Subfield-algebraic geometry

Abstract

In this monograph, we lay the foundations for a new theory that generalizes real algebraic geometry. Let $R|K$ be a field extension, where $R$ is a real closed field and $K$ is an ordered subfield of $R$. The main objective is to study $K$-algebraic subsets of $R^n$, i.e., those subsets of $R^n$ that are the zero loci of polynomials with coefficients in $K$. Real algebraic geometry already covers the case when $K$ is also a real closed field. Our goal is to extend real algebraic geometry to the case when $K$ is not real closed, for example when $K$ is the field $\mathbb{Q}$ of rational numbers. Several new geometric phenomena appear. There is no complex counterpart to this generalized real algebraic geometry. The reason is as follows. If $C|K$ is a field extension with $C$ algebraically closed and $X$ is a $K$-algebraic subset of $C^n$, then Hilbert's Nullstellensatz implies that the ideal of polynomials with coefficients in $C$ that vanish on~$X$ is generated by the ideal of polynomials with coefficients in $K$ that vanish on $X$. In the real realm, this is false in general, for example when we consider field extensions $R|K$ with $R$ real closed and $K=\mathbb{Q}$. This monograph also presents some applications of the theory developed. Here is an example. The celebrated Nash-Tognoli theorem states that every compact smooth manifold $M$ is diffeomorphic to a nonsingular real algebraic set $M'$, called algebraic model of $M$. The theory developed here provides the theoretical basis to prove that the algebraic model $M'$ of $M$ can be chosen to be $\mathbb{Q}$-algebraic and $\mathbb{Q}$-nonsingular. This guarantees for the first time that, up to smooth diffeomorphisms, every compact smooth manifold can be encoded both globally and locally involving only finitely many exact data.