Polynomials whose nth powers have prescribed multiple-of-nth-degree coefficients

TL;DR

This paper proves that, under mild conditions, for any given field elements, there exists a degree-m polynomial whose nth power has prescribed coefficients at multiples of n, with a constructive proof for the case n=2.

Contribution

It establishes the existence of polynomials with prescribed coefficients in their nth powers and offers a more constructive proof specifically for the case n=2.

Findings

Existence of such polynomials under mild hypotheses

Constructive proof for the case n=2

Generalization to arbitrary field elements

Abstract

We show under a mild hypothesis that given field elements $a_0, \dots, a_m \in K$, there always exists a degree-$m$ polynomial whose $n$th power whose degree-$jn$ coefficient is equal to $a_j$ for $0 \leq j \leq m$. We provide an alternate proof for the $n = 2$ case which is more constructive.