On highly regular embeddings

TL;DR

This paper establishes new lower bounds on the dimensions needed for specific types of embeddings from R^d to R^N, extending previous results and sharpening existing bounds for k-regular, l-skew, and combined embeddings.

Contribution

It provides improved lower bounds on embedding dimensions for k-regular, l-skew, and combined embeddings, advancing the theoretical understanding of these mappings.

Findings

New lower bounds for embedding dimensions N

Extension of previous results by Chisholm and Ghomi-Tabachnikov

Sharper bounds for k-regular-l-skew embeddings

Abstract

Given parameters k, l, and d, we give new lower bounds on the dimensions N such that there are maps from R^d to R^N that are k-regular, l-skew embeddings, or k-regular-l-skew embeddings. This extends and sharpens results due to Chisholm (1979) and Ghomi-Tabachnikov (2008).