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.
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 Perron-Frobenius 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 . We
introduce a new dioid, namely a variant of the standard max-plus algebra whose elements carry