Constructions of k-regular maps using finite local schemes

A continuous map Rm→RN or Cm→CN 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, but there are very few results on the existence of such maps for particular values m and k. Using 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≤9, and we provide explicit examples for k≤5. We also provide upper bounds for arbitrary m and k.