Corps et anneaux de Rolle

Abstract:

Brown, Craven and Pelling have proved that if the polynomials over an ordered field $K$ satisfy Rolle’s theorem, they satisfy it for any ordering on $K$. We say that such a field is a Rolle field. We prove that this is a first order property in the language of rings, and that the theory of Rolle fields is decidable. Then we give a common generalisation of these fields and the real closed rings defined by Cherlin and Dickmann: The polynomials over an ordered ring $A$ satisfy Rolle’s theorem iff $A$ is the valuation ring of a henselian valued field with Rolle residue field and $m$-divisible value group for all odd $m$.