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38
Planar functions and planes of LenzBarlotti class
 II, Des. Codes Cryptogr
, 1997
"... Dedicated to Professor Lenz on the occasion of his 80th birthday Abstract. Planar functions were introduced by Dembowski and Ostrom ([4]) to describe projective planes possessing a collineation group with particular properties. Several classes of planar functions over a finite field are described, i ..."
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Dedicated to Professor Lenz on the occasion of his 80th birthday Abstract. Planar functions were introduced by Dembowski and Ostrom ([4]) to describe projective planes possessing a collineation group with particular properties. Several classes of planar functions over a finite field are described, including a class whose associated affine planes are not translation planes or dual translation planes. This resolves in the negative a question posed in [4]. These planar functions define at least one such affine plane of order 3 e for every e ≥ 4 and their projective closures are of LenzBarlotti type II. All previously known planes of type II are obtained by derivation or lifting. At least when e is odd, the planes described here cannot be obtained in this manner. 1.
Global Construction of General Exceptional Covers  With Motivation For Applications To Encoding
 in: Finite Fields: Theory, Applications, and Algorithms American Mathematical Socity
, 1994
"... The paper [FGS] uses the classification of finite simple groups and covering theory in positive characteristic to solve Carlitz's conjecture (1966). We consider only separable polynomials; their derivative is nonzero. Then, f # Fq [x] is exceprional if it acts as a permutation map on infinitely ..."
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Cited by 16 (12 self)
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The paper [FGS] uses the classification of finite simple groups and covering theory in positive characteristic to solve Carlitz's conjecture (1966). We consider only separable polynomials; their derivative is nonzero. Then, f # Fq [x] is exceprional if it acts as a permutation map on infinitely many finite extensions of the finite field Fq , q = p a for some prime p. Carlitz's conjecture says f must be of odd degree (if p is odd). The main theorem of [FGS; Theorem 14.1] restricts the list of possible geometric monodromy groups of exceptional indecomposable polynomials (§1.1): either p = 2 or 3 or these must be affine groups. The proof of Carlitz's conjecture motivates considering general exceptional covers of nonsingular projective algebraic curves. For historical reasons we sometimes call these Schur covers [Fr2]. Suppose # : X # P 1 is an exceptional cover over Fq . Then, for some integer s, there is au/IL x x x # X(F q t ) over each z # P 1 (F q t ) foreac h integer t with (t, s) = 1. In particular /5/ X(F q t ) = q t +1 when (t, s) = 1. We include a complete proof that exceptionality is equivalent to a statement about the geometric/arithmetic monodromy pair of the cover. Theorem 2.5 shows all geometric/arithmetic monodromy pairs satisfying necessary conditions (§1.1§1.2) derive from covers over Fp for all suitably large primes p. Other topics: (i) How modular curve points over finite fields explicitly produce rational function exceptional covers of prime degree (Corollary 3.5). (ii) How fiber products produce abundant general exceptional covers (Lemma 3.7). (iii) How MüllerChenMatthews produced exceptional polynomials with nonsolvable monodromy group ($1.7). (iv) How general exceptional covers realize curves of high genus over Fq with q small and X(F q t # ) large for...
A generalized Lucas sequence and permutation binomials
 Proc. Amer. Math. Soc
, 2006
"... Abstract. Let p be an odd prime and q = pm. Let l be an odd positive integer. Let p ≡ −1 (mod l) or p ≡ 1 (mod l) and l  m. By employing l−1 ..."
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Abstract. Let p be an odd prime and q = pm. Let l be an odd positive integer. Let p ≡ −1 (mod l) or p ≡ 1 (mod l) and l  m. By employing l−1
HODGESTICKELBERGER POLYGONS FOR LFUNCTIONS OF EXPONENTIAL SUMS OF P (x s)
"... Abstract. Let Fq be a finite field of cardinality q and characteristic p. Let P (x) be any onevariable Laurent polynomial over Fq of degree (d1, d2) respectively and p ∤ d1d2. For any fixed s ≥ 1 coprime to p, we prove that the qadic Newton polygon of the Lfunctions of exponential sums of P (x s) ..."
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Abstract. Let Fq be a finite field of cardinality q and characteristic p. Let P (x) be any onevariable Laurent polynomial over Fq of degree (d1, d2) respectively and p ∤ d1d2. For any fixed s ≥ 1 coprime to p, we prove that the qadic Newton polygon of the Lfunctions of exponential sums of P (x s) has a tight lower bound which we call HodgeStickelberger polygon, depending only on the d1, d2, s and the residue class of (p mod s). This HodgeStickelberger polygon is a certain weighted convolution of the Hodge polygon for Lfunction of exponential sums of P (x) and the Newton polygon for the Lfunction of exponential sums of x s (which is precisely given by the classical Stickelberger theory). We have an analogous HodgeStickelberger lower bound for multivariable Laurent polynomials as well. For any ν ∈ (Z/sZ) × , we show that there exists a Zariski dense open subset Uν defined over Q such that for every Laurent polynomial P in Uν(Q) the qadic Newton polygon of L(P (x s)/Fq; T) converges to the HodgeStickelberger polygon as p approaches infinity and p ≡ ν mod s. As a corollary, we obtain a tight lower bound for the qadic Newton polygon of the numerator of the zeta function of an ArtinSchreier curve given by affine equation y p − y = P (x s). This estimates the qadic valuations of reciprocal roots of the zeta function of the ArtinSchreier curve. 1.
Permutation binomials over finite fields
 TRANS. AMER. MATH. SOC
, 2007
"... We prove that, if x m + ax n permutes the prime field Fp, where m> n> 0 and a ∈ F ∗ p, then gcd(m − n, p − 1)> √ p − 1. Conversely, we prove that if q ≥ 4 and m> n> 0 are fixed and satisfy gcd(m − n, q − 1)> 2q(log log q) / log q, then there exist permutation binomials over Fq of ..."
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We prove that, if x m + ax n permutes the prime field Fp, where m> n> 0 and a ∈ F ∗ p, then gcd(m − n, p − 1)> √ p − 1. Conversely, we prove that if q ≥ 4 and m> n> 0 are fixed and satisfy gcd(m − n, q − 1)> 2q(log log q) / log q, then there exist permutation binomials over Fq of the form x m + ax n if and only if gcd(m, n, q − 1) = 1.
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.
A new family of exceptional polynomials in characteristic two
 ARXIV
, 2008
"... We produce a new family of polynomials f(x) over fields k of characteristic 2 which are exceptional, in the sense that f(x) â f(y) has no absolutely irreducible factors in k[x, y] except for scalar multiples of x â y; when k is finite, this condition is equivalent to saying there are infinitely ..."
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We produce a new family of polynomials f(x) over fields k of characteristic 2 which are exceptional, in the sense that f(x) â f(y) has no absolutely irreducible factors in k[x, y] except for scalar multiples of x â y; when k is finite, this condition is equivalent to saying there are infinitely many finite extensions â/k for which the map Î± â¦ â f(Î±) is bijective on â. Our polynomials have degree 2 eâ1 (2 e â 1), where e> 1 is odd. We also prove that this completes the classification of indecomposable exceptional polynomials of degree not a power of the characteristic.
CLASSES OF PERMUTATION POLYNOMIALS BASED ON CYCLOTOMY AND AN ADDITIVE ANALOGUE
"... Abstract. I present a construction of permutation polynomials based on cyclotomy, an additive analogue of this construction, and a generalization of this additive analogue which appears to have no multiplicative analogue. These constructions generalize recent results of José Marcos. Dedicated to Mel ..."
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Cited by 5 (3 self)
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Abstract. I present a construction of permutation polynomials based on cyclotomy, an additive analogue of this construction, and a generalization of this additive analogue which appears to have no multiplicative analogue. These constructions generalize recent results of José Marcos. Dedicated to Mel Nathanson on the occasion of his sixtieth birthday 1.
PLANAR FUNCTIONS AND PERFECT NONLINEAR MONOMIALS OVER FINITE FIELDS
"... Abstract. The study of finite projective planes involves planar functions, namely, functions f: Fq → Fq such that, for each a ∈ F ∗ q, the function c ↦ → f(c + a) − f(c) is a bijection on Fq. Planar functions are also used in the construction of DESlike cryptosystems, where they are called perfect ..."
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Abstract. The study of finite projective planes involves planar functions, namely, functions f: Fq → Fq such that, for each a ∈ F ∗ q, the function c ↦ → f(c + a) − f(c) is a bijection on Fq. Planar functions are also used in the construction of DESlike cryptosystems, where they are called perfect nonlinear functions. We determine all planar functions on Fq of the form c ↦ → c t, under the assumption that q ≥ (t − 1) 4. This resolves two conjectures of Hernando, McGuire and Monserrat. Our arguments also yield a new proof of a conjecture of Segre and Bartocci about monomial hyperovals in finite Desarguesian projective planes. 1.
Some families of permutation polynomials over finite fields, Int
 Hill Center, Department of Mathematics, Rutgers University
"... Abstract. We give necessary and sufficient conditions for a polynomial of the form x r (1 + x v + x 2v + · · · + x kv) t to permute the elements of the finite field Fq. Our results yield especially simple criteria in case (q − 1) / gcd(q − 1, v) is a small prime. 1. ..."
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Abstract. We give necessary and sufficient conditions for a polynomial of the form x r (1 + x v + x 2v + · · · + x kv) t to permute the elements of the finite field Fq. Our results yield especially simple criteria in case (q − 1) / gcd(q − 1, v) is a small prime. 1.