 
Summary: PRIMITIVE MATRICES OVER POLYNOMIAL SEMIRINGS
SHIGEKI AKIYAMA AND HORST BRUNOTTE
Abstract. An extension of the definition of primitivity of a real nonnegative matrix to ma
trices with univariate polynomial entries is presented. Based on a suitably adapted notion of
irreducibility an analogue of the classical characterization of real nonnegative primitive matri
ces by irreducibility and aperiodicity for matrices with univariate polynomial entries is given.
In particular, univariate polynomials with nonnegative coefficients which admit a power with
strictly positive coefficients are characterized. Moreover, a primitivity criterion based on almost
linear periodic matrices over dioids is presented.
1. Introduction
In this paper, we are interested in the behavior of powers of matrices over a commutative
semiring. Inside this semiring we fix a subset which we think of as the 'nonnegative' elements
and which in turn contains another subset which we think of as the 'positive' elements. Given a
matrix with only 'nonnegative' entries we ask for conditions which guarantee that a power of this
matrix has only 'positive' entries. In this case we talk of a primitive matrix.
We answer the aforementioned question in two different ways. Our first approach is a generaliza
tion of the classical PerronFrobenius Theorem on real matrices with nonnegative coefficients:
Here primitivity is characterized by irreducibility and aperiodicity. Our second approach is based
on the theory of almost linear periodic sequences over dioids as introduced by Gavalec [9]. We
introduce a new dioid, namely a variant of the standard maxplus algebra whose elements carry
