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On points at infinity of real spectra of polynomial rings

Abstract : Let R be a real closed field and A=R[x_1,...,x_n]. Let sper A denote the real spectrum of A. There are two kinds of points in sper A : finite points (those for which all of |x_1|,...,|x_n| are bounded above by some constant in R) and points at infinity. In this paper we study the structure of the set of points at infinity of sper A and their associated valuations. Let T be a subset of {1,...,n}. For j in {1,...,n}, let y_j=x_j if j is not in T and y_j=1/x_j if j is in T. Let B_T=R[y_1,...,y_n]. We express sper A as a disjoint union of sets of the form U_T and construct a homeomorphism of each of the sets U_T with a subspace of the space of finite points of sper B_T. For each point d at infinity in U_T, we describe the associated valuation v_{d*} of its image d* in sper B_T in terms of the valuation v_d associated to d. Among other things we show that the valuation v_{d*} is composed with v_d (in other words, the valuation ring R_d is a localization of R_{d*} at a suitable prime ideal).
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Contributor : Daniel Schaub <>
Submitted on : Monday, July 16, 2007 - 6:43:57 PM
Last modification on : Wednesday, June 9, 2021 - 10:00:08 AM
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François Lucas, Daniel Schaub, Mark Spivakovsky. On points at infinity of real spectra of polynomial rings. The Michigan Mathematical Journal, Michigan Mathematical Journal, 2008, 57, pp.587-599. ⟨10.1307/mmj/1220879425⟩. ⟨hal-00162963v2⟩



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