On points at infinity of real spectra of polynomial rings

TL;DR

This paper investigates the structure of points at infinity in the real spectrum of polynomial rings over real closed fields, providing a decomposition and describing associated valuations through homeomorphisms.

Contribution

It introduces a decomposition of the real spectrum into subsets and characterizes valuations at infinity via homeomorphisms with finite points of related spectra.

Findings

Decomposition of sper A into disjoint sets U_T

Homeomorphism between U_T and subspaces of sper B_T

Valuation v_{d*} is composed with v_d

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).