On homogeneous polynomials determined by their partial derivatives

TL;DR

This paper proves that for a generic homogeneous polynomial, the polynomial can be uniquely identified (up to a constant) from the vector space generated by its partial derivatives of a certain order, under specific degree conditions.

Contribution

It establishes a new theoretical result linking the polynomial's degree and the order of derivatives needed for unique determination.

Findings

Homogeneous polynomial is determined by derivatives of order up to d/2 - 1.

The result holds for a generic polynomial, indicating broad applicability.

Provides a mathematical foundation for polynomial identification from derivatives.

Abstract

We prove that a generic homogeneous polynomial of degree $d$ is determined, up to a nonzero constant multiplicative factor, by the vector space spanned by its partial derivatives of order $k$ whenever $k\leq\frac{d}{2}-1$.