Subexponential estimates in Shirshov's theorem on height. (Q2901896)

The celebrated Shirshov height theorem is one of the corner stones of combinatorial PI-theory. It states that any finitely generated PI-algebra \(A\) is of bounded height \(h\). This means that there exists a constant \(h\) depending on the number \(l\) of the generators of \(A\) and the degree \(n\) of the polynomial identity satisfied by \(A\) such that the elements of \(A\) are linear combinations of products \(u_1^{k_1}\cdots u_r^{k_r}\), where \(u_i\) are words of the generators of length \(\leq n-1\) and \(r\leq h\).NEWLINENEWLINE The proof of the theorem easily follows from the lemma of Shirshov in the theory of combinatorics of words: There is a constant \(N=N(n,d,l)\) such that every word of length \(>N\) contains a subword of the form \(u^d\) or of the form \(u_1\cdots u_n\), where \(u_1\succ\cdots\succ u_n\) in the lexicographical ordering. There are many improvements of the Shirshov theorem concerning the length of the words \(u_i\) in the products \(u_1^{k_1}\cdots u_r^{k_r}\), estimates for the height \(h\), etc. Up till now all estimates for \(h\) are exponential.NEWLINENEWLINE In the paper under review the authors show that the height is bounded subexponentially and \(h\leq\Phi(n,l)=4^Al\cdot n^B\), where \(A=21\log_34+17\), \(B=30\log_34+10+12\log_3n\) (and \(n\) and \(l\) are the degree of the polynomial identity and the number of the generators of \(A\), respectively). The authors obtain subexponential estimates also for the essential height \(h_{\text{ess}}\) which considers products of the form \(v_0u_1^{k_1}v_1\cdots v_{r-1}u_r^{k_r}v_r\), \(r\leq h_{\text{ess}}\) for words \(u_i,v_i\) of bounded length.NEWLINENEWLINE The proofs are based on careful analysis of the occurrence of degrees of subwords in the words in \(l\) letters. One of the new moments is the use of the Dilworth theorem which, up till now was involved in PI-theory for multilinear words only (and was a key step in the transparent proof of Latyshev of the theorem of Regev for the exponential growth of the codimension sequence of PI-algebras).