A unified approach to embeddings of a line in 3-space

TL;DR

This paper introduces a new method using residual coordinates to analyze embeddings of a line in 3-space, providing new proofs and generalizations of existing partial results in affine geometry.

Contribution

It presents a novel approach with residual coordinates that unifies and extends previous results on embeddings of affine lines in 3-space.

Findings

Reproved all known partial results on embeddings of lines in 3-space

Generalized results of Bhatwadekar-Roy and Kuroda

Provided a new framework for analyzing embeddings using residual coordinates

Abstract

While the general question of whether every closed embedding of an affine line in affine $3$-space can be rectified remains open, there have been several partial results proved by several different means. We provide a new approach, namely constructing (strongly) residual coordinates, that allows us to give new proofs of all known partial results, and in particular generalize the results of Bhatwadekar-Roy and Kuroda on embeddings of the form $(t^n,t^m,t^l+t)$.