Linear independence of powers for polynomials

TL;DR

This paper proves that for a set of linearly independent polynomials over a field of characteristic zero, sufficiently high powers of these polynomials remain linearly independent, extending understanding of polynomial power independence.

Contribution

It establishes a new lower bound on the power degree ensuring linear independence of polynomial powers, generalizing previous results in polynomial algebra.

Findings

For any set of linearly independent polynomials, their sufficiently high powers are also linearly independent.

The minimal power degree depends on the number of polynomials and their pairwise relations.

Provides a theoretical foundation for polynomial power independence in algebraic contexts.

Abstract

Given $k \ge 2$ polynomials in $d \ge 1$ variables with coefficients in a field of characteristic $0$, such that no two are linearly dependent, we show that for any integer $r$ greater than $\max\left\{k {k-1 \choose 2}, 2\right\}$, the $r$-th powers of the polynomials are linearly independent.