This is a brief foray into algebra from a 2006 REU project at Texas A&M.
Given two polynomials , we write if there is a differential operator such that .
The relation is reflexive and transitive, but is not antisymmetric. If both and hold, we say that and are -equivalent, denoted .
A polynomial is -homogeneous if it is -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 -homogeneous polynomials is that they are refinable, meaning that one has a nontrivial identity of the form where , , and only finitely many of the coefficients are nonzero. The value of does not matter as long as . Conversely, every -refinable polynomial is -homogeneous.