Intersective Polynomials Arising from Sums of Powers

TL;DR

This paper establishes necessary and sufficient conditions for certain polynomials derived from sums of powers to have roots modulo all positive integers, advancing understanding of their solvability in modular arithmetic.

Contribution

It provides a complete characterization of when these sum-of-powers polynomials are solvable modulo every positive integer, a novel result in number theory.

Findings

Identifies conditions for roots modulo all positive integers

Characterizes polynomials based on sum of powers and parameter choices

Advances understanding of polynomial solvability in modular arithmetic

Abstract

Given a natural number $n \geq 2$, an integer $k$ and for a judiciously chosen $l = l(n)$ we give necessary and sufficient conditions for the polynomial $f_{n,k} = \big( \sum_{i=1}^{l} x_{i}^{n} \big) - k$ to have roots modulo every positive integer.