Polynomials with minimal value set over Galois rings.
A quasi-permutation polynomial is a polynomial which is a bijection from one subset of a finite field onto another with the same number of elements. This is a natural generalization of the familiar permutation polynomials. Basic properties of quasi-permutation polynomials are derived. General criteria for a quasi-permutation polynomial extending the well-known Hermite's criterion for permutation polynomials as well as a number of other criteria depending on the permuted domain and range are established....
Introduction. Soit q une puissance d’un nombre premier p et soit le corps fini à q éléments. Une certaine analogie entre l’arithmétique de l’anneau ℤ des entiers rationnels et celle de l’anneau a conduit à étendre à de nombreuses questions de l’arithmétique classique. L’équirépartition modulo 1 est une de ces questions. Le corps des nombres réels est alors remplacé par le corps des séries de Laurent formelles, complété du corps des fractions rationnelles pour la valuation à l’infini et...
We are concerned with solving polynomial equations over rings. More precisely, given a commutative domain A with 1 and a polynomial equation antn + ...+ a0 = 0 with coefficients ai in A, our problem is to find its roots in A.We show that when A = B[x] is a polynomial ring, our problem can be reduced to solving a finite sequence of polynomial equations over B. As an application of this reduction, we obtain a finite algorithm for solving a polynomial equation over A when A is F[x1, ..., xN] or F(x1,...