Image of ideals under linear K-derivations and the LNED conjecture

Abstract

Let K be a field of characteristic zero and K[X] = K[x1, x2,...,xn] be the polynomial algebra in n variables over K. We show that, for a linear K-derivation d of K[X] and the maximal ideal m = (x1, x2,...,xn) of K[X], if d(m) is a MathieuZhao subspace of K[X], then the image of every m-primary ideal under d forms a Mathieu-Zhao subspace of K[X]. Additionally, we observe that the image of all monomial ideals under the K-derivation d = f∂x1 of K[X], for f ∈ K[X] forms an ideal of K[X]. Finally, we prove that the image of certain monomial ideals under a linear locally nilpotent K-derivation of K[x1, x2, x3] defined by d = x2∂x1 + x3∂x2 forms a Mathieu-Zhao subspace