Georgia Institute of Technology College of Sciences

Differential operator rings can be described as polynomial rings over differential operators. We will study two of them: first the relatively simple ring of differential operators R with rational function coefficients, and then the more complicated ring D with polynomial coefficients, or the Weyl algebra. It turns out that these rings are non-commutative because of the way differential operators act on smooth functions. Despite this, with a bit of work we can show properties similar to the regular polynomial rings, such as division, the existence of Gröbner bases, and Macaulay's theorem. As an example application, we will describe the holonomic gradient descent algorithm, and show how it can be used to efficiently solve computationally heavy problems in statistics.