Bounds for fixed point free elements in a transitive group and applications to curves over finite fields

We investigate VI, the cardinality of the value set of a polynomial f of degree n over a finite field of cardinality q. It has been shown that if f is not bijective. then V-f less than or equal to q - (q - 1)/n. Polynomials do exist which essentially achieve that bound. We do prove that if the degree of f is prime to the characteristic and f is not bijective, then asymptotically V-f less than or equal to (5/6)g. We consider related problems for curves and higher dimensional varieties. This problem is related to the number of fixed point free elements in finite groups, and we prove some results in that setting as well.