Subgroups generated by rational functions in finite fields

Abstract

Abstract. For a large prime p, a rational function ψ ∈ F p (X) over the finite field F p of p elements, and integers u and H ≥ 1, we obtain a lower bound on the number consecutive values ψ(x), x = u+1, . . . , u+H that belong to a given multiplicative subgroup of F * p .

Sparsity of curves and additive and multiplicative expansion of rational maps over finite fields

“…We recall that for subgroups G, H Ă Fp, Corvaja and Zannier [10] haven give a nontrivial on N F pG, Hq. Using their result we obtain a bound on N F pI, Gq with an interval I and a subgroup G. It extends the result of Karpinski, Mérai and Shparlinski [15] who bound N F pI, Gq with F pX, Y q " f pXq´Y gpXq, see also [11,14,17,18].…”

“…We study some geometric properties of polynomial maps in finite fields. In particular, we continue investigating the introduced in [11] question of expansion of dynamical systems generated by polynomial and rational function maps in positive characteristic, see [5][6][7][8][9][14][15][16][17][18]20] and the reference therein for recent results, methods and applications. Here we consider both additive and multiplicative expansion, and also study more general compositions of several maps.…”

Polynomial values in small subgroups of finite fields

“…It is shown in [16] that if f ∈ F p [X] is a polynomial of degree d ≥ 2, then for any interval I = {u + 1, . .…”

Polynomial Values in Subfields and Affine Subspaces of Finite Fields

“…There are also related results where bounds on the size of the intersection # (f (A) ∩ B) are given, where A and B are some 'interesting' sets and f (A) = {f (a) : a ∈ A} is the value set of a polynomial f on A; see [8,10] for the case when both sets A and B are intervals of consecutive integers and [13,24] for the case when A is such an interval and B is a multiplicative subgroup of F p . Unfortunately in the very interesting case when both sets A and B are subgroups of F q no results are known.…”