Image of the Jacobian map for A = k[X]/(X^p) in characteristic p

Determine the exact image of the Jacobian map f ↦ jac_σ(u_f), where A = k[X]/(X^p) with k of characteristic p and u_f: A → A is the automorphism defined by u_f(x) = f ∈ rad A \ rad^2 A. In particular, establish whether the necessary conditions for a unit a = ∑_{i=0}^{p-1} a_i x^i to be a Jacobian—namely ∑_{i=0}^{p-1} (−1)^i a_i = 1 and a_0 ≠ 0—are also sufficient.

Background

For the smallest non-semisimple group algebras in characteristic p, the paper analyzes A = k[X]/(X^p), computes Jacobians of automorphisms u_f via an explicit formula, and derives necessary conditions on a unit a to arise as a Jacobian. Empirical evidence from Gröbner basis elimination for small p suggests these conditions may be sufficient, but a proof is lacking.

Resolving this problem would characterize the Jacobian image precisely, clarify the structure of automorphism-induced Jacobians in local truncated polynomial algebras, and potentially inform broader classes of modular group algebras.