Global and local ℚ-algebrization problems in real algebraic geometry

In 2020 Parusiński and Rond proved that every algebraic set X ⊂ ℝ^n is homeomorphic to an algebraic set X’ ⊂ ℝ^n which is described globally (and also locally) by polynomial equations whose coefficients are real algebraic numbers. In general, the following problem was widely open: Open Problem. Is every real algebraic set homeomorphic to a real algebraic set defined by polynomial equations with rational coefficients? The aim of my PhD thesis is to provide classes of real algebraic sets that positively answer to above Open Problem. In Chapter 1 I introduce a new theory of real and complex algebraic geometry over subfields recently developed by Fernando and Ghiloni. In particular, the main notion to outline is the so called ℝ|ℚ-regularity of points of a ℚ-algebraic set X ⊂ ℝ^n. This definition suggests a natural notion of a ℚ-nonsingular ℚ-algebraic set X ⊂ ℝ^n. The study of ℚ-nonsingular ℚ-algebraic sets is the main topic of Chapter 2. Then, in Chapter 3 I introduce ℚ-algebraic approximation techniques a là Akbulut-King developed in collaboration with Ghiloni and the main consequences we proved, that are, versions ‘over ℚ’ of the classical and the relative Nash-Tognoli theorems. Last results can be found in in Chapters 3 & 4, respectively. In particular, we obtained a positive answer to above Open Problem in the case of compact nonsingular algebraic sets. Then, after extending ‘over ℚ’ the Akbulut-King blowing down lemma, we are in position to give a complete positive answer to above Open Problem also in the case of compact algebraic sets with isolated singularities in Chapter 4. After algebraic Alexandroff compactification, we obtained a positive answer also in the case of non-compact algebraic sets with isolated singularities. Other related topics are investigated in Chapter 4 such as the existence of ℚ-nonsingular ℚ-algebraic models of Nash manifolds over every real closed field and an answer to the ℚ-algebrization problem for germs of an isolated algebraic singularity. Appendices A & B contain results on Nash approximation and an evenness criterion for the degree of global smoothings of subanalytic sets, respectively.