Examples of $k$-regular maps and interpolation spaces

TL;DR

This paper constructs explicit polynomial maps from complex 3-space to higher-dimensional complex spaces that are k-regular, advancing the understanding of such maps with minimal target dimension using algebraic geometry.

Contribution

It provides new explicit examples of k-regular polynomial maps with small target dimension using algebraic geometry techniques.

Findings

Constructed a 4-regular polynomial map from a73 to a311.

Constructed a 5-regular polynomial map from a73 to a314.

Advances the explicit construction of k-regular maps with minimal target dimension.

Abstract

A continous map $f: \mathbb{C}^n \rightarrow \mathbb{C}^N$ is $k$-regular if the image of any $k$ points spans a $k$-dimensional subspace. It is an important problem in topology and interpolation theory, going back to Borsuk and Chebyshev, to construct $k$-regular maps with small $N$ and only a few nontrivial examples are known so far. Applying tools from algebraic geometry we construct a 4-regular polynomial map $\mathbb{C}^3\rightarrow \mathbb{C}^{11}$ and a 5-regular polynomial map $\mathbb{C}^3\rightarrow \mathbb{C}^{14}$.