On the Bound of Inverse Images of a Polynomial Map

TL;DR

This paper establishes an upper bound on the number of solutions to polynomial equations over finite fields and complex numbers, generalizing previous bounds to higher-degree polynomials and different fields.

Contribution

It provides a new upper bound for the number of solutions where polynomials take roots of unity, extending prior results to higher degrees and various fields.

Findings

Derived an upper bound for solutions over finite fields.

Extended bounds to polynomials of degree greater than one.

Applicable to both finite fields and complex numbers.

Abstract

Let $f_1(x),\ldots,f_n(x)$ be some polynomials. The upper bound on the number of $x\in\mathbb F_p$ such that $f_1(x),\ldots,f_n(x)$ are roots of unit of order $t$ is obtained. This bound generalize the bound of the paper \cite{V-S} to the case of polynomials of degrees greater than one. The bound is obtained over fields of positive characteristic and over the complex field.