# Linear independence of powers

## TL;DR

This paper proves that for r elements in an integral domain over an algebraically closed field, if any two are linearly independent, then their e-th powers are also linearly independent for some e between 1 and r!.

## Contribution

It establishes a bound on the exponent e ensuring the linear independence of powers of elements in an integral domain, extending previous understanding.

## Key findings

- Existence of an integer e between 1 and r! for which powers are linearly independent.
- Generalization of linear independence properties to powers of elements.
- Provides a new bound related to the factorial of the number of elements.

## Abstract

Given r elements in an integral domain over an algebraically closed field such that any two are linearly independent, we show that there is an integer e between 1 and r! such the eth powers of these elements are linearly independent.