Conference abstracts

Session S05 - Rings and Algebras

July 29, 17:30 ~ 17:55

## On $D$ algebras.

### Leonid Makar-Limanov

### Wayne State University, USA - lml@wayne.edu

Consider an algebraic function $z$ of $n$ variables $x_1, x_2, \dots, x_n$. Denote by $D(z)$ a subalgebra of the field ${\Bbb C}(x_1, x_2, \dots, x_n)[z]$ which is generated by $x_1, x_2, \dots, x_n; \ z$ and all partial derivatives of $z$. I am interested in properties of algebras $D(z)$.

In my talk I will discuss the following conjectural dichotomy:

If $z \in {\Bbb C}[x_1, \dots, x_n]$ then (obviously) $D(z) = {\Bbb C}[x_1, \dots, x_n]$, but if $z \not\in {\Bbb C}[x_1, \dots, x_n]$ then $D(z)$ cannot be embedded into a polynomial ring.