A criterion for membership in archimedean semirings
Abstract
We prove an extension of the classical Real Representation Theorem (going back to Krivine, Stone, Kadison, Dubois and Becker and often called KadisonDubois Theorem). It is a criterion for membership in subsemirings (sometimes called preprimes) of a commutative ring. Whereas the classical criterion is only applicable for functions which are positive on the representation space, the new criterion can under certain arithmetic conditions be applied also to functions which are only nonnegative. Only in the case of preorders (i.e., semirings containing all squares), our result follows easily from recent work of Scheiderer, Kuhlmann, Marshall and Schwartz. Our proof does not use (and therefore shows) the classical criterion. We illustrate the usefulness of the new criterion by deriving a theorem of Handelman from it saying inter alia the following: If an odd power of a real polynomial in several variables has only nonnegative coefficients, then so do all sufficiently high powers.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 September 2004
 DOI:
 10.48550/arXiv.math/0409443
 arXiv:
 arXiv:math/0409443
 Bibcode:
 2004math......9443S
 Keywords:

 Mathematics  Commutative Algebra;
 13J25;
 13J30;
 16Y60;
 26C99;
 54H10
 EPrint:
 23 pages. See also: http://www.mathe.unikonstanz.de/homepages/schweigh/