ИСТИНА

This talk is based on a joint work with Anton Shafarevich. The Makar-Limanov in- variant ML(X) of an affine variety X is the intersection of all kernels of locally nilpotent derivations on K[X]. The Derksen invariant HD(X) is the subalgebra in K[X] generated by all kernels of locally nilpotent derivations. We investigate modified Makar-Limanov and modified Derksen invariants of an affine algebraic variety. The modified Makar-Limanov invariant ML∗(X) is the intersection of kernels of all locally nilpotent derivations with slices and the modified Derksen invariant HD∗(X) is the subalgebra generated by these kernels. We prove that for every variety ML(X) = ML∗(X), if there exists a locally nilpotent derivation with a slice. Also we 2 construct an example of a variety admitting a locally nilpotent derivation with a slice such that HD∗(X) ̸= HD(X).