Constructions of k-regular maps using finite local schemes

TL;DR

This paper constructs k-regular maps from R^m or C^m to R^N or C^N using algebraic geometry, relating their existence to the dimension of certain Gorenstein schemes, and provides explicit bounds and examples for small k.

Contribution

It introduces a novel algebraic geometric approach to constructing k-regular maps and relates their minimal embedding dimension to Gorenstein schemes in the Hilbert scheme.

Findings

Explicit dimension computations for Gorenstein schemes for k<10

Construction of k-regular maps for small k and m

Upper bounds on N for general m and k

Abstract

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.