Exceptional Polynomials Research Articles

Lenstra′s Proof of the Carlitz-Wan Conjecture on Exceptional Polynomials: An Elementary Version

We give a proof, following an argument of Lenstra, of the conjecture of Carlitz (1966) as generalized by Wan (1993). This says that there are no exceptional polynomials of degree n over Fq if (n, q − 1) > 1. Fried, Guralnick, and Saxl previously proved Carlitz′s conjecture: there are no exceptional polynomials of even degree over fields of odd order.

Exceptional Polynomials over Finite Fields

A recently discovered family of indecomposable polynomials of nonprime power degree over F2 (which include a class of exceptional polynomials) is set against the background of the classical families and their monodromy groups are obtained without recourse to the classification of finite simple groups.

Discrete Reflectionless Potentials, Quantum Algebras, and q-Orthogonal Polynomials

Using Darboux transformations for the lattice Schrödinger equation, we construct two families of discrete reflectionless potentials with j and 2j bound states (solitons). These potentials are related to the exceptional Askey-Wilson polynomials. In the continuous limit they are reduced to the well-known solvable potential u(x) = −j(j + 1)/cosh2x. The limit j → ∞ for the lattice systems may be defined in two different ways. In the first case, discrete spectrum grows exponentially fast, and the continuous spectrum band, −2 ≤ λ ≤ 2, is preserved. In the second case, the limiting potentials are related to the q−1-Hermite polynomials. Their spectra are purely discrete and consist of one or two geometric series accumulating near the zero. These series are generated by the q-Weyl algebra and its "square root" version. In contrast to the continuous reflectionless potentials with quantum algebraic symmetries, the derived discrete potentials are expressed in terms of the elementary functions.

A Class of Exceptional Polynomials

A Class of Exceptional Polynomials

On the Periods of Generalized Fibonacci Recurrences

We give a simple condition for a linear recurrence $\pmod {2^w}$ of degree r to have the maximal possible period ${2^{w - 1}}({2^r} - 1)$. It follows that the period is maximal in the cases of interest for pseudorandom number generation, i.e., for three-term linear recurrences defined by trinomials which are primitive $\pmod 2$ and of degree $r > 2$. We consider the enumeration of certain exceptional polynomials which do not give maximal period, and list all such polynomials of degree less than 15.

A class of exceptional polynomials

We present a class of indecomposable polynomials of non prime-power 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 $PS{L_2}({2^k})$, where $k \geq 3$ 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 $PS{L_2}({3^k})$, $k \geq 3$ , and odd in fields of characteristic 3.

On the periods of generalized Fibonacci recurrences

We give a simple condition for a linear recurrence ( mod 2 w ) \pmod {2^w} of degree r to have the maximal possible period 2 w − 1 ( 2 r − 1 ) {2^{w - 1}}({2^r} - 1) . It follows that the period is maximal in the cases of interest for pseudorandom number generation, i.e., for three-term linear recurrences defined by trinomials which are primitive ( mod 2 ) \pmod 2 and of degree r > 2 r > 2 . We consider the enumeration of certain exceptional polynomials which do not give maximal period, and list all such polynomials of degree less than 15.

Schur covers and Carlitz’s conjecture

We use the classification of finite simple groups and covering theory in positive characteristic to solve Carlitz’s conjecture (1966). An exceptional polynomialf over a finite field\({\mathbb{F}}_q \) is a polynomial that is a permutation polynomial on infinitely many finite extensions of\({\mathbb{F}}_q \). Carlitz’s conjecture saysf must be of odd degree (ifq is odd). Indeed, excluding characteristic 2 and 3, arithmetic monodromy groups of exceptional polynomials must be affine groups.We don’t, however, know which affine groups appear as the geometric metric monodromy group of exceptional polynomials. Thus, there remain unsolved problems. Riemann’s existence theorem in positive characteristic will surely play a role in their solution. We have, however, completely classified the exceptional polynomials of degree equal to the characteristic. This solves a problem from Dickson’s thesis (1896). Further, we generalize Dickson’s problem to include a description of all known exceptional polynomials.Finally: The methods allow us to consider coversX→\({\mathbb{P}}^1 \) that generalize the notion of exceptional polynomials. These covers have this property: Over each\({\mathbb{F}}_{q^t } \) point of\({\mathbb{P}}^1 \) there is exactly one\({\mathbb{F}}_{q^t } \) point ofX for infinitely manyt. ThusX has a rare diophantine property whenX has genus greater than 0. It has exactlyq t+1 points in\({\mathbb{F}}_{q^t } \) for infinitely manyt. This gives exceptional covers a special place in the theory of counting rational points on curves over finite fields explicitly. Corollary 14.2 holds also for a primitive exceptional cover having (at least) one totally ramified place over a rational point of the base. Its arithmetic monodromy group is an affine group.

Permutation polynomials and primitive permutation groups

By STEPSEN D. COHEN 1. Introduction. We present here progress that has been made through the application of the theory of primitive permutation groups to the conjecture of L. Carlitz (1966) on the non-existence of permutation polynomials of even degree over a finite field Fr of odd order q = p~. Featured recently as unsolved problem P9 in [9], the conjecture can be stated as follows. C o nj e c t u re C.. Given an even positive integer n, there exists a constant c. such that, ifq is odd and q > c,, there does not exist a permutation polynomial of degree n over The weaker version of C, that has the added condition p ~e n was proved by Hayes [8] but polynomials whose degree is divisible by p are mucli more difficult to handle. Here, as far as published results ha this direction are concerned, C, has been established for all even n 1 in Fq[y] is called exceptional over Fq if every irreducible factor of q~y (x, y) in F~ [x, y] is not absolutely irreducible, which means that it becomes reducible in Fq [x, y], where Fq denotes the algebraic closure of Fq. The merit of introducing exceptional polynomials is, of course, the fact that for each even n, Conjecture C, is implied by the following statement. E,. There are no exceptional polynomials of degree n over F~ for any odd q. We shall prove that the set of even integers n for which E, (and hence C,) is true is infinite and includes all small values by establishing the following theorem.

On a Conjecture of Carlitz

A conjecture of Carlitz on permutation polynomials is as follow: Given an even positive integer n, there is a constant Cn, such that if Fq is a finite field of odd order q with q > Cn, then there are no permutation polynomials of degree n over Fq. 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.

On Exceptional Polynomials

Let f(x) be a polynomial of degree d ≥ 2 defined over the finite field kq with q = pn elements. LetIf f*(x, y) has no irreducible factor over kq which is absolutely irreducible, f is called an exceptional polynomial [1]. Davenport and Lewis have noted that when d is small compared with p, a permutation (substitution) polynomial is necessarily an exceptional polynomial. It is the purpose of this paper to prove the converse; that is, we will show the existence of a constant a(d), depending only on d, such that if f(x.) is an exceptional polynomial over kq, where p ≥ a(d), then f(x) is a permutation polynomial.

On a conjecture of Davenport and Lewis concerning exceptional polynomials

On a conjecture of Davenport and Lewis concerning exceptional polynomials