Modified Makar-Limanov and Derksen invariants

TL;DR

This paper explores modified invariants related to affine algebraic varieties, showing conditions under which they coincide with classical invariants and providing a counterexample where they differ.

Contribution

It introduces modified Makar-Limanov and Derksen invariants, analyzing their properties and relationships with existing invariants in affine algebraic geometry.

Findings

Modified Makar-Limanov invariant equals classical when a slice exists.

Constructed example where modified Derksen invariant differs from Derksen invariant.

Proved conditions for invariants to coincide or differ.

Abstract

We investigate modified Makar-Limanov and Derksen invariants of an affine algebraic variety. The modified Makar-Limanov invariant is the intersection of kernels of all locally nilpotent derivations with slices and the modified Derksen invariant is the subalgebra generated by these kernels. We prove that modified Makar-Limanov invariant coincide with Makar-Limanov invariant if there exists a locally nilpotent derivation with a slice. Also we construct an example of a variety admitting a locally nilpotent derivation with a slice such that modified Derksen invariant does not coincide with Derksen invariant.