A question of Joseph Ritt from the point of view of vertex algebras.

Let k be a field of characteristic zero. This paper studies a problem proposed by Joseph F. Ritt in 1950. Precisely, we prove that (1) If p ⩾ 2 is an integer, for every integer i ∈ N , the nilpotency index of the image of T i in the ring k { T } / [ T p ] equals (i + 1) p − i. (2) For every pair of integers (i , j) , the nilpotency index of the image of T i U j in the ring k { T } / [ T U ] equals i + j + 1. [ABSTRACT FROM AUTHOR]