Studies on polynomial rings through locally nilpotent derivations/ Nikhilesh Dasgupta

Thesis (Ph.D.) - Indian Statistical Institute, 2019

Includes bibliography

Introduction -- Preliminaries -- On algebraic characterization of the affine three space -- On Nice and Quasi-Nice Derivations

Guided by Prof. Neena Gupta

The main aim of the thesis is to investigate the following problems :
(i) To find an algebraic characterization of the polynomial ring k[X, Y, Z]
over an algebraically closed field k of characteristic zero (in particular,
an algebraic characterization of the affine three space).
(ii) To determine the structure of the kernel of a nice derivation on the
polynomial ring R[X, Y, Z] over a PID R containing Q; in particular,
the structure of the kernel of a nice derivation on k[X1, X2, X3, X4] of
rank 3, where k is a field of characteristic zero.
The first problem will be discussed in Chapter 3 under the heading “On
algebraic characterization of the affine three space” while the second problem
will be taken up in Chapter 4 entitled “On Nice and Quasi-Nice Derivations”.
Sections 1.2 and 1.3 of this chapter present an overview of the main results
of Chapters 3 and 4, along with their contexts. In Chapter 2, we give the
necessary definitions (Section 2.1) and state some well-known results on locally
nilpotent derivations (Section 2.2) and on polynomial rings and projective
modules (Section 2.3).