[Paper Review] Estimations of the particular periodicity in case of the extremal periods in Shirshov's Height theorem

Mikhail Kharitonov|arXiv (Cornell University)|Aug 31, 2011

semigroups and automata theory4 references3 citations

TL;DR

This paper provides improved upper bounds on the essential height of l-generated PI-algebras of polynomial growth degree n by analyzing periodic subwords in non-n-reducible words. Using combinatorial techniques on word structure, including lexicographic comparability and Dilworth's theorem, it establishes tight estimates for subwords with periods 2, 3, and n−1, leading to a subexponential bound of $\Upsilon(n,l) = 8(l+1)^n n^6$ on the essential height, significantly improving prior estimates and advancing the proof of Shirshov's Height Theorem.

ABSTRACT

Let us recall the well-known Shirshov's Height Theorem. "Let A be a finitely generated algebra of degree d. Then there exists a finite set Y which is the subset of A that A has and an integer h' = h(A) such that A has Shirshov's height h' over set Y. For Y we may take the set of words of length

Motivation & Objective

• To improve the upper bound on the essential height of l-generated PI-algebras with polynomial identities of degree n.
• To analyze the structure of non-n-reducible words by focusing on lexicographically comparable subwords with specific periods (2, 3, and n−1).
• To refine the proof strategy of Shirshov's Height Theorem via combinatorial word analysis and periodicity estimation.
• To bridge the gap between known lower bounds (e.g., $\sim (l-1)n^2/4 + 1$) and the upper bounds in the literature.

Proposed method

• Application of Dilworth’s theorem to bound the number of lexicographically comparable subwords with period 2, 3, and n−1.
• Use of Rosamond (k-graph) representations to model word trajectories and analyze periodicity in subwords.
• Recursive application of bounds via lemmas on $\beth(t,l,n)$, leveraging logarithmic depth and exponentiation to control growth.
• Codification of small-period subwords into a framework that supports the derivation of the main height bound.
• Use of contradiction arguments to show that too many non-comparable high-period subwords force n-reducibility.
• Establishment of a chain of inequalities involving $\beth(t,l,n)$, leading to the final bound $\Upsilon(n,l) < 8(l+1)^n n^6$.

Experimental results

Research questions

• RQ1 What is the maximal number of lexicographically comparable subwords with period 2 in any monoid of an l-generated PI-algebra of degree n?
• RQ2 How many subwords with period 3 can exist in such algebras without forcing n-reducibility?
• RQ3 Can the structure of subwords with period n−1 be used to derive tighter bounds on essential height?
• RQ4 What is the tightest possible upper bound on the essential height of l-generated PI-algebras with polynomial identities of degree n?
• RQ5 How do the new bounds compare with known lower bounds based on Gelfand–Kirillov dimension and Amitsur–Levitzky theorems?

Key findings

• The number of lexicographically comparable subwords with period 2 is at most $\frac{(2l-1)(n-1)(n-2)}{2}$ in any monoid of the algebra.
• The number of such subwords with period 3 is at most $(2l-1)(n-1)(n-2)$.
• For period $n-1$, the maximum number of such subwords is bounded by $(l-2)(n-1)$.
• The essential height of an l-generated PI-algebra of degree n is less than $\Upsilon(n,l) = 8(l+1)^n n^6$, a subexponential bound.
• The bound $\Upsilon(n,l)$ improves upon earlier estimates such as $\Phi(l,n) = 2^{87}l \cdot n^{12\log_3 n + 48}$.
• The results support a refined proof of Shirshov’s Height Theorem via periodic word decomposition and combinatorial control of subword structure.