Linear independence of powers for polynomials

Alexandru Crăciun

doi:10.48550/arxiv.2507.10163

July 14, 2025Open Access

Linear independence of powers for polynomials

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Key Points

The $r$-th powers of $k$ polynomials are shown to be linearly independent if $r$ exceeds $k {k-1 race 2}$ and 2.
Characterizing polynomials in $d$ variables, no two of which are linearly dependent, is fundamental to this result.
This finding relies on the assumption of a field with characteristic $0$, which is essential for polynomial behavior.
Implications of linear independence are significant for understanding polynomial relationships and representations.

Implication

This analysis shows that high powers of polynomials remain linearly independent in specified conditions.

Abstract

Given k ≥ 2 polynomials in d ≥ 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 k-1 2, 2\, the r-th powers of the polynomials are linearly independent.

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Cite This Study

Alexandru Crăciun (2025) studied this question.

synapsesocial.com/papers/68de6f4283cbc991d0a22e9d https://doi.org/10.48550/arxiv.2507.10163

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Linear independence of powers for polynomials

Key Points

Implication

Abstract

Discussion

Cite This Study

Linear independence of powers for polynomials

Key Points

Implication

Abstract

Discussion

Cite This Study

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