On semigroup orbits of polynomials and multiplicative orders

TL;DR

This paper investigates the behavior of polynomial semigroup orbits modulo large primes, demonstrating that under certain conditions, these orbits do not contain many elements with small multiplicative order, extending prior research in the area.

Contribution

It extends previous work by establishing new restrictions on the multiplicative orders of elements in polynomial semigroup orbits modulo large primes.

Findings

Semigroup orbits of polynomials have limited elements of small multiplicative order.

Under natural restrictions, the number of small-order elements in these orbits is bounded.

The results generalize and extend earlier findings by Shparlinski (2017).

Abstract

We show, under some natural restrictions, that some semigroup orbits of polynomials cannot contain too many elements of small multiplicative order modulo a large prime $p$, extending previous work of Shparlinski (2017).