Subfield-algebraic geometry

This paper is devoted to the study of the geometric properties of algebraic sets in an affine or projective space over a field $L$ that derive from polynomials with coefficients in a fixed subfield $K$ of $L$. Here we develop the foundational elements and results of the new theory resulting from this study, which we call `subfield-algebraic geometry'. Our main goal is to study the real case, more precisely the case in which $L$ is a real closed field and $K$ is an ordered subfield of $L$. The most interesting phenomena appear when $K$ is not a real closed field. A main example to keep in mind is the one in which $L$ is the field $\mathbb{R}$ of real numbers or the field $\overline{\mathbb{Q}}^r$ of real algebraic numbers and $K$ is the field $\mathbb{Q}$ of rational numbers.