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

Cesaratto, Eda; Matera, Guillermo; Pérez, Mariana; Privitelli, Melina

doi:10.48550/arxiv.1306.1744

Cited by 1 publication

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“…Our approach to prove Theorem 1.1 shares certain similarities with that of [CMPP13]. Indeed, we express the quantity V(d, s, a) in terms of the number χ a r of certain "interpolating sets" with d − s + 1 ≤ r ≤ d. More precisely, for f a := T d + a d−1 T d−1 + • • • + a d−s T d−s , we define χ a r as the number of r-element subsets of F q at which f a can be interpolated by a polynomial of degree at most d − s − 1.…”

Section: Introductionmentioning

confidence: 87%

“…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:…”

Section: Introductionmentioning

confidence: 92%

“…[LN83]). This paper is a continuation of [CMPP13] and is concerned with results on the average value set of certain families of polynomials of F q [T ].…”

Section: Introductionmentioning

confidence: 99%

“…According to this result, we have to determine the asymptotic behavior of χ a r for d − s + 1 ≤ r ≤ d. In [CMPP13] we introduce an affine F q -variety V a r ⊂ A r in such a way that the number of q-rational points of V a r with pairwise distinct coordinates agrees with the number χ a r . In the sequel we follow a different approach, considering the incidence variety consisting of the set of triples (b, b 0 , α 1 , .…”

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confidence: 99%

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

2014

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Abstract. We obtain an estimate on the average cardinality of the value set of any family of monic polynomials ofis the average second moment on any family of monic polynomials of F q [T ] of degree d with s consecutive coefficients fixed as above. Finally, we show that, where V 2 (d, 0) denotes the average second moment of all monic polynomials in F q [T ] of degree d with f (0) = 0. All our estimates hold for fields of characteristic p > 2 and provide explicit upper bounds for the constants underlying the O-notation in terms of d and s with "good" behavior. Our approach reduces the questions to estimate the number of F q -rational points with pairwise-distinct coordinates of a certain family of complete intersections defined over F q . A critical point for our results is an analysis of the singular locus of the varieties under consideration, which allows to obtain rather precise estimates on the corresponding number of F q -rational points.

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Section: Introductionmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 92%

“…[LN83]). This paper is a continuation of [CMPP13] and is concerned with results on the average value set of certain families of polynomials of F q [T ].…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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