A_1x,\ldots,A_{r-1}x \] are all nowhere zero. For $r=2$ this recovers the theorem of Alon and Tarsi over nonprime finite fields. The proof tracks one monomial in the product of the coordinate forms. Frobenius powers keep every exponent below $p^r$, and finite-field polynomial nonvanishing supplies the required vector. The same method treats rectangular matrices with independent rows and prescribed forbidden values. The method also gives an extension-degree criterion for representable matroids and clarifies an unpublished prime-field conjecture of M. J. Moghaddamzadeh. Projective-geometric examples explain why the analogous field-size statement fails over proper extensions and, translated back to matrices, give lower bounds for the large-field problem.