Improving Razmyslov's (1974) bound: the Nagata–Higman index is at most n² − 2 for n ≥ 3, and other miscellanea

Let d(n) be the least m such that every associative algebra over a field of characteristic zero satisfying xn = 0 also satisfies x1x2⋯xm = 0 (the Nagata–Higman index). Razmyslov proved d(n) ≤ n2 in 1974, and the Kuzmin (1975) conjecture is d(n) = n(n+1)/2. We prove d(n) ≤ n2 − 1 for n ≥ 2 and d(n) ≤ n2 − 2 for n ≥ 3; in particular d(6) ≤ 34. The proof uses Formanek's criterion in cycle form and studies, one isotypic block of SN at a time, the span of the N-cycles inside the block: a block with two corners meets this span exactly when its shape is a self-conjugate hook, and for every k ≥ 2 the block of (k+1, 2, 1k−1), which has three corners, meets it too. The next step, d(n) ≤ n2 − 3, reduces to one statement about the block of (nn−2, n−1, n−2); for every n ≥ 4 we prove the part of it that concerns the constituents with at most two rows of length 3, and for n = 5 the whole statement (computer-assisted). Through the hook theorem the question becomes one about the polynomial identities of Kemer's superalgebras Mk,l over the Grassmann algebra: lower bounds for their PI degrees give upper bounds for d(n), and the lower half of the Samoilov (2010) conjecture on these PI degrees would give d(n) < (8/9)n2 for n ≥ 3.