| Résume | Let R be a real closed field and let K be any subfield of R. Given a subset X of R^n, we say that X is a K-algebraic set if it is the zero set in R^n of a family of polynomials in the subring K[x] := K[x_1, ... , x_n] of R[x].
We are interested in studying the algebraic geometry of K-algebraic sets X ⊂ R^n using only polynomials in K[x], and comparing it with the usual algebraic geometry of X ⊂ R^n in which polynomials of the entire ring R[x] are used.
This study generates a new ‘hybrid’ real algebraic geometry, we call subfield-algebraic geometry, which is particularly rich and interesting in the case K is not a real closed subfield of R, such as K = Q.
I will present here an overview of the foundational concepts and results of this geometry. If time permits, I will also describe some deep applications to the Q-algebraicity problem.
This seminar is based on two papers, one co-written with José F. Fernando and one with Enrico Savi. |
