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Towards a cell decomposition for rational functions. (English) Zbl 0632.93009

The paper deals with the decomposition of the space rat n of strictly proper rational functions \(g(s)=q(s)/p(s)\) of degree n into continued fraction cells. For \(\ell,m=\overline{0,n}\) with \(\ell +m=n\), denoting by \(\sigma (H_ g)\) the signature of the \(n\times n\) Hankel matrix associated to g (the Cauchy index) and by rat(\(\ell,m)\) the connected component \(\{\) \(g: g\in rat n\), \(\sigma (H_ g)=\ell -m\}\), this decomposition is conjectured to be a cell decomposition in the topological sense.
Several combinatorial formulas pertaining to the cell decomposition are presented the last section of the paper is devoted to the effect of certain scalings on this decomposition.
Reviewer: O.Pastravanu

MSC:

93B15 Realizations from input-output data
30B70 Continued fractions; complex-analytic aspects
54H10 Topological representations of algebraic systems
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