Cubic polynomials over algebraic number fields

We prove that every cubic form in 16 variables over an algebraic number field represents zero, generalizing the corresponding result of Davenport for cubic forms over the rationals. (This has already been proved for cubic forms in 17 or more variables by Ryavec.) We present this result as a special case of a “local-implies-global” theorem for cubic polynomials.     

Permutation polynomials from trace functions over finite fields

In this paper, we propose several classes of permutation polynomials based on trace functions over finite fields of characteristic 2. The main result of this paper is obtained by determining the number of solutions of certain equations over finite fields.     

Normal binomials over algebraic number fields

In this paper, normal and weakly normal binomials over an arbitrary algebraic number field will be characterized. Explicit results on the possible degrees of such binomials are given. Several examples conclude the paper.     

Permutation polynomials over finite fields from a powerful lemma

Using a lemma proved by Akbary, Ghioca, and Wang, we derive several theorems on permutation polynomials over finite fields. These theorems give not only a unified treatment of some earlier constructions of permutation polynomials, but also new specific permutation polynomials over Fq. A number of earlier theorems and constructions of permutation polynomials are generalized. The results presented in this paper demonstrate the power of this lemma when it is employed together with other techniques.     

Algebraic leaves of algebraic foliations over number fields

Summary — We prove an algebraicity criterion for leaves of algebraic foliations defined over number fields. Namely, consider a number field K embedded in C, a smooth algebraic variety X over K, equipped with a K-rational point P, and F an algebraic subbundle of the its tangent bundle T_X, defined over K. Assume moreover that the vector bundle F is involutive, i.e., closed unter Lie bracket. Then it defines an holomorphic foliation of the analytic mainfold X(C), and one may consider its leaf ℱ through P. We prove that ℱ is algebraic if the following local conditions are satisfied: i) For almost every prime ideal p of the ring of integers 𝒪_K of the number field K, the p-curvature of the reduction modulo p of the involutive bundle F vanishes at P (where p denotes the characteristic of the residue field 𝒪_K / p ). ii) The analytic manifold ℱ satisfies the Liouville property; this arises, in particular, if ℱ is the image by some holomorphic map of the complement in a complex algebraic variety of a closed analytic subset. This algebraicity criterion unifies and extends various results of D. V. and G. V. Chudnovsky, André, and Graftieaux, and also admits new consequences. For instance, applied to an algebraic group G over K, it shows that a K-Lie subalgebra h of Lie G is algebraic if and only if for almost every non-zero prime ideal p of 𝒪_K , of residue characteristic p, the reduction modulo p of h is a restricted Lie subalgebra of the reduction modulo p of Lie G (i.e., is stable under p-th powers). This solves a conjecture of Ekedahl and Shepherd-Barron. The algebraicity criterion above follows from a more basic algebraicity criterion concerning smooth formal germs in algebraic varieties over number fields. The proof of the latter relies on “transcendence techniques”, recast in a modern geometric version involving elementary concepts of Arakelov geometry, and on some analytic estimates, related to the First Main Theorem of higher-dimensional Nevanlinna theory.

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Résumé — Nous établissons un critère d'algébricité concernant les feuilles des feuilletages algébriques définis sur un corps de nombres. Soit en effet K un corps de nombres plongé dans C, X une variété algébrique lisse sur K, munie d'un point K-rationnel P, et F un sous-fibré du fibré tangent T_X, défini sur K. Supposons de plus que le fibré vectoriel F soit involutif, i.e.., stable par crochet de Lie. Il définit alors un feuilletage holomorphe de la variété analytique X(C) et l'on peut considérer la feuille ℱ de ce feuilletage passant par P. Nous montrons que ℱ est algébrique lorque les conditions locales suivantes son satisfaites: i) Pour presque tout idéal premier p de l'annneau des entiers 𝒪_K de K, la réduction modulo p du fibré F est stablé par l'opération de puissance p-ième (où p désigne la caractéristique du corps résiduel 𝒪_K / p ). ii) La variété analytique ℱ satisfait à la propriété de Liouville; cela a lieu, par exemple, lorsque ℱ est l'image par une application holomorphe du complémentaire d'un sous-ensemble analytique fermé dans une variété algébrique. Ce critère d'algébricité unifie et généralise divers résultats de D. V. and G. V. Chudnovsky, André et Graftieaux. Il conduit aussi à de nouvelles conséquences. Par exemple, appliqué à un groupe algébrique G sur K, il montre qu'une sous-algèbre de Lie h de Lie G, définie sur K, est algébrique si et seulement si, pour presque tout idéal premier p de 𝒪_K , de caractéristique résiduelle p, la réduction modulo p de h est une sous-p-algèbre de Lie de la réduction modulo p de Lie G (i.e., est stable par puissance p-ième). Cet énoncé résout une conjecture d'Ekedahl et Shepherd-Barron. Le critère d'algébricité ci-dessus découle d'un critère d'algébricité plus général, concernant les germes de sous-variétés formelles des variétés sur les corps de nombres. La démonstration de ce dernier repose sur des “techniques de transcendance”, reformulées dans une version géométrique utilisant diverses notions élémentaires de géométrie d'Arakelov, et sur des estimations analytiques reliées au premier théorème fondamental de la théorie de Nevanlinna en dimension supérieure.

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Manucsrit re?u le 27 septembre 2000.     

Permutation polynomials with low differential uniformity over finite fields of odd characteristic

In this paper, we propose a construction of functions with low differential uniformity based on known perfect nonlinear functions over finite fields of odd characteristic. For an odd prime power q, it is proved that the proposed functions over the finite field Fq are permutations if and only if q≡3(mod 4).     

Permutation polynomials of degree 6 or 7 over finite fields of characteristic 2

In Dickson (1896–1897) [2], the author listed all permutation polynomials up to degree 5 over an arbitrary finite field, and all permutation polynomials of degree 6 over finite fields of odd characteristic. The classification of degree 6 permutation polynomials over finite fields of characteristic 2 was left incomplete. In this paper we complete the classification of permutation polynomials of degree 6 over finite fields of characteristic 2. In addition, all permutation polynomials of degree 7 over finite fields of characteristic 2 are classified.