Note on the mean value of $V \left( f \right)$

Counting polynomials with a given number of zeros in a finite field

“…For the sake of completeness, although some other authors have at least partly considered such questions previously, we shall also briefly consider the case of distinct zeros: Regarding M,(n, k), we remark that this was studied previously by Zsigmondy [6] when q is prime, and by Uchiyama [5] and Cohen [I] when k = 0. Further, the average 1 and the variance 1 -q-' are special cases of results of Schmidt [4, pp.…”

Explicit estimates for the number of rational points of singular complete intersections over a finite field

“…A variant of this problem, considered by Uchiyama and Cohen, asks for results on the average cardinality of the value set of the set of polynomials f ∈ F q [T ] of given degree were some coefficients are fixed. For this problem, in [Uch55] and [Coh72] the authors obtain the following result. Consider the family of monic polynomials of F q [T ] of degree n, were s consecutive coefficients are fixed, with 1 ≤ s ≤ n − 2.…”

“…Concerning the behavior of V(f ) for "large" sets of elements of F q [T ], Birch and Swinnerton-Dyer established the following significant result [BS59]: for fixed n ≥ 1, if f is a generic polynomial of degree n, then V(f ) = µ n q + O(q 1/2 ), where µ n := n r=1 (−1) r−1 /r! and the constant underlying the O-notation depends only on n. Results on the average value V(n, 0) of V(f ) when f ranges over all monic polynomials in F q [T ] of degree n with f (0) = 0 were obtained by Uchiyama [Uch55] and improved by Cohen [Coh73]. More precisely, in [Coh73,§2] it is shown that V(n, 0) = n r=1 (−1) r−1 q r q 1−r = µ n q + O(1).…”

On the value set of small families of polynomials over a finite field, II

“…On the other hand, in [Uch56] it is shown that, for p := char(F q ) > d and assuming the Riemann hypothesis for L-functions, one has V 2 (d, 0) = µ 2 d q 2 + O(q). It must be observed that no explicit expression for the constant underlying the O-notation is provided in [Uch56].…”

“…where the constant underlying the O-notation depends only on d and s (see [Uch55b], [Coh72]). In a previous paper [CMPP13] we obtain the following explicit estimate for q > d and 1 ≤ s ≤ d 2 − 1:…”