On Rolle fields, with examples
As it is known, a field \(K\) is called a Rolle field if it admits a total order \(\leq\), compatible with the field structure and the following property is satisfied: if \(f\in K[X]\), \(a,b\in K\) and \(a<b\), \(f(a)= f(b)\), then there exists \(c\in K\) such that \(a<c<b\) and the derivative \(f'(c)= 0\). The paper focuses on henselian valued Rolle fields, concrete examples are provided by Rolle fields \(K= [[ k^{S,\leq} ]]\) of generalized power series, which have been studied in several papers of the author [with \textit{G. A. Elliott}, Arch. Math. 54, 365-371 (1990; Zbl 0688.13006) and Manuscr. Math. 75, 115-150 (1992; Zbl 0767.12001)]. Here \(S\) is a torsion-free abelian group, \(\leq\) is a compatible subtotal order, \(k\) is a field and \(K\) consists of all generalized power series with coefficients in \(k\), exponents in \(S\) and support which is artenian and narrow, i.e. quasi-well-ordered. A basic result is the analogue of the theorem of Brown, Craven and Pelling for henselian valued fields \((K,v)\): \(K\) is a Rolle field if and only if the residue field \(\overline {K}\) has the Rolle property and the value group is odd-divisible. This allows to give many concrete examples of Rolle fields of generalized power series (as indicated in the final section of the paper). In \S 3, one finds results concerning Rolle fields admitting only finitely many compatible total orders. As a consequence, it is shown how to obtain, for each \(n\geq 1\), examples of Rolle fields of generalized power series admitting exactly \(2^ n\) compatible total orders.
scientific article; zbMATH DE number 769519
scientific article