# A relation between polynomials

This is a brief foray into algebra from a 2006 REU project at Texas A&M.

Given two polynomials $P,Q \in \mathbb C[z_1,\dots,z_n]$, we write $Q\preccurlyeq P$ if there is a differential operator $T\in \mathbb C[\frac{\partial}{\partial z_1},\dots, \frac{\partial}{\partial z_n}]$ such that $Q=T P$.

The relation $\preccurlyeq$ is reflexive and transitive, but is not antisymmetric. If both $Q\preccurlyeq P$ and $Q\preccurlyeq P$ hold, we say that $P$ and $Q$ are $\partial$-equivalent, denoted $P\thicksim Q$.

A polynomial is $\partial$-homogeneous if it is $\partial$-equivalent to a homogeneous polynomial. Obviously, any polynomial in one variable has this property. Polynomials in more than one variable usually do not have it.

The interesting thing about $\partial$-homogeneous polynomials is that they are refinable, meaning that one has a nontrivial identity of the form $P(z)=\sum_{j\in\mathbb Z^n} c_{j} P(\lambda z-j)$ where $c_{j}\in \mathbb C$, $j\in \mathbb Z^n$, and only finitely many of the coefficients $c_j$ are nonzero. The value of $\lambda$ does not matter as long as $|\lambda|\ne 0,1$. Conversely, every $\lambda$-refinable polynomial is $\partial$-homogeneous.