## Geometry and Representation Theory Seminar

### Monday, February 5th, 2018

**Time:** 4:30 p.m. **Place:** Jeffery Hall 319

**Speaker:** David Wehlau (Queen's University)

**Title:** Khovanski Bases and Derivations

**Abstract: ** Let $R=K[x_1,,\dots,x_m,y_1,\dots,y_m,z_1,\dots,z_m]$ be a polynomial algebera over a field $K$ of characteristic zero, Let $\Delta$ be the locally nilpotent derivation on $R$ determined by $\Delta(z_i) = y_i$, $\Delta(y_i) = x_i$ and $\Delta(x_i)=0$ for $i=1,2,\dots,m$. This is an example of

a Weitzenb\"ock derivation. We exhibit a minimal set of generators $\mathcal G$ for the algebra of

constants $R^\Delta = \ker \Delta$. We also construct a Khovanski (or sagbi) basis for this algebra.

Even though this basis is infinite our proof yields an algorithm to express any element of $R^\Delta$ as

a polynomial in the elements of $\mathcal G$. In particular, this method shows how the classical techniques of polarization and restitution may be used in combination with Khovanski bases to yield a constructive method for expressing elements of a subalgebra as a polynomials in its generators.