Branches and Completions for Real Algebraic Curves

Author

Arthur E. Bukowski

Publication Date

5-5-1972

Abstract

Let Q be a real ideal in a commutative ring A over an ordered field k. Do the classical results of cummutative rings hold for the ideal Q, the realradical of Q, and the real primes of A? We deal with these questions in Section 1. We prove that the minimal realprimes of Q are the minimal primes of Hence the minimal real primes of A are the minimal primes of some a_iϵ A}. A finitely generated realsimple ring isin fact a field and so any maximal realideal in a finitely generated ring A|k is a maximal ideal. An example shows that the classical lying-over theorem of Cohen-Seidenberg is not true for real primes in general. A projective reelnullstellensatz is then proved and the relationship between the affine and projective realradicals is shown. In Section 2 we develop a theory for the completion of an ordered field via Cauchy sequences. Assuming an ordered field k is first countable we show k and its order topology satisfies some universal mapping properties with respect to completions. k has a unique completion and we know what this completion is. In fact k is complete if and only if k has no proper dense extensions, i.e., there does not exist a proper extension K of k with kdense In K. Section 3 deals with real algebraic curves over an ordered ground field. We show some properties of power series over a microbial, complete realclosed field. If a real branch representation exists for a point P on a realcurve Γ then this branch is convergent and represents a homeomorphism of Γ onto an interval of 0 in k. We show also that if a polynomial F(X, Y) changes sign on a real curve Γ in k⁽²⁾ then F(X, Y) has zeroon Γ.

Degree Name

Mathematics

Level of Degree

Doctoral

Department Name

Mathematics & Statistics

First Committee Member (Chair)

Donald Ward Dubois

Second Committee Member

Clifford Ray Qualls

Third Committee Member

Theodore N. Guinn

Fourth Committee Member

Richard Clyde Metzler

Language

English

Document Type

Dissertation

Recommended Citation

Bukowski, Arthur E.. "Branches and Completions for Real Algebraic Curves." (1972). https://digitalrepository.unm.edu/math_etds/220