Continuous k-regular maps

    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 N_0 of N for which such maps exist. The methods of algebraic topology provide lower bounds for N_0, 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_0 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. The problem has an equivalent phrasing in the language of interpolation theory. Its generalisations lead to further open questions, and to an idea of a more general framework.

Joint work with Tadeusz Januszkiewicz, Joachim Jelisiejew, Mateusz Michałek.