Tropical recurrent sequences

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.Comment: it is described when the entropy is positiv

Entropy of tropical holonomic sequences

We introduce tropical holonomic sequences of a given order and calculate their entropy in case of the second order

Entropy of radical ideal of a tropical prevariety

The entropy of a tropical ideal is introduced. The radical of a tropical ideal consists of all tropical polynomials vanishing on the tropical prevariety determined by the ideal. We prove that the entropy of the radical of a tropical bivariate polynomial with vanishing coefficients equals zero. Also we prove that the entropy of a zero-dimensional tropical prevariety vanishes. An example of a non-radical tropical ideal having a positive entropy is exhibited

Энтропия тропических рекуррентных последовательностей

Рассматривается энтропия тропических рекуррентных последовательностей. В первой части работы мы приводим и доказываем точные оценки для энтропии. Во второй части работы рассматривается энтропия тропических булевых векторов. В этом случае вводится тропический булевский граф. Мы доказываем, что нахождение энтропии в этом случае сводится к нахождению оптимального цикла в тропическом булевском графе. Наконец, для тропических булевых векторов мы доказываем, что размерность пространства рекуррентных последовательностей длины s как функция от s является квазилинейной функцией.Entropy of tropical recurrent sequences is considered. In the first part of work we provide and prove sharp estimates for entropy. In the second part of work entropy of tropical boolean vectors is considered. In this case zero-infinity graph is introduced. We prove that finding the entropy in this case reduces to finding the optimal cycle in the zero-infinity graph. Finally, for tropical boolean vectors we prove that the dimension of the space of recurrent sequences of length s as a function of s is a quasi-linear function