Geodesic
On Extremal Polynomials
Canadian mathematical bulletin, Tome 10 (1967) no. 4, pp. 585-594

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Let p denote a prime number and let κp denote the finite field ofp elements. Let f(x) ∊ κp[x] be of fixed degree d ≥ 2. We supposethat p is also fixed, large compared with d, say, p ≥ p0(d). ByV(f) we denote the number of distinct values of f(x), x ∊ κp. Wecall f maximal if V(f) = p and quasi-maximal if V(f) = p + O(1). Clearly amaximal polynomial is quasi-maximal but it is not known under whatconditions the converse holds. 
Williams, Kenneth S. On Extremal Polynomials. Canadian mathematical bulletin, Tome 10 (1967) no. 4, pp. 585-594. doi: 10.4153/CMB-1967-057-8
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     author = {Williams, Kenneth S.},
     title = {On {Extremal} {Polynomials}},
     journal = {Canadian mathematical bulletin},
     pages = {585--594},
     year = {1967},
     volume = {10},
     number = {4},
     doi = {10.4153/CMB-1967-057-8},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1967-057-8/}
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