  A continuous map from R^m to R^N or from C^m to C^N is called k-regular if
the images of any $k$ points are linearly independent. Given integers m and k a
problem going back to Chebyshev and Borsuk is to determine the minimal value of
N for which such maps exist. The methods of algebraic topology provide lower
bounds for N, however there are very few results on the existence of such maps
for particular values m and k. Using the methods of algebraic geometry we
construct k-regular maps. We relate the upper bounds on N with the dimension of
the locus of certain Gorenstein schemes in the punctual Hilbert scheme. The
computations of the dimension of this family is explicit for k<10, and we
provide explicit examples for k<6. We also provide upper bounds for arbitrary m
and k.
