When Does a Polynomial over a Finite Field Permute the Elements of the Field?

Lidl, Rudolf; Mullen, Gary L.

doi:10.1080/00029890.1988.11971991

Cited by 77 publications

(80 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…These arithmetic monodromy groups contain a group of the form V × s C with C cyclic and acting irreducibly on V = F a p (see §1.c). This generalizes Theorem 8.1 and Dickson's conjecture by showing these are the semi-linear polynomials of Cohen ([C2]; [LMu;discussion following P9]). These two theorems describe all affine groups known to be arithmetic monodromy groups of exceptional indecomposable polynomials.…”

Section: Conjecture (Carlitz 1966 [Lmu; P9]): If P Is Odd Exceptionmentioning

confidence: 87%

Schur covers and Carlitz’s conjecture

Fried

Guralnick

Saxl³

1993

Israel J. Math.

124

View full text Add to dashboard Cite

Abstract:We use the classification of finite simple groups and covering theory in positive characteristic to solve Carlitz's conjecture (1966). An exceptional polynomial f over a finite field F q is a polynomial that is a permutation polynomial on infinitely many finite extensions of F q . Carlitz's conjecture says f must be of odd degree (if q 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 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 covers X → P 1 that generalize the notion of exceptional polynomials. These covers have this property: Over each F q t point of P 1 there is exactly one F q t point of X for infinitely many t. Thus X has a rare diophantine property when X has genus greater than 0. It has exactly q t + 1 points in F q t for infinitely many t. 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 an indecomposable exceptional cover having (at least) one totally ramified place over a rational point of the base. Its arithmetic monodromy group is an affine group.• Supported by NSA grant MDA 14776 and BSF grant 87-00038.• 2 First author supported by the Institute for Advanced Studies in Jerusalem and IFR Grant #90/91-15.• 3 Supported by NSF grant DMS 91011407. Dedication:To the contributions of John Thompson to the classification of finite simple groups; and to the memory of Daniel Gorenstein and the success of his project to complete the classification.

show abstract

Section: Conjecture (Carlitz 1966 [Lmu; P9]): If P Is Odd Exceptionmentioning

confidence: 87%

Schur covers and Carlitz’s conjecture

Fried

Guralnick

Saxl³

1993

Israel J. Math.

124

View full text Add to dashboard Cite

show abstract

“…Now this curve has the same form as (5) and satisfies that h + 1 -p < p -1 . The above argument applies except for the following two cases: h + l-p = p -1 or t = 1.…”

Section: The Carlitz Conjecture For N = 21mentioning

confidence: 99%

“…If f(x) ^ g(x ) for any polynomial g(x), there is a smallest odd positive integer t such that ah+l ¿ 0. We move the point P = (-1, 1, 0) to the origin (x -► x + 1), then (3) takes the form (up to a nonzero constant) (5) x + axz +bz H-= 0, where a and b are nonzero elements of F . If h < p -1, we make the birational transform x->xz (z ^ z) and remove the trivial factor z .…”

Section: The Carlitz Conjecture For N = 21mentioning

confidence: 99%

See 1 more Smart Citation

Permutation polynomials and resolution of singularities over finite fields

Wan¹

1990

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…For a good discussion of the maln conjectures and open problems, and for some references concerning applications, the reader is referred to the recent article [5]. Any reader whose interest is more than superficially aroused (by the present paper and/or [5]) should consult Chapter 7 of [2], where she/he will find an extended discussion together wlth voluminous historical and bibliographic notes. …”

Section: Introductionmentioning

confidence: 99%