# Anna Torstensson: Describing subalgebras of k[x] by conditions

**Time: **
Mon 2022-11-28 15.00 - 16.00

**Location: **
Zoom

**Video link: **
Meeting ID: 642 5442 0032

**Participating: **
Anna Torstensson, Lund

### Abstract

Subalgebras of the polynomial ring (in one or several variables) are most commonly described by a set of generators. If we are lucky it is given by generators that form a so called SAGBI basis. A SAGBI basis gives a more direct description of the leading terms occurring in the subalgebra, which in turn makes way for an algorithm for checking if a certain polynomial belongs to the subalgebra.

In this talk we will examine an alternative way to describe subalgebras (that are of finite codimension) in k[x]. This description involves only values of a polynomial and its (directional) derivatives in a finite number of points. It therefore gives a very direct and fast way to check subalgebra membership. We will see examples in one and several variables and discuss some advantages of this way of looking at subalgebras. I will also give you some insight into the ideas of the proof that such a description exists for all subalgebras of finite codimension in k[x].

Finally I will point to some (of the many) related open questions.

Main reference: Subalgebras in K[x] of small codimension Rode Grönkvist, Erik Leffler, Anna Torstensson & Victor Ufnarovski in Applicable Algebra in Engineering, Communication and Computing (2022)

(Can be found here: https://link.springer.com/article/10.1007/s00200-022-00573-4 )