Formula omitted] solutions of semialgebraic or definable equations

Abstract

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 Cm 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 Rn, 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 Cr solution G(x), then there is a Cm definable solution. Likewise in problem (1), given a closed definable subset X of Rn, we find r=r(m) such that if g:X→R is definable and extends to a Cr function on Rn, then there is a Cm definable extension.