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Exceptional covers and bijections on rational points
 Int. Math. Res. Not. IMRN 2007, art. ID rnm004
"... Abstract. We show that if f: X − → Y is a finite, separable morphism of smooth curves defined over a finite field Fq, where q is larger than an explicit constant depending only on the degree of f and the genus of X, then f maps X(Fq) surjectively onto Y (Fq) if and only if f maps X(Fq) injectively i ..."
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Cited by 7 (5 self)
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Abstract. We show that if f: X − → Y is a finite, separable morphism of smooth curves defined over a finite field Fq, where q is larger than an explicit constant depending only on the degree of f and the genus of X, then f maps X(Fq) surjectively onto Y (Fq) if and only if f maps X(Fq) injectively into Y (Fq). Surprisingly, the bounds on q for these two implications have different orders of magnitude. The main tools used in our proof are the Chebotarev density theorem for covers of curves over finite fields, the Castelnuovo genus inequality, and ideas from Galois theory. 1.
Residue classes free of values of Euler’s function
 In: Gy}ory K (ed) Proc Number Theory in Progress, pp 805–812. Berlin: W de Gruyter
, 1999
"... Dedicated to Andrzej Schinzel on his sixtieth birthday By a totient we mean a value taken by Euler’s function φ(n). Dence and Pomerance [DP] have established Theorem A. If a residue class contains at least one multiple of 4, then it contains ..."
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Cited by 6 (1 self)
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Dedicated to Andrzej Schinzel on his sixtieth birthday By a totient we mean a value taken by Euler’s function φ(n). Dence and Pomerance [DP] have established Theorem A. If a residue class contains at least one multiple of 4, then it contains
Algebraic Geometry over a field of positive characteristic
"... Lectures given by Prof. J.W.P. Hirschfeld.
Abstract
Curves over finite fields not only are interesting structures in themselves, but they are
also remarkable for their application to coding theory and to the study of the geometry of
arcs in a finite pl ..."
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Lectures given by Prof. J.W.P. Hirschfeld.
Abstract
Curves over finite fields not only are interesting structures in themselves, but they are
also remarkable for their application to coding theory and to the study of the geometry of
arcs in a finite plane. In this note, the basic properties of curves and the number of their
points are recounted.
Algebraic curves and maximal arcs
 JOURNAL OF ALGEBRAIC COMBINATORICS
, 2008
"... Abstract A lower bound on the minimum degree of the plane algebraic curves containing every point in a large pointset K of the Desarguesian plane PG(2,q) is obtained. The case where K is a maximal (k, n)arc is considered in greater depth. ..."
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Abstract A lower bound on the minimum degree of the plane algebraic curves containing every point in a large pointset K of the Desarguesian plane PG(2,q) is obtained. The case where K is a maximal (k, n)arc is considered in greater depth.
LOWDEGREE PLANAR MONOMIALS IN CHARACTERISTIC TWO
"... ABSTRACT. Planar functions over finite fields give rise to finite projective planes and other combinatorial objects. They exist only in odd characteristic, but recently Zhou introduced an even characteristic analogue which has similar applications. In this paper we determine all planar functions on ..."
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ABSTRACT. Planar functions over finite fields give rise to finite projective planes and other combinatorial objects. They exist only in odd characteristic, but recently Zhou introduced an even characteristic analogue which has similar applications. In this paper we determine all planar functions on Fq of the form c ↦ → ac t, where q is a power of 2, t is an integer with 0 < t ≤ q 1/4, and a ∈ F ∗ q. This settles and sharpens a conjecture of Schmidt and Zhou. 1.
ON THE SURJECTIVITY OF ENGEL WORDS ON PSL(2, q)
"... Abstract. We investigate the surjectivity of the word map defined by the nth Engel word on the groups PSL(2, q) and SL(2, q). For SL(2, q), we show that this map is surjective onto the subset SL(2, q)\{−id} ⊂ SL(2, q) provided that q ≥ q0(n) is sufficiently large. Moreover, we give an estimate for ..."
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Abstract. We investigate the surjectivity of the word map defined by the nth Engel word on the groups PSL(2, q) and SL(2, q). For SL(2, q), we show that this map is surjective onto the subset SL(2, q)\{−id} ⊂ SL(2, q) provided that q ≥ q0(n) is sufficiently large. Moreover, we give an estimate for q0(n). We also present examples demonstrating that this does not hold for all q. We conclude that the nth Engel word map is surjective for the groups PSL(2, q) when q ≥ q0(n). By using the computer, we sharpen this result and show that for any n ≤ 4, the corresponding map is surjective for all the groups PSL(2, q). This provides evidence for a conjecture of Shalev regarding Engel words in finite simple groups. In addition, we show that the nth Engel word map is almost measure preserving for the family of groups PSL(2, q), with q odd, answering another question of Shalev. Our techniques are based on the method developed by Bandman, Grunewald and Kunyavskii for verbal dynamical systems in the group SL(2, q). 1.