The values of a polynomial over a finite field

Cohen, Stephen D.

doi:10.1017/s0017089500001981

Cited by 10 publications

(4 citation statements)

References 9 publications

(7 reference statements)

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“…To calculate these multiplicities, we need the following lemma. Particular cases of the lemma were considered in Zsigmondy [25], and in Cohen [5]. The complete result appears in A. Knopfmacher and J. Knopfmacher [8].…”

Section: Resultsmentioning

confidence: 99%

On the spectrum of Wenger graphs

Cioabă

Lazebnik

2014

Journal of Combinatorial Theory, Series B

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Let q = p e , where p is a prime and e ≥ 1 is an integer. For m ≥ 1, let P and L be two copies of the (m + 1)-dimensional vector spaces over the finite field F q . Consider the bipartite graph W m (q) with partite sets P and L defined as follows: a point (p) = (p 1 , p 2 , . . . , p m+1 ) ∈ P is adjacent to a line [l] = [l 1 , l 2 , . . . , l m+1 ] ∈ L if and only if the following m equalities hold: l i+1 + p i+1 = l i p 1 for i = 1, . . . , m. We call the graphs W m (q) Wenger graphs. In this paper, we determine all distinct eigenvalues of the adjacency matrix of W m (q) and their multiplicities. We also survey results on Wenger graphs.

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Section: Resultsmentioning

confidence: 99%

On the spectrum of Wenger graphs

Cioabă

Lazebnik

2014

Journal of Combinatorial Theory, Series B

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“…It is clear that, for lower bounds, there are obvious limits to the results you can expect to obtain -obviously V (f ) ≥ 1 with equality possible, while for polynomials of given degree d, V (f ) ≥ 1 + q−1 d is clear. That said, we have the following deep result by Cohen [2] concerning the average lower bound of V (f ).…”

Section: Polynomials Over Finite Fields and N 2 (F )mentioning

confidence: 74%

On the Number of Distinct Values of a Class of Functions with Finite Domain

Coulter

Senger

2014

Ann. Comb.

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Abstract. By relating the number of images of a function with finite domain to a certain parameter, we obtain both an upper and lower bound for the image set. Even though the arguments are elementary, the bounds are, in some sense, best possible. These bounds are then applied in several contexts. In particular, we obtain the first non-trivial upper bound for the image set of a planar function over a finite field.

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“…Results on the average value V(d, 0) of V(f ) when f ranges over all monic polynomials in F q [T ] of degree d with f (0) = 0 were obtained by Uchiyama [24] and improved by Cohen [9]. More precisely, in [9, §2] it is shown that…”

Section: Introductionmentioning

confidence: 99%

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

Cesaratto¹,

Matera²,

Pérez³

et al. 2014

Journal of Combinatorial Theory, Series A

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We obtain an estimate on the average cardinality of the value set of any family of monic polynomials of F q [T ] of degree d for which s consecutive coefficients a d−1 , . . . , a d−s are fixed. Our estimate holds without restrictions on the characteristic of F q and asserts thatis such an average cardinality, µ d := d r=1 (−1) r−1 /r! and a := (a d−1 , . . . , d d−s ). We provide an explicit upper bound for the constant underlying the O-notation in terms of d and s with "good" behavior. Our approach reduces the question to estimate the number of F q -rational points with pairwise-distinct coordinates of a certain family of complete intersections defined over F q . We show that the polynomials defining such complete intersections are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning the singular locus of the varieties under consideration, from which a suitable estimate on the number of F q -rational points is established.

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The values of a polynomial over a finite field

Abstract: The object of this paper is to derive, using a version of the large sieve for function fields due to J. Johnsen [6], explicit lower boundsfor the average number of distinct values taken by a polynomial over a finite field.

Cited by 10 publications

References 9 publications

On the spectrum of Wenger graphs

On the spectrum of Wenger graphs

On the Number of Distinct Values of a Class of Functions with Finite Domain

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

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