Popescu's nested approximation theorem states the following: Let F be a polynomial system of equations in two sets of variables x,y with a formal power series solution \hat{y}(x) (i.e.,F(x,\hat{y}(x)=0) that is nested (i.e., each \hat{y}_i(x) only depends on the first x_1,...,x_{n_i}). Then for any natural number c there exists an algebraic power series y(x) such that
(1) F(x,y(x))=0,
(2) y(x) and \hat{y}(x) agree up to degree c and
(3) y(x) is nested.
In general, the proof of this theorem requires Popescu's theorem on smoothing morphisms. This talk will first give a more elementary way to prove the theorem if there are only two nests and the first nest is only dependent on x_1.
The second part of the talk will be an outline of the general proof.
Two Proof Approaches to Popescu's Nested Approximation Theorem
07.05.2024 09:45 - 11:15
Organiser:
F. Fürnsinn, H. Hauser
Location: