Cᵐ solutions of semialgebraic equations and the Whitney extension problem

Jean-Baptiste Campesato
University of Toronto

Date(s) : 18/02/2021   iCal
14 h 00 min - 15 h 00 min

We address the question of whether geometric conditions on the given data can be preserved by a solution in

(1) the Whitney extension problem, which consists in determining whether a function g:X→ℝ defined on a closed subset X⊂ℝⁿ admits a Cᵐ extension on ℝⁿ, and,

(2) the Brenner-Fefferman-Hochster-Kollár problem, about the existence of a Cᵐ solution to A(x)G(x)=F(x), where A is a matrix of functions on ℝⁿ, and the unknown is a vector-valued function G.

In a joint work with E. Bierstone and P.D. Milman, we prove that, for both problems, when the data are semialgebraic (or, more generally, definable in a suitable o-minimal structure), the existence of a solution implies the existence of a semialgebraic (or definable) solution. Our results involve a certain loss of differentiability.

More precisely, for (1), we prove that given a semialgebraic closed subset X⊂ℝⁿ, there exists r:ℕ→ℕ such that if a semialgebraic function g:X→ℝ is the restriction of a Cʳ⁽ᵐ⁾ function then it is the restriction of a semialgebraic Cᵐ function.

For (2), we prove that given A a matrix of semialgebraic functions, there exists r:ℕ→ℕ such that if F is semialgebraic and A(x)G(x)=F(x) admits a Cʳ⁽ᵐ⁾ solution, then there exists a Cᵐ solution which is semialgebraic.

Slides: http://www.math.toronto.edu/campesat/docs/krakow.pdf


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