Examples of $k$-regular maps and interpolation spaces

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 $\mathbb{C}^3\rightarrow \mathbb{C}^{14}$.