The number of directions determined by a function over a finite field.
Let \(GT(q)\) be the finite field with \(q= p^n\) elements, \(p\) prime. Let \(f\) be a function from \(GT(q)\) to \(GT(q)\), let \[ D= \Biggl\{{f(y)- f(x)\over y-x}\,\Biggl|\,x,y\in GF(q),\, x\neq y\Biggr\} \] be the set of directions determined by the function \(f\) and \(N=| D|\). The objective of this article is to prove that number \(N\) of directions determined by a function over \(GT(q)\) is either 1, at least \((q+3)/2\), or between \(q/s+1\) and \((q-1)/(s-1)\) for some \(s\) where \(GT(s)\) is a subfield of \(GT(q)\). The fact that if \(s> 2\), then the graph of \(f\) is \(GT(s)\)-linear, is proved too.
scientific article; zbMATH DE number 2032718