TL;DR
This paper introduces tropical recurrent sequences, explores conditions for their periodicity, and presents algorithms and bounds related to their existence and entropy, advancing understanding of tropical linear recurrences.
Contribution
It studies the existence of non-periodic tropical recurrent sequences, provides an algorithm for testing this, and introduces the concept of tropical entropy with bounds.
Findings
Existence of non-periodic sequences depends on the vector properties.
An algorithm is developed to test for non-periodic sequences with integer vectors.
Bounds on the tropical entropy of a vector are established.
Abstract
Tropical recurrent sequences are introduced satisfying a given vector (being a tropical counterpart of classical linear recurrent sequences). We consider the case when Newton polygon of the vector has a single (bounded) edge. In this case there are periodic tropical recurrent sequences which are similar to classical linear recurrent sequences. A question is studied when there exists a non-periodic tropical recurrent sequence satisfying a given vector, and partial answers are provided to this question. Also an algorithm is designed which tests existence of non-periodic tropical recurrent sequences satisfying a given vector with integer coordinates. Finally, we introduce a tropical entropy of a vector and provide some bounds on it.
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