# C^m solutions of semialgebraic (or definable) equations

We address the question of whether geometric conditions on the given data

can be preserved by a solution in (1) the Whitney extension problem, and

(2) the Brenner-Fefferman-Hochster-Kollár problem, both for C^m functions.

Our results involve a certain loss of differentiability.

Problem (2) concerns the solution of a system of linear equations

A(x)G(x)=F(x), where A is a matrix of functions on R^n, and F, G are

vector-valued functions. Suppose the entries of A(x) are semialgebraic

(or, more generally, definable in a suitable o-minimal structure).

Then we find r=r(m) such that, if F(x) is definable and the system admits

a C^r solution G(x), then there is a C^m definable solution.

Likewise in problem (1), given a closed definable subset X of R^n, we find

r=r(m) such that if g:X→R is definable and extends to a C^r function on

R^n, then there is a C^m definable extension.

(Joint work with Edward Bierstone and Pierre Milman)