Polynomial values in small subgroups of finite fields

Abstract

Abstract. For a large prime p, and a polynomial f over a finite field F p of p elements, we obtain a lower bound on the size of the multiplicative subgroup of F * p containing H ≥ 1 consecutive values f (x), x = u + 1, . . . , u + H, uniformly over f ∈ F p [X] and an u ∈ 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].…”

“…The group size G Ψ,u pNq generated by the first N elements has been investigated [11,17,18]. For example, in [17, Theorem 1.1] it is proved that for a polynomial f pXq P F q rXs of degree d, satisfying some mild conditions, and for initial value u P F q , we have for any ν ě 1 that…”

Values of rational functions in small subgroups of finite fields and the identity testing problem from powers

“…It remains to bound L. Assume that A`B " D for some A P A 0 , B P B 0 pAq and D P E 8dh and write A 0`B0`D8 " A 8`B8`D0 . Then we have (9) A 0 ď B 8`D0 .…”

“…In the prime field case, Gómez-Pérez and Shparlinski [5] and Shparlinski [9] provided lower bounds on E f pIq for intervals I Ă F p and polynomials f pXq P F p rXs.…”

Multiplicative energy of polynomial images of intervals modulo $q$

“…In this paper we give a result in the spirit of Shparlinski's theorem, [12,Theorem 7], through the application of the Graham-Ringrose Theorem as improved by Chang [2]. In [12] lies an improvement to an earlier result of Shparlinski and Gómez-Pérez, [7,Theorem 7], which discusses the greatest lower bound for the order of a subgroup of a finite field containing the image of an interval of consecutive integers under a rational function.…”