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# Images of dominant endomorphisms of affine space

Viktor Balch Barth<EMAIL_ADDRESS>Department of Mathematics, UiO,
Norway.  Tuyen Trung Truong<EMAIL_ADDRESS>Department of Mathematics,
UiO, Norway.

###### Abstract

A basic problem in the study of algebraic morphisms is to determine which sets
can be realised as the image of an endomorphism of affine space. This paper
extends the results previously obtained by the first author on the question of
existence of surjective maps
$F\colon\mathbb{A}^{n}\rightarrow\mathbb{A}^{n}\setminus Z$, where $Z$ is an
algebraic subvariety of $\mathbb{A}^{n}$ of codimension at least 2. In
particular, we show that for any (affine) algebraic variety $Z$ of dimension
at most $n-2$, there is an algebraic variety $W\subset\mathbb{A}^{n}$
birational to $Z$ and a surjective algebraic morphism
$\mathbb{A}^{n}\rightarrow\mathbb{A}^{n}\setminus W$. We also propose a
conjectural approach towards resolving unknown cases.

†† _Key words_ : affine space, endomorphism, Hilbert Nullstellensatz,
surjectivity

##  1 Introduction

We work over an algebraically closed field $k$ of characteristic zero. Let
$\mathbb{A}^{n}$ be the affine space of dimension $n$. A basic problem in the
study of algebraic morphisms is the following:

######  Question A.

What subsets of $\mathbb{A}^{n}$ can be realised as the image of a dominant
endomorphism $F\colon\mathbb{A}^{n}\rightarrow\mathbb{A}^{n}$?

A classical general result is Chevalley’s theorem [6, Théorème 3] which
asserts that the image of an algebraic map is constructible (see the appendix
of this paper for some more detail on it). Aside from that, Question A is
largely open, except for an elementary argument that $\mathbb{A}^{n}\setminus
F(\mathbb{A}^{n})$ cannot contain a hypersurface (see e.g. [1, 21]).

This paper concerns a nontrivial subproblem of Question A, aiming to show the
existence of and construct explicitly surjective morphisms
$F\colon\mathbb{A}^{n}\rightarrow\mathbb{A}^{n}\setminus Z$, where $Z$ is a
closed algebraic subvariety of codimension at least $2$. This question can
naturally be stated as a stronger version of Hilbert’s effective
Nullstellensatz as follows. Let $(z_{1},\ldots,z_{n})$ be the coordinates in
the domain, $(w_{1},\ldots,w_{n})$ the coordinates in the range,
$F=(F_{1},\ldots,F_{n})$ the supposed endomorphism, and
$Z=\\{g_{1}=\ldots=g_{m}=0\\}$. We want to check the two conditions:
$F(\mathbb{A}^{n})\subset\mathbb{A}^{n}\setminus Z$, and if
$w\in\mathbb{A}^{n}\setminus Z$ then $F^{-1}(w)$ is non-empty. The first
condition is equivalent to showing that
$\\{w_{1}-F_{1}=\ldots=w_{n}-F_{n}=0=g_{1}=\ldots=g_{m}\\}=\emptyset$, which
can be interpreted (by Hilbert’s Nullstellensatz) as the existence of
polynomials $h_{1},\ldots,h_{n},h_{n+1},\ldots,h_{n+m}$ (in variables
$z_{1},\ldots,z_{n},w_{1},\ldots,w_{n}$) so that $h_{1}(w_{1}-F_{1})+\ldots
h_{n}(w_{n}-F_{n})+h_{n+1}g_{1}+\ldots+h_{n+m}g_{m}$ is identically $1$. The
second condition is equivalent to showing that there do not exist polynomials
$h_{1},\ldots,h_{n},h_{n+1}$ (in variables
$z_{1},\ldots,z_{n},t_{1},\ldots,t_{m}$) and $w_{1},\ldots,w_{n}$ so that
$h_{1}(w_{1}-F_{1})+\ldots+h_{n}(w_{n}-F_{n})+h_{n+1}(t_{1}g_{1}+\ldots+t_{m}g_{m}-1)$
is identically $1$, where $t_{1},\ldots,t_{m}$ are new variables, which again
can be reduced to Hilbert’s Nullstellensatz. Note that
$\mathbb{A}^{n}\setminus Z$ is a rational variety, and hence the existence of
surjective maps $\mathbb{A}^{n}\rightarrow\mathbb{A}^{n}\setminus Z$ is
interesting also from the viewpoint of birational geometry. So far, strongest
evidences for the existence of such maps come from (algebraic) Oka theory.

Over the complex numbers, such an $\mathbb{A}^{n}\setminus Z$ is a holomorphic
(sub)elliptic manifold [15], and hence by [11] there are holomorphic
surjective maps $\mathbb{A}^{n}\rightarrow\mathbb{A}^{n}\setminus Z$. In the
algebraic setting, however, this was previously known only for the case
$Z\subset\mathbb{A}^{n-2}\subset\mathbb{A}^{n}$, by the work of the first
author [4] (and some earlier works on dimension $2$, [18, 8]). On the other
hand, [21] showed that for the more general class of algebraic manifolds $X$
which are algebraically subelliptic (see [15] or [10] for a precise
definition, and note the recent result that algebraic ellipticity and
subellipticity are equivalent for smooth varieties [19]) we always have
surjective morphisms $\mathbb{A}^{\dim(X)+1}\rightarrow X$. (An earlier result
[1] showed that, for the special class of flexible manifolds $X$—which
includes the manifolds of the form $\mathbb{A}^{n}\setminus Z$ considered in
this paper—we always have surjective morphisms $\mathbb{A}^{N}\rightarrow X$,
where in general $N>\dim(X)$). For the case where $X$ is an algebraically
(sub)elliptic compact manifold, Forstnerič showed the existence of a
surjective algebraic map $\mathbb{A}^{\dim(X)}\rightarrow X$, [11, Theorem
1.6]. See [3] for many examples of algebraically (sub)elliptic compact
manifolds. Whether any algebraically (sub)elliptic manifold $X$ always admits
algebraic surjective morphisms $f\colon\mathbb{A}^{\dim X}\to X$ is an open
question asked by several authors [11, 12, 22], or indeed how we can
characterize those varieties $X$ to which there is such a surjective map [1,
Problem 2]. In the case $X=\mathbb{A}^{n}\setminus Z$, the results in this
paper provide a solution to a weaker version of this question, see Corollary
1.2.

A natural idea to construct surjective morphisms
$\mathbb{A}^{n}\rightarrow\mathbb{A}^{n}\setminus Z$ is to start from a
surjective morphism $g\colon\mathbb{A}^{N}\rightarrow\mathbb{A}^{n}\setminus
Z$ (which exists by the above mentioned results [1, 21]), then find an
appropriate linear subspace $H\subset\mathbb{{A}}^{N}$ of dimension $n$ such
that $F=g|_{H}$ is also surjective. In general, we may need to consider
subvarieties $H$ of $\mathbb{{A}}^{N}$ isomorphic to $\mathbb{A}^{n}$, and not
just linear subspaces. One would expect that a generic choice of $H$ will not
yield a surjective map $g|_{H}$, and one must therefore look for special
subvarieties $H$. To this end, it will be very useful—at least in the first
stages—to be able to efficiently test if the restrictions $g|_{H}$ to some
reasonable choices of $H$ are surjective.

The main idea from [21] is as follows. If $X$ is a (sub)elliptic manifold of
dimenison $n$, then for every $x\in X$ there is an algebraic morphism
$F_{x}\colon\mathbb{A}^{n}\rightarrow X$—coming from the associated sprays, by
using results from Forstneric [9]—which is dominant and contains $x$ in its
image. By Noetherian property, there is a finite number of such maps
$F_{x_{1}},\ldots,F_{x_{N}}\colon\mathbb{A}^{n}\rightarrow X$, such that the
union of the images of $F_{x_{1}},\ldots,F_{x_{N}}$ is $X$. We think about
$F_{x_{i}}$ as a map from a hyperplane $H_{i}\subset\mathbb{A}^{n+1}$, where
$H_{i}$ and $H_{j}$ are disjoint if $i\not=j$, and then extend these to a
morphism $F\colon\mathbb{A}^{n+1}\rightarrow X$ whose image contains
$\bigcup_{i=1}^{N}F_{x_{i}}(H_{i})=X$.

In the case where $X$ is of the form $\mathbb{A}^{n}\setminus Z$, where $Z$ is
an algebraic subvariety of codimension at least 2, a more convenient
construction/procedure is given in [1], using the Gromov–Winkelmann theorem
[25, Proposition 1], [14, p. 72, Exercise (b’)]. See Section 3 for more
detail. This procedure yields an explicit surjective map, but typically with
$N>n$. In our experiments, we find that there are cases where $N$ can be
chosen to be $n+1$, but there are cases where it is not obvious so.

Our main result is Theorem 1.1.

######  Theorem 1.1.

Let $Z\subset\\{0\\}\times\mathbb{A}^{n-1}\subset\mathbb{A}^{n}$ be an
algebraic subvariety. Then there exists a surjective algebraic map
$F\colon\mathbb{A}^{n}\to\mathbb{A}^{n}\setminus Z$.

In the proof we explicitly construct $F$. It is possible to do this in
different ways. In Theorem 2.4, we give a construction which requires some
mild assumptions on the choice of coordinate system, but achieves a map $F$ of
lower degree.

Computational tools have been indispensable for us to find first non-trivial
examples which in turn provided inspiration and ideas for the above mentioned
theorems. The proofs of the above theorems are the generalizations of the
proof we found first for a special case of dimension 3, by experimenting with
the help of Macaulay2, see Example 3.1 and Appendix A.2 for detail. Theorem
1.1 is obtained by first constructing a surjective morphism
$\mathbb{A}^{m+2n-2}\rightarrow\mathbb{A}^{n}\setminus Z$ (where $m$ is an
upper bound for the number of equations defining $Z$, and hence we can always
choose $m\leq n$ by [20, 24, 7]), and then restricting to an appropriate
subvariety $H$ of $\mathbb{A}^{m+2n-2}$ which is isomorphic to
$\mathbb{A}^{n}$. Theorem 2.4 is obtained by first constructing a surjective
morphism $\mathbb{A}^{n+m}\rightarrow\mathbb{A}^{n}\setminus Z$, and then
restricting to an appropriate subvariety $H\cong\mathbb{A}^{n}$ of
$\mathbb{A}^{n+m}$.

We obtain the following consequence, which can be viewed as almost a solution
to the question about the existence of such surjective morphisms
$\mathbb{A}^{n}\rightarrow\mathbb{A}^{n}\setminus Z$.

######  Corollary 1.2.

Let $Z$ be an (affine) algebraic variety of dimension at most $n-2$. Then
there are an algebraic variety $W\subset\mathbb{A}^{n}$ birational to $Z$ and
a surjective morphism $F\colon\mathbb{A}^{n}\rightarrow\mathbb{A}^{n}\setminus
W$.

###### Proof.

It is a classical result that there is an algebraic variety
$W\subset\\{0\\}\times\mathbb{A}^{n-1}\subset\mathbb{A}^{n}$ birational to $Z$
(see [23, Theorem 1.8], [17, Theorem I.4.9]). Theorem 1.1 furnishes a
surjective algebraic morphism
$F\colon\mathbb{A}^{n}\rightarrow\mathbb{A}^{n}\setminus W$. ∎

Given a variety $X$, we denote by $\operatorname{Aut}(X)$ its group of
automorphisms. A nontrivial regular action $\varphi\colon\mathbb{G}_{a}\times
X\to X$ by the additive group $\mathbb{G}_{a}=(k,+)$ gives rise to a subgroup
$G$ of $\operatorname{Aut}(X)$, which we call a _$\mathbb{G}_{a}$ -subgroup_.
We consider $X=\mathbb{A}^{n}\setminus Z$ and fix some point
$p\in\mathbb{A}^{n}\setminus Z$. By [1, Proposition 2, Corollary 1] (see also
[2, Corollary 1.11]), one can find a finite sequence of
$\mathbb{G}_{a}$-subgroups $G_{i}$ of
$\operatorname{Aut}(\mathbb{A}^{n}\setminus Z)$ such that

$(G_{N}\cdot\ldots\cdot G_{1}).p=\mathbb{A}^{n}\setminus Z.$

The sequence of groups $G_{1},\ldots,G_{N}$ furnishes a surjective morphism
$F\colon\mathbb{A}^{N}\to\mathbb{A}^{n}\setminus Z$, where
$F\colon(g_{1},g_{2},\ldots,g_{N})\mapsto(g_{N}\ldots g_{1}.p)$.

If $W$ is an algebraic subvariety of $\mathbb{A}^{n}$, we denote by
$\operatorname{Aut}(\mathbb{A}^{n},W)$ the set of algebraic automorphisms of
$\mathbb{A}^{n}$ fixing $W$ pointwise. Towards a complete solution for the
existence of surjective algebraic maps
$\mathbb{A}^{n}\rightarrow\mathbb{A}^{n}\setminus Z$, we propose the following
conjectural approach:

######  Conjecture 1.3.

Let $Z\subset\mathbb{A}^{n}$ be an algebraic subvariety of codimension at
least $2$. Then there exist

1) $N$ hypersurfaces $W_{1},\ldots,W_{N}\subset\mathbb{A}^{n}$ containing $Z$,
with $\bigcap_{i}W_{i}=Z$,

2) $\mathbb{G}_{a}$-subgroups
$G_{i}\subset\operatorname{Aut}(\mathbb{A}^{n},W_{i})$,

3) A point $p\in\mathbb{A}^{n}\setminus Z$, and

4) A subvariety $H\subset\mathbb{A}^{N}$ isomorphic to $\mathbb{A}^{n}$,

such that for $\psi:=(G_{N}\cdot\ldots\cdot
G_{1}).p\colon\mathbb{A}^{N}\rightarrow\mathbb{A}^{n}\setminus Z$, the
restricted map $\psi|_{H}$ is surjective.

In the setting of Conjecture 1.3, since $Z\subset W_{i}$,
$\operatorname{Aut}(\mathbb{A}^{n},W_{i})$ is naturally a subgroup of
$\operatorname{Aut}(\mathbb{A}^{n}\setminus Z)$, for all $i$. We will
sometimes write $G_{i,a_{i}}=G_{i}$ to emphazise the parameter $a_{i}$. While
similar to the setting in [1], which looks at 1-parameter subgroups of
$\operatorname{Aut}(\mathbb{A}^{n}\setminus Z)$, the above formulation of
Conjecture 1.3 aims for a more concrete strategy to look for surjective
morphisms $\mathbb{A}^{n}\rightarrow\mathbb{A}^{n}\setminus Z$ in practical
settings.

Both proofs of Theorems 1.1 and 2.4 follow the strategy in Conjecture 1.3.

Plan of the paper. In Section 2 we prove the main theoretical results. In
Section 3 we present some computational experiments, which are the inspiration
for Theorems 1.1 and 2.4 and are evidences to support Conjecture 1.3. In the
appendix we present some algorithms used in the experiments, and illustrations
of how one can write code in Macaulay2 to perform the experiments.

Acknowledgements. The authors are partially supported by the Young Research
Talents grant 300814 from Research Council of Norway. The first author thanks
Mateusz Michałek for some help with the code of the paper [16], and thanks
Finnur Lárusson for some discussions on the problem while at University of
Adelaide several years ago. The authors thank John Christian Ottem for
interest and useful comments.

##  2 Proofs of main results

We now prove our main theorem, Theorem 1.1. Then we state and prove an
alternate version, giving a map of lower degree, Theorem 2.4.

###### Proof of Theorem 1.1.

We begin by constructing a surjective map
$\psi\colon\mathbb{A}^{m+2n-2}\to\mathbb{A}^{n}\setminus Z$ by using group
actions of $\mathbb{G}_{a}$-subgroups of
$\operatorname{Aut}(\mathbb{A}^{n}\setminus Z)$. $Z$ can be written as
$Z=V(w_{1},q_{1},\ldots,q_{m})$, where each $q_{i}$ is a polynomial in
$w_{2},\ldots,w_{n}$. We use $2n-2+m$ groups $G_{i}$, and write $\varphi_{i}$
for the group action $\varphi_{i}\colon G_{i}\times\mathbb{A}^{n}\setminus
Z\to\mathbb{A}^{n}\setminus Z$. We use the following $\varphi_{i}$’s.

$\displaystyle\varphi_{i}\colon(a_{i},w_{1},\ldots,w_{n})$
$\displaystyle\mapsto(w_{1},\ldots,w_{i},w_{i+1}+a_{i}w_{1},w_{i+2},\ldots,w_{n})$
$\displaystyle\varphi_{i+n-1}\colon(c_{i},w_{1},\ldots,w_{n})$
$\displaystyle\mapsto(w_{1}+c_{i}q_{i}(w_{2},\ldots,w_{n}),w_{2},\ldots,w_{n})$
$\displaystyle\varphi_{i+n-1+m}\colon(b_{i},w_{1},\ldots,w_{n})$
$\displaystyle\mapsto(w_{1},\ldots,w_{i},w_{i+1}+b_{i}w_{1},w_{i+2},\ldots,w_{n}).$

We fix a point $p=(1,0,\ldots,0)$ and compose the group actions to obtain
$\psi=(G_{2n-2+m}\ldots
G_{1}).(p)\colon\mathbb{A}^{2n-2+m}\to\mathbb{A}^{n}\setminus Z$, with the
following explicit formula.

(1)
$\psi=(1+\sum_{j=1}^{j=m}c_{j}q_{j}(a_{1},\ldots,a_{n-1}),a_{1}+b_{1}w_{1},\ldots,a_{n-1}+b_{n-1}w_{1}),$

where $w_{1}$ is shorthand for the expression in the first coordinate,
$w_{1}=1+\sum_{j=1}^{j=m}c_{j}q_{j}.$ We now find an
$\mathbb{A}^{n}\subset\mathbb{A}^{m+2n-2}$ such that the restriction of $\psi$
to $\mathbb{A}^{n}$ is still surjective.

Let $d=\max(\deg(q_{j}))$, and set $p_{k}:=d^{k}+d^{k-1}+\ldots+1$. The
subvariety $H$ defined by $c_{j}=c_{1}^{(j-1)(1+p_{n-1})}$ for all $j\neq 1$,
and $b_{i}=c_{1}^{p_{i-1}}$ for all $i$, i.e.
$H=V(c_{j}-c_{1}^{(j-1)(1+p_{n-1})},b_{i}-c_{1}^{p_{i-1}})$, is isomorphic to
$\mathbb{A}^{n}$. We define $F=\psi|_{H}$ and will prove that $F(a_{i},c_{1})$
is surjective. Below, we write $c$ in place of $c_{1}$.

Let $F\colon\mathbb{A}^{n}\to\mathbb{A}^{n}\setminus Z$ be

(2)
$F(a_{i},c)=(1+cq_{1}+\sum_{j=2}^{j=m}c^{(j-1)(1+p_{n-1})}q_{j}(a_{1},\ldots,a_{n-1}),\\\
a_{1}+cw_{1},a_{2}+c^{d+1}w_{1},\ldots,a_{i-1}+c^{p_{i-2}}w_{1},\ldots,a_{n-1}+c^{p_{n-2}}w_{1}),$

where again, $w_{1}$ is shorthand for
$w_{1}=1+cq_{1}+\sum_{j=2}^{j=m}c^{(j-1)(1+p_{n-1})}q_{j}(a_{1},\ldots,a_{n-1})$.
We check whether a point $(w_{1},\ldots,w_{n})$ is in the image, by checking
whether the preimage $F^{-1}(w_{1},\ldots,w_{n})$ is empty. We have $n$
equations:

(3) $\displaystyle w_{1}$
$\displaystyle=1+cq_{1}+c^{1+p_{n-1}}q_{2}+\ldots+c^{1+(m-1)p_{n-1}}q_{m}$
$\displaystyle w_{i}$ $\displaystyle=a_{i-1}+c^{p_{i-2}}w_{1}.$

The last $n-1$ equations are solved for $a_{i}$ to obtain
$a_{i-1}=w_{i}-c^{p_{i-2}}w_{1}$. It only remains to solve the following
polynomial equation in $c$:

(4) $\displaystyle 0$
$\displaystyle=cq_{1}+c^{1+p_{n-1}}q_{2}+\ldots+c^{1+(m-1)p_{n-1}}q_{m}-w_{1}+1,$

where now
$q_{j}=q_{j}(a_{1},\ldots,a_{n-1})=q_{j}(w_{2}-cw_{1},\ldots,w_{n}-c^{p_{n-2}}w_{1}).$

This has no solutions if and only if for all $j$,
$q_{j}(w_{2}-cw_{1},\ldots,w_{n}-c^{p_{n-2}}w_{1})\equiv 0$. This happens when
$w_{1}=0$ and $q_{j}(w_{2},\ldots,w_{n})=0$ for all $j$. If any of those
conditions are not met, then Eq. (4) is non-constant polynomial, and has
solutions in $c$, since $k$ is algebraically closed. We have now computed that
the preimage of a point $(w_{1},\ldots,w_{n})$ is empty if and only if
$(w_{1},\ldots,w_{n})\in Z$, and conclude that
$F\colon\mathbb{A}^{n}\to\mathbb{A}^{n}\setminus Z$ is surjective. ∎

######  Remark 2.1.

An upper bound for the degree of $F$ constructed in the above proof is
$mp_{n-1}=md^{n-1}+md^{n-2}+\ldots+m$.

We now state and prove an alternate version of our main theorem, which can
provide maps of smaller degrees. In order to do that, we first need the
following lemma, which is probably well known.

######  Lemma 2.2.

For any set of nonconstant polynomials $\\{q_{j}(x_{1},\ldots,x_{n})\\}$,
where each $q_{j}$ has degree $d_{j}$, we can always find a linear change of
coordinates from $x_{i}$ to $y_{i}$ in such a way that each $q_{j}$ contains a
monomial $a_{j}y_{n}^{d_{j}}$ for some constants $a_{j}\not=0$.

###### Proof.

One can define new variables $y_{i}=x_{i}+A_{i}x_{n}$, $(1\leq i\leq n-1)$,
and $y_{n}=x_{n}$. For a general choice of $A_{i}$’s, the conclusion of the
Lemma is satisfied. Indeed, any monomial of degree $d$ in the $x_{i}$’s will
be transformed by this linear map to a homogeneous polynomial in the $y_{i}$’s
which contains a monomial $ay_{n}^{d}$. We just need to choose the numbers
$A_{i}$’s so that the coefficient of $y_{n}^{d_{j}}$ (which is a polynimial in
the $A_{i}$’s) is non-zero. ∎

######  Remark 2.3.

What Lemma 2.2 achieves is that for any fixed tuple $(y_{1},\ldots,y_{n-1})$,
one can change the value of $q_{j}(y_{1},\ldots,y_{n})$ by changing $y_{n}$.

######  Theorem 2.4.

Let $Z$ be a subvariety
$Z\subset\left(\\{0\\}\times\mathbb{A}^{n-1}\right)\subset\mathbb{A}^{n}$,
defined by
$Z=V(w_{1},q_{1}(w_{2},\ldots,w_{n}),\ldots,q_{m}(w_{2},\ldots,w_{n}))$, where
each $q_{j}$ contains a monomial $a_{j}w_{n}^{d_{j}}$ in the sense of Lemma
2.2. Then there exists a surjective function
$F\colon\mathbb{A}^{n}\to\mathbb{A}^{n}\setminus Z$ of degree $\leq m(d+1)+1$,
where $d=\max(\deg(q_{j}))$.

###### Proof.

By the same procedure as in the proof of 1.1, we construct the map $\psi$ in
Eq. (1). However, we then restrict to the subvariety defined by
$c_{j}=c_{1}^{(j-1)(d+1)+1}$ for all $j$, $b_{i}=0$ for all $i<n-1$, and
$b_{n-1}=c_{1}$, and obtain the following map.

(5) $\displaystyle F(a_{i},c)$
$\displaystyle=(1+\sum_{j=1}^{j=m}c^{(j-1)(d+1)+1}q_{j},a_{1},a_{2},\ldots,a_{n-1}+cw_{1}),$

where $q_{j}=q_{j}(a_{1},\ldots,a_{n-1})$. Computing the preimage of a point
$w$ reduces to solving the following polynomial equation in $c$:

(6) $\displaystyle 0$
$\displaystyle=cq_{1}+\ldots+c^{(m-1)(d+1)+1}q_{m}-w_{1}+1,$

where $q_{j}=q_{j}(w_{2},\ldots,w_{n}-cw_{1})$. This has no solutions if and
only if for all $j$, $q_{j}(w_{2},w_{3},\ldots,w_{n}-cw_{1})\equiv 0$. This
happens when $w_{1}=0$ and all $q_{j}(w_{2},\ldots,w_{n})=0$. Lemma 2.2
ensures that Eq. 6 is solvable otherwise. ∎

##  3 Some explicit computational examples

In this section we present some experiments which are the inspiration for
Theorems 1.1 and 2.4, as well as give support to Conjecture 1.3.

To be able to perform the computations, we need the following tools:

Tool 1: Algorithms to compute the image of algebraic morphisms. We will in
particularly use the ideas from [5]. For the readers’ convenience, we briefly
recall this in the appendix.

Tool 2: Algorithms to construct surjective morphisms
$\mathbb{A}^{N}\rightarrow\mathbb{A}^{n}\setminus Z$, where in general $N$ can
be much bigger than $n$.

In Section 1, we mentioned the general result by [21] on the existence of
surjective algebraic maps $\mathbb{A}^{n+1}\rightarrow X$, for algebraic
subelliptic manifolds $X$ of dimension $n$. However, this construction
(because of some non-intrinsic features) may yield complicated maps which make
computations inefficient in the cases we consider.

Instead, here we use the ideas from [25], and subsequent developments in [2,
1] for flexible varieties. Here we briefly recall the idea for our setting
$X=\mathbb{A}^{n}\setminus Z$. We can start with $\mathbb{G}_{a}$-subgroups
$G_{1},\ldots,G_{n}$ of $\operatorname{Aut}(X)$ (constructed explicitly in
[25] by utilising general linear projections to hyperplanes of
$\mathbb{A}^{n}$), so that the orbit of one point by
$F_{n}=(G_{n}\cdot\ldots\cdot G_{1})$ contains a Zariski-open subset in $X$.
Then, for any point $x_{1}$ in $X$ not contained in the image, we compose
$F_{n}$ with the action of another 1-parameter subgroup $G_{n+1}(a_{n+1},.)$
to cover $x_{1}$. Since the action $G_{i}(0,.)$ is the identity map on $X$, it
follows that the image of $F_{n+1}=(G_{n+1}\cdot F_{n}).x_{0}$ is strictly
bigger than that of $F_{n}$ and contains $x_{1}$. Composing with more such
maps, we can even arrange that the map is dominant over $x_{1}$. We
inductively do this whenever there are still some missing points. This process
will terminate after a finite number of steps, thanks to Chevalley’s theorem
and the Noetherian property.

In [25], an explicit way to construct such 1-parameter families of
automorphisms for $X=\mathbb{A}^{n}\setminus Z$ is given. It works as follows.
For a generic coordinate system $z_{1},\ldots,z_{n}$, we consider the
projection $\pi_{i}$ from $\mathbb{A}^{n}$ to the hyperplane $\\{z_{i}=0\\}$,
and let $f_{i}(z_{1},\ldots,z_{i-1},z_{i+1},\ldots,z_{n})$ be a polynomial
which vanishes on $\pi_{i}(Z)$. Then we construct the
$\mathbb{G}_{a}$-subgroup
$G_{i}(a_{i},.)\in\operatorname{Aut}(\mathbb{A}^{n},\\{f_{i}=0\\})$ by the
following formula:

(7)
$G_{i}(a_{i},z)=(z_{1},\ldots,z_{i-1},z_{i}+a_{i}f_{i}(z_{1},\ldots,z_{i-1},z_{i+1},\ldots,z_{n}),z_{i+1},\ldots,z_{n}).$

We write $W_{i}=\\{f_{i}=0\\}$, for the fixed point set of $G_{i}$. In
experiments we find that there are cases (even in small dimensions like $3$)
the construction by Winkelman [25] may yield very complicated polynomials
making computations with Macaulay2 unfeasible. In such cases, we may need to
work with other 1-parameter families of automorphisms. Conjecture 1.3 was
stated to cover such cases.

Now we are ready to present our experiments, following the strategy proposed
in Conjecture 1.3. This first example inspired Theorems 1.1 and 2.4.

######  Example 3.1 (Smooth cubic curve).

Let $E=V(w_{2}^{2}-w_{3}^{3}-w_{3},w_{1})\subset\mathbb{A}^{3}$ be a smooth
plane cubic curve. We want to construct a surjective morphism
$\mathbb{A}^{3}\rightarrow\mathbb{A}^{3}\setminus E$.

According to the general strategy, we find $\mathbb{G}_{a}$-actions on
$\mathbb{A}^{3}$ that fix the curve $E$. By using the idea of projection to
hyperplanes from [25], we find actions $\varphi_{i}\colon
G_{i}\times\mathbb{A}^{3}\setminus E\to\mathbb{A}^{3}\setminus E$.

$\displaystyle\varphi_{1}:(t,w)$
$\displaystyle\mapsto(w_{1},w_{2}+tw_{1},w_{3})$
$\displaystyle\varphi_{2}:(t,w)$
$\displaystyle\mapsto(w_{1},w_{2},w_{3}+tw_{1})$
$\displaystyle\varphi_{3}:(t,w)$
$\displaystyle\mapsto(w_{1}+t(w_{2}^{2}-w_{3}^{3}-w_{3}),w_{2},w_{3}).$

Set $\varphi_{4}=\varphi_{2}$, and note that the fixed hypersurfaces are
$W_{1}=V(w_{2}^{2}-w_{3}^{3}-w_{3})$ and $W_{2}=W_{3}=W_{4}=V(w_{1})$. We pick
$p=(1,0,0)\in\mathbb{A}^{3}\setminus E$ and composing group actions
$(G_{4}\cdot G_{3}\cdot G_{2}\cdot G_{1}).p$ to obtain a surjective map to
$\mathbb{A}^{3}\setminus E$.

$\displaystyle\varphi_{1}:(a,1,0,0)$ $\displaystyle\mapsto(1,a,0)$
$\displaystyle\varphi_{2}:(b,1,a,0)$ $\displaystyle\mapsto(1,a,b)$
$\displaystyle\varphi_{3}:(c,1,a,b)$
$\displaystyle\mapsto(1+c(a^{2}-b^{3}-b),a,b)$
$\displaystyle\varphi_{4}:(d,1+c(a^{2}-b^{3}-b),a,b)$
$\displaystyle\mapsto(1+c(a^{2}-b^{3}-b),a,b+d(1+c(a^{2}-b^{3}-b))).$

The resulting map
$\psi\colon\mathbb{A}^{4}_{a,b,c,d}\to\mathbb{A}_{w_{1},w_{2},w_{3}}^{3}\setminus
E$ is given by

(8) $\displaystyle\psi(a,b,c,d)$
$\displaystyle\mapsto(1+c(a^{2}-b^{3}-b),a,b+d+cd(a^{2}-b^{3}-b)).$

We claim $\psi$ is surjective. Moreover, the restriction $F=\psi|_{d=c}$ is
surjective, and hence we obtain a surjective morphism
$\colon\mathbb{A}^{3}\rightarrow\mathbb{A}^{3}\setminus E$. This is checked
with the Macaulay2 code in Appendix A.2, and can also be seen directly by
using the proof of Theorem 2.4.

The next example is not literally covered by Theorems 1.1 and 2.4, but can be
covered by these theorems after a change of coordinates.

######  Example 3.2 (Twisted cubic).

Let $A$ be the curve $A=V(w_{2}^{2}-w_{1},w_{2}^{3}-w_{3}).$ The approach of
constructing $\mathbb{G}_{a}$-actions by projection to coordinate hyperplanes
as in [25] does not obviously result in a surjective map
$\mathbb{A}^{4}\to\mathbb{A}^{3}\setminus A$.

Instead, we may change coordinates appropriately to reduce to the case where
Theorems 1.1 and 2.4 apply. Let $\tau:\mathbb{A}^{3}\to\mathbb{A}^{3}$ be the
automorphism

$\tau:(w_{1},w_{2},w_{3})\mapsto(w_{1}-w_{2}^{2},w_{2},w_{3}-w_{2}^{3}),$

and its inverse is $\tau^{-1}=(w_{1}+w_{2}^{2},w_{2},w_{3}+w_{2}^{3})$. Notice
that $\tau(A)=(0,w_{2},0)$. We now use Theorem 2.4 with $q_{1}=w_{3}$. This
yields

(9) $f({a},{b},c)=(1+c{b},{a},{b}+c(1+c{b})),$

which has image $\mathbb{A}^{3}\setminus V(w_{2})$. Hence $F=\tau^{-1}\circ f$
has image $\mathbb{A}^{3}\setminus A$.

We may also arrive at $F$ as follows. For each group action $\varphi_{i}\colon
G_{i}\times X\to X$ used to build $f$, we define
$\tilde{\varphi}_{i}=\tau^{-1}(\varphi_{i}(G_{i},\tau(X)))$. In this example

(10) $\displaystyle\varphi_{1}(t)$ $\displaystyle=(w_{1}+tw_{3},w_{2},w_{3})$
$\displaystyle\tilde{\varphi}_{1}$
$\displaystyle=(w_{1}+t(w_{3}-w_{2}^{3}),w_{2},w_{3})$ (11)
$\displaystyle\varphi_{3}(t)$ $\displaystyle=(w_{1},w_{2},w_{3}+tw_{1})$
$\displaystyle\tilde{\varphi}_{3}$
$\displaystyle=(w_{1},w_{2},w_{3}+t(w_{1}-w_{2}^{2}))$ (12)
$\displaystyle\varphi_{2}(t)$ $\displaystyle=(w_{1},w_{2}+tw_{1},w_{3}).$

Finally, we have

(13)
$\tilde{\varphi}_{2}=(w_{1}-w_{2}^{2}+(w_{2}+{a}(w_{1}-w_{2}^{2}))^{2},w_{2}+{a}(w_{1}-w_{2}^{2}),\\\
w_{3}-w_{2}^{3}+(w_{2}+{a}(w_{1}-w_{2}^{2}))^{3})$

Note that the $\tilde{\varphi}_{i}$’s respectively fix the subvarieties
$\tilde{W}_{1}=V(w_{3}-w_{2}^{3})$, and
$\tilde{W}_{2}=\tilde{W}_{3}=V(w_{1}-w_{2}^{2})$. We pick a point $p=(1,0,0)$
and act (by slight abuse of notation) by
$\psi=(\tilde{\varphi}_{3}\cdot\tilde{\varphi}_{1}\cdot\tilde{\varphi}_{3}\cdot\tilde{\varphi}_{2}).p:\mathbb{A}^{4}\to\mathbb{A}^{3}\setminus
A$. Then $F\vcentcolon=\psi|_{c=d}$ is given by

(14) $F({a},{b},c)=(1+c{b}+{a}^{2},{a},{b}+c(1+c{b})+{a}^{3}),$

which has image $\mathbb{A}^{3}\setminus A$. We can also use Macaulay2,
similar to the code in the appendix, to verify that indeed we have a
surjective map $\mathbb{A}^{3}\rightarrow\mathbb{A}^{3}\setminus A$.

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##  Appendix A Algorithmically computing the image

In this appendix we briefly summarize an algorithm to compute the image of an
algebraic morphism which is convenient for our setting, and illustrate how we
can write code in Macaulay2 to run the experiments in Section 3.

###  A.1 An algorithm for computing the image of an algebraic map

We recall here the algorithm in [5] for computing the constructible image of
an algebraic map, which has been convenient for our purposes. It is also worth
mentioning the Macaulay2-library TotalImage described in [16], which has been
very useful for maps of lower degree.

Given a polynomial map $f\colon Z\rightarrow\mathbb{A}^{n}$, where
$Z\subset\mathbb{A}^{N}$ is a closed algebraic subvariety. We want to decide
$f(Z)$, which is a constructible set by Chevalley’s theorem.

We can assume that $Z$ is irreducible. We note that $\overline{f(Z)}$ is a
closed algebraic subvariety of $\mathbb{A}^{n}$. The main idea is to find a
proper algebraic subvariety $W\subset\overline{f(Z)}$ so that
$\overline{f(Z)}\setminus f(Z)\subset W$. Then, we have:

$\displaystyle f(Z)=[\overline{f(Z)}\setminus W]\sqcup[W\cap
f(Z)]=[\overline{f(Z)}\setminus W]\sqcup[f(f^{-1}(W)\cap Z)].$

Since $f^{-1}(W)\cap Z$ is a closed algebraic subvariety of $\mathbb{A}^{N}$
of smaller dimension than $Z$, we can proceed inductionaly to have a complete
decomposition of $f(Z)$ into sets of the form $V_{1}\setminus V_{2}$, where
both $V_{1}$ and $V_{2}$ are closed algebraic subvarities of $\mathbb{A}^{n}$.

We can first reduce to the case where the dimensions of $Z$ and $f(Z)$ are the
same. If not, then we can choose a random linear subspace
$L\subset\mathbb{A}^{N}$ of appropriate dimension such that $L\cap Z$ has the
same dimension as $f(Z)$ and $f(L\cap Z)$. (This is the same as checking that
$\overline{f(Z)}$ has the same dimension as $\overline{f(L\cap Z)}$ and $L\cap
Z$.)

With the assumption as in the previous paragraph, we can explicitly find one
such $W$ as follows. We let
$\Gamma_{f}\subset\mathbb{A}^{N}\times\mathbb{A}^{n}$ be the graph of $f$, let
$H=\mathbb{P}^{N}\setminus\mathbb{A}^{N}$ be the hyperplane at infinity,
$\overline{\Gamma}_{f}\subset\mathbb{P}^{N}\times\mathbb{A}^{n}$ be the
closure of $\Gamma_{f}$, and
$\Gamma_{f}^{\infty}=\overline{\Gamma}_{f}\cap(H\times\mathbb{A}^{n})$. Then
$W$ can be chosen as the projection of $\Gamma_{f}^{\infty}$ to the second
factor $\mathbb{A}^{n}$.

The above algorithm gives a constructive proof of Chevalley’s theorem
mentioned before.

Implementation detail: The above algorithm can be implemented in different
computer algebra systems. In our experiments later, we will use the free
software Macaulay2 [13].

We note that the map $f$ can be identified with the projection to
$\mathbb{A}^{n}$ from the graph $\Gamma_{f}$. As such, the closure of the
image of $f$ can be computed by the elimination technique, using Gröbner
basis.

The closure $\overline{\Gamma}_{f}\subset\mathbb{P}^{N}\times\mathbb{A}^{n}$
can be computed by homogenizing, then saturating with respect to the ideal of
the hyperplane at infinity.

The closure of the image of the projection to $\mathbb{A}^{n}$ from
$\Gamma_{f}^{\infty}$ again can be computed by the elimination of variables.

###  A.2 Illustrating code in Macaulay2

Here we present the code in Macaulay2 used to verify Example 3.1. It executes
the algorithm described above. Note that it is not a complete implementation
of the algorithm in [5].

⬇

r = QQ[a,b,c,e,w1,w2,w3, Degrees => {1,1,1,1,0,0,0}]

affineGraph = ideal(1+c*(a^2-b^3-b)-w1,a-w2,b+c+c^2*(a^2-b^3-b)-w3)

homogenized = homogenize(affineGraph,e)

graphClosure = saturate(homogenized, e)

graphAtInfty = graphClosure + ideal(e)

We have now computed graphAtInfty, which is
$\Gamma_{f}^{\infty}\subset\mathbb{P}^{2}\times\mathbb{A}^{3}$. We now compute
its projection down to $\mathbb{A}^{3}$ from the different open affine subsets
by elimination.

⬇

h1 = graphAtInfty + ideal(a-1)

h1elim = eliminate({a,b,c,e}, h1)

h2 = graphAtInfty + ideal(b-1)

h2elim = eliminate({a,b,c,e}, h2)

h3 = graphAtInfty + ideal(c-1)

h3elim = eliminate({a,b,c,e}, h3)

After eliminating $a,b,c,e$ from h1 and h2, the resulting ideals are just
$(1)$. With h3 on the other hand, we get a nonzero ideal, whose radical is
$(w_{1})$. We now add this to our original graph ideal to get $\Gamma_{1}$.

⬇

idealToAdd = radical h3elim

gamma1 = affineGraph + idealToAdd

Then we repeat the whole process replacing affineGraph with gamma1.

⬇

homogenized = homogenize(gamma1,e)

graphClosure = saturate(homogenized, e)

graphAtInfty = graphClosure + ideal(e)

h1 = graphAtInfty + ideal(a-1)

h1elim = eliminate({a,b,c,e}, h1)

h2 = graphAtInfty + ideal(b-1)

h2elim = eliminate({a,b,c,e}, h2)

h3 = graphAtInfty + ideal(c-1)

h3elim = eliminate({a,b,c,e}, h3)

idealToAdd = radical h3elim

gamma2 = gamma1 + idealToAdd

homogenized = homogenize(gamma2,e)

This final homogenized ideal is $(1)$, meaning that the algorithm finished,
and that the added ideal $(w_{1},w_{2}^{2}-w_{3}^{3}-w_{3})$ describes the
complement of the image, $Z=V(w_{1},w_{2}^{2}-w_{3}^{3}-w_{3})$.