| Résume | Let K be a subfield of ℝ. We say that an algebraic set V ⊂ ℝ^n is K-algebraic if it can be described by global polynomial equations with coefficients in K. Denote by ℚ̅^r the field of real algebraic numbers, that is, the real closure of ℚ.
In 2020, Parusínski and Rond proved that every algebraic set V ⊂ ℝ^n is homeomorphic to a ℚ̅^r-algebraic set V ⊂ ℝ^n n via a Zariski equisingular semialgebraic and arc-analytic deformation h_t : ℝ^n → ℝ^n. The latter result and the example of Teissier of an irrational Whitney equisingular class motivate the following open problem:
ℚ-algebraicity problem: (Parusínski, 2021) Is every algebraic set V ⊂ ℝ^n homeomorphic to some ℚ-algebraic set V' ⊂ ℝ^m, with m ≥ n?
In general, the fact that ℚ is not a real closed field is a crucial difficulty.
The aim of the talk is to introduce the ℚ-algebraicity problem and to explain how our new approximation techniques over ℚ, inspired by Nash-Tognoli and Akbulut-King results, allowed us to provide a positive answer for nonsingular real algebraic sets, also in a relative setting, and real algebraic sets with isolated singularities. Time permitting, I will give some insights on current investigations.
The talk is based on two works, one of those in collaboration with Ghiloni. |
