Surjective morphisms from affine space to its Zariski open subsets

TL;DR

This paper constructs explicit surjective morphisms from affine space onto certain open subsets, extending to affine varieties, demonstrating the existence of such maps with constructive methods.

Contribution

It provides a constructive proof for surjective morphisms from affine space to open subvarieties, extending results to affine varieties via Noether's normalization.

Findings

Constructed explicit surjective morphisms onto open subsets of affine space.

Extended results to affine varieties using Noether's normalization.

Demonstrated surjective maps from affine space to complements of algebraic sets.

Abstract

We prove constructively the existence of surjective morphisms from affine space onto certain open subvarieties of affine space of the same dimension. For any algebraic set $Z\subset \mathbb{A}^{n-2}\subset \mathbb{A}^{n}$, we construct an endomorphism of $\mathbb{A}^{n}$ with $\mathbb{A}^{n} \setminus Z$ as its image. By Noether's normalization lemma, these results extend to give surjective maps from any $n$-dimensional affine variety $X$ to $\mathbb{A}^{n} \setminus Z$.