Locally Nilpotent Derivations and the Cancellation Problem in Affine Algebraic Geometry

Abstract

Let K be a field of characteristic zero and let R [n] denote the polynomial ring in n variables over a ring R for any n ∈ N , n > 0. We present some basic theory for the study of locally nilpotent derivations as an effective tool in algebraic geometry. Using this tool, we examine the Cancellation Problem in affine algebraic geometry, which asks: Let A be a K -algebra such that A[1] = K [n+1]. Does it follow that A = K [n]? This problem is open for n > 2. We present the solutions to the cases n = 1 and n = 2, in the latter case essentially following the algebraic method of Crachiola and Makar-Limanov [9]. We examine a potential counterexample, R = K [X, Y, Z, T]/⟨X + X ²Y + Z² + T³⟩, referred to as Russell's Cubic. We show that while R closely resembles a polynomial ring in 3 variables, we have that R ≠ K&sqbl0;3&sqbr0; , a result due to Makar-Limanov [25]. This is achieved by showing that the Derksen invariant of R is not equal to the Derksen invariant of K&sqbl0;3&sqbr0; . It is unknown if R[1] is a polynomial ring in 4 variables over K , nonetheless, we examine some properties of R [1] which highlight its similarities with K&sqbl0;4&sqbr0; .