# Algebras of linear growth and the dynamical Mordell-Lang conjecture

## TL;DR

This paper explores the connection between algebraic growth properties, the dynamical Mordell-Lang conjecture, and the rationality of Hilbert series in finitely presented algebras, revealing conditions under which these properties hold or fail.

## Contribution

It establishes a link between the automaton property of algebras of linear growth and the dynamical Mordell-Lang conjecture, providing new insights into Hilbert series rationality.

## Key findings

- Positive answer depends on the field being an algebraic extension of its prime subfield.
- Existence of graded algebras with irrational Hilbert series.
- Infinite variety of Hilbert series for quadratic algebras with 5 generators.

## 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 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 finite characteristic case give examples of algebras with irrational Hilbert series.