$k$-regular embeddings of the plane

Abstract:

A map $f:X \to {R^n}$ is said to be k-regular if whenever ${x_1}, \ldots ,{x_k}$ are distinct points of X, then $f({x_1}), \ldots ,f({x_k})$ are linearly independent. Such maps are of interest in the theory of Cebyšev approximation. In this paper, configuration spaces and homological methods are used to show that there does not exist a k-regular map of ${R^2}$ into ${R^{2k - \alpha (k) - 1}}$ where $\alpha (k)$ denotes the number of ones in the dyadic expansion of k. This result is best possible when k is a power of 2.