Geodesic
Distinct Values of a Polynomial in Subsets of a Finite Field
Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 1483-1488

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If A is a set with only a finite number of elements, we write |A| for the number of elements in A. Let p be a large prime and let m be a positive integer fixed independently of p. We write [pm ] for the finite field with pm elements and [pm ]′ for [pm ] – {0}. We consider in this paper only subsets H of [pm ] for which |H| = h satisfies 1.1 If f(x) ∈ [pm, x] we let N(f; H) denote the number of distinct values of y in H for which at least one of the roots of f(x) = y is in [pm ]. We write d(d ≥ 1) for the degree of f and suppose throughout that d is fixed and that p ≧ p 0(d), for some prime p 0, depending only on d, which is greater than d. 
Williams, Kenneth S. Distinct Values of a Polynomial in Subsets of a Finite Field. Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 1483-1488. doi: 10.4153/CJM-1969-162-0
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