Elsevier

Advances in Mathematics

Volume 343, 5 February 2019, Pages 141-156
Advances in Mathematics

Algebras of linear growth and the dynamical Mordell–Lang conjecture

https://doi.org/10.1016/j.aim.2018.11.016Get rights and content
Under an Elsevier user license
open archive

Abstract

Ufnarovski remarked in 1990 that it is unknown whether any finitely presented associative algebra of linear growth is automaton, that is, whether the set of normal words in the algebra form a regular language. If the algebra is graded, then the rationality of the Hilbert series of the algebra follows from the affirmative answer to Ufnarovski's question. Assuming that the ground field has a positive characteristic, we show that the answer to Ufnarovskii's question is positive for graded algebras if and only if the basic field is an algebraic extension of its prime subfield. Moreover, in the “only if” part we show that there exists a finitely presented graded algebra of linear growth with irrational Hilbert series. In addition, over an arbitrary infinite basic field, the set of Hilbert series of the quadratic algebras of linear growth with 5 generators is infinite.

Our approach is based on a connection with the dynamical Mordell–Lang conjecture. This conjecture describes the intersection of an orbit of an algebraic variety endomorphism with a subvariety. We show that the positive answer to Ufnarovski's question implies some known cases of the dynamical Mordell–Lang conjecture. In particular, the positive answer for a class of algebras is equivalent to the Skolem–Mahler–Lech theorem which says that the set of the zero elements of any linear recurrent sequence over a zero characteristic field is the finite union of several arithmetic progressions. In particular, the counter-examples to this theorem in the positive characteristic case give examples of algebras with irrational Hilbert series.

MSC

16W50
14L30
11B37
68Q45

Keywords

Graded algebra
Hilbert series
Growth
Linear recurrence
Dynamical Mordell–Lang conjecture
Automaton algebra

Cited by (0)

The article was prepared within the framework of the Academic Fund Program at the National Research University Higher School of Economics (HSE) in 2019–2020 (grant 19-01-031) and by the Russian Academic Excellence Project “5–100”. Partially supported by the grant 18-01-00908 of the Russian Basic Research Foundation.