Counting integers representable as images of polynomials modulo $n$

TL;DR

This paper develops a method to count how many integers modulo n can be expressed as polynomial values, focusing on sums and differences of squares, with applications to specific quadratic forms.

Contribution

It introduces a new approach to determine the count of representable integers for certain polynomial forms, especially sums of powers and quadratic forms.

Findings

Method effectively computes the count for polynomials of the form c_1x_1^k + ... + c_t x_t^k

Applied to sums and differences of squares, including x^2+y^2 and x^2-y^2

Provides explicit counts for integers representable by these quadratic forms

Abstract

Given a polynomial $f(x_1,x_2,\ldots, x_t)$ in $t$ variables with integer coefficients and a positive integer $n$, let $\alpha(n)$ be the number of integers $0\leq a<n$ such that the polynomial congruence $f(x_1, x_2, \ldots, x_t)\equiv a\ (mod\ n)$ is solvable. We describe a method that allows to determine the function $\alpha$ associated to polynomials of the form $c_1x_1^k+c_2x_2^k+\cdots+c_tx_t^k$. Then we apply this method to polynomials that involve sums and differences of squares, mainly to the polynomials $x^2+y^2, x^2-y^2$ and $x^2+y^2+z^2$.