The distribution of polynomials over finite fields

Cohen, Stephen D.

doi:10.4064/aa-17-3-255-271

Cited by 103 publications

(95 citation statements)

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“…The former of these was first established in full by the first author [3] and applies even when the degree of the polynomial is divisible by the characteristic p, which is particulars relevant here. A more elementary proof has recently been given by D. Wan [19].…”

Section: Preliminariesmentioning

confidence: 99%

A class of exceptional polynomials

Cohen¹,

Matthews²

1994

Trans. Amer. Math. Soc.

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Abstract. We present a class of indecomposable polynomials of non primepower degree over the finite field of two elements which are permutation polynomials on infinitely many finite extensions of the field. The associated geometric monodromy groups are the simple groups PSL2(2k), where k > 3 and odd.(The first member of this class was previously found by P. Müller [17]. This realises one of only two possibilities for such a class which remain following deep work of Fried, Guralnick and Saxl [7]. The other is associated with PSL2CSk), k > 3 , and odd in fields of characteristic 3.

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Section: Preliminariesmentioning

confidence: 99%

A class of exceptional polynomials

Cohen¹,

Matthews²

1994

Trans. Amer. Math. Soc.

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“…We define ( , ℎ) to be the number of irreducible polynomials of the form ℎ ( /ℎ), where is a linear monic polynomial in [ ]. Now we state a particular case of Theorem 3 in [4].…”

Section: Counting Irreducible Polynomialsmentioning

confidence: 99%

“…We start with a result due to S. D. Cohen [4] regarding the number of irreducible polynomials of a certain form over finite fields. Before stating it we need to introduce some notation.…”

Section: Counting Irreducible Polynomialsmentioning

confidence: 99%

Counting squarefree discriminants of trinomials under abc

Mukhopadhyay

Murty

Kotyada

2009

Proc. Amer. Math. Soc.

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“…This polynomial y(y) is irreducible over E as is well known (see van der Waerden [14]) and has the root y = x in fi(x). Therefore, fi(jc) is a simple algebraic extension field of E of degree d, and for any two roots y x , y 2 of <p(y) in B(x), there exists an ^-automorphism of S2(JC) which maps y y •-+ y 2 , by a fundamental theorem on the extension of isomorphisms, …”

Section: Ii) Suppose P Is Odd and P 2 + N If (X Y) Has A Linear Facmentioning

confidence: 99%

“…Further, his results show that, except for "accidents", there are no permutation polynomials of degree 2, 4, and 6 except when the characteristic of the field is 2. In an address before the 376 WanDaquing [2] Mathematical Association of America, Professor L. Carlitz suggested that this behaviour is perhaps characteristic: that is, Carlitz suggested the conjecture stated in the abstract. Dickson's results show that the conjecture is true if n = 2, 4, or 6.…”

Section: Introductionmentioning

confidence: 99%

On a Conjecture of Carlitz

Wan

1987

J Aust Math Soc A

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A conjecture of Carlitz on permutation polynomials is as follow: Given an even positive integer n, there is a constant C n , such that if F q is a finite field of odd order q with q > C n , then there are no permutation polynomials of degree n over F . The conjecture is a well-known problem in this area. It is easily proved if n is a power of 2. The only other cases in which solutions have been published are n = 6 (Dickson [5]) and n = 10 (Hayes [7]); see Lidl [11], Lausch and Nobauer [9], and Lidl and Niederreiter [10] for remarks on this problem. In this paper, we prove that the Carlitz conjecture is true if n = 12 or n = 14, and give an equivalent version of the conjecture in terms of exceptional polynomials.

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The distribution of polynomials over finite fields

Cited by 103 publications

References 0 publications

A class of exceptional polynomials

A class of exceptional polynomials

Counting squarefree discriminants of trinomials under abc

On a Conjecture of Carlitz

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