Abstract
Let Xo⊂R n be an irreducible analytic germ and Ω the order space of its field of meromorphicfunetion germs. A formal half-branch in Xo is a kind of C∞-map germ c∶[0,ε)→Xo; an ordering αεΩ is centered at c if it contains the functions which are positive on c. We obtain a partition Ω1,...,Ωd, d=dim Xo, of the set Ω* of central (i.e.: centered at some half-branch) orderings, according to the “dimension” of half-branches. Then we show that all Ωe, e= 1,.,d, as well as the set Ω\Ω* of noncentral orderings, are dense in Ω. Finally, we solve the 17th Hubert Problem for analytic germs.
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Ruiz, J.M. Central orderings in fields of real meromorphic function germs. Manuscripta Math 46, 193–214 (1984). https://doi.org/10.1007/BF01185201
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DOI: https://doi.org/10.1007/BF01185201