Examples of $k$-regular maps and interpolation spaces

A continous map $f:{\mathbb{C}}^n\to {\mathbb{C}}^N$ 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 ${\mathbb{C}}^3\to {\mathbb{C}}^{11}$ and a 5-regular polynomial map ${\mathbb{C}}^3\to {\mathbb{C}}^{14}$.