Minimal degree of a standard polynomial identity in Fn(Ap)

Determine the minimal integer k for which the standard alternating polynomial Sk(x1, …, xk) = ∑_{σ∈Sk} (−1)^σ x_{σ(1)}···x_{σ(k)} is a polynomial identity for the relatively free algebra Fn(Ap) of rank n in the variety Ap of Lie-nilpotent associative algebras of index p over a field of characteristic zero.

Background

By Kemer’s theorem, every finitely generated PI-algebra over an infinite field satisfies some standard identity Sd. In Section 5 the authors ask for the minimal degree of such a standard identity specifically for Fn(Ap).

They provide supporting bounds and patterns from their explicit determinations for p = 3 and p = 4, and formulate a conjecture giving a parity-dependent formula for the minimal degree once n is sufficiently large relative to p.