A nowhere-zero point in liner mappings

We s(ate the following conjecture and prove it for the case where q is a proper prime power: Let A be a nonsingular n by n matrix over the finite fieM GF~, q~-4, then there exists a vector x in (GFa) ~ such that both x and ~4x have no zero component. In this note we consider-the following conjecture: Conjecture 1. Let A be a nonsingular n by n matrix over the finite field GFa, q~_4, then there exists a vector x in (GF~) n such that both x and A x have no zero component. Notice that there are easy examples showing that the assertion of the con, jecture is false for q~_3. We have reached this conjecture while trying to generalize some simple properties of sparse graphs to more general matroids. Specifically: a graph whose edge set is the union of two forests is clearly 4-colorable. In general, the chromatic number of a matroid whose dement set is the union of two independent sets can be bigger. This claim can be verified by checking the chromatic polynomial of the uniform matroid Un,2~ (see [4] for the relevant definitions). However, if such a matroid is representable over a field GFq for which conjecture 1 holds then its chromatic number is at most q, since the conjecture implies that its critical number over GFq is 1 ([4], Chapter !5.5). The conjecture also seems, to be of interest for its own. The case q = 5 was stated as an open problem by F. Jaeger [3]. All we could do so far is to prove the following partial result given in Theorem 1 below. Our proof resembles the ones given in [2],:[1], but contains several additional ideas.-Theorem 1. Conjecture 1 holds for the case where q is not a prime, that is q = p ~ fo~ a prime p and k ~_2.