Nonnegative companion matrices and star-height of \(\mathbb{N}\)-rational series
A formal power series is called \(\mathbb{N}\)-rational if it can be obtained from polynomials with coefficients in \(\mathbb{N}\) by using sums, Cauchy products, and star operations \(P^*\), where for \(P\) such that \(P(0)= 0\), \[ P^*= {1\over 1-P}= \sum_{n\geq 0} P^n. \] The star-height of such a series is the minimal number of star operations needed to construct it as above. It is known that the star-height of any \(\mathbb{N}\)-rational series in one variable is at most 2 (see [\textit{A. Salomaa} and \textit{M. Soittola}, Automata-theoretic aspects of formal power series, Springer (1978; Zbl 0377.68039) and \textit{T. Katayama}, \textit{M. Okamoto} and \textit{H. Enomoto}, Inf. Control 36, 85-101 (1978; Zbl 0378.68042)]). Since the \(\mathbb{N}\)-rational series of star-height \(0\) are exactly the polynomials, an interesting question is to characterize the \(\mathbb{N}\)-rational series of star-height \(1\). The author proves the nice following result: If an \(\mathbb{N}\)-rational series has star-height \(1\), then its real positive roots \(\lambda_i\) are such that \(\forall i\exists k_i\) such that \(\lambda^{k_i}_i\) is a Perron number that has no positive real algebraic conjugate. This condition is also sufficient if the series has a dominating root. (A Perron number is a real algebraic number \(>1\) strictly greater than all the moduli of its conjugates. The author proposes the term of Handelman numbers for Perron numbers satisfying the extra condition above).
scientific article; zbMATH DE number 1177589
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