Akad~miai Kind6 ~ Springer-Verlag
COMnrSATO~aCA9 (4) (1989) 393---395
A NOWHERE-ZERO POINT IN LINEAR MAPPINGS
N. ALON and M. TARSI
Received March 29, 1988
Wes(ate the followingconjectureand proveit for the casewhereq is a proper prime power:
Let A be anonsingular nby n matrix over the finitefieM GF~, q~--4,then there exists a vector
x in (GFa)~such that both x and ~4xhave 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 Ax 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 in-
dependent sets can be bigger. This claim can be verified by checking the chromatic
polynomial of the uniform matroid Un,2~ (see  for the relevant definitions).