# Model theory of complex numbers with polynomials functions

Benjamin Castle  Department of Mathematics, Ben Gurion University of the
Negev, Be’er-Sheva 84105, Israel<EMAIL_ADDRESS>and  Chieu-Minh Tran
Department of Mathematics, National University of Singapore, Singapore
<EMAIL_ADDRESS>
###### Abstract.

Let $\mathbb{C}$ be the set of complex numbers, and let $\mathcal{P}$ be a
collection of complex polynomial maps in several variables. Assuming at least
one $P\in\mathcal{P}$ depends on at least two variables, we classify all
possibilities for the structure $(\mathscr{M};\mathcal{P})$ up to definable
equivalence. In particular, outside a short list of exceptions, we show that
$(\mathscr{M};\mathcal{P})$ always defines $+$ and $\times$. Our tools include
Zilber’s Restricted Trichotomy, as well as the classification of symmetric
non-expanding pairs of polynomials over $\mathbb{C}$ from arithmetic
combinatorics. Along the way, we also give a new condition for a reduct
$\mathscr{M}=(M,...)$ of a smooth curve over an algebraically closed field to
recover all constructible subsets of powers of $M$.

###### 2020 Mathematics Subject Classification:

Primary 03C45; Secondary 11B30

The first author was supported NSF Grant DMS 1800692, ISF grant No. 555/21,
and a Ben Gurion University Kreitman Fellowship.

The second author was supported by Anand Pillay’s NSF Grant-2054271.

##  1\. Introduction

###  1.1. Background and result

Suppose $P$ and $Q$ are complex polynomial maps in several variables. Using
mathematical logic, one can formalize the notion ‘$Q$ can be defined in terms
of $P$’: this says that the graph of $Q$ is first-order definable in the
structure on $\mathbb{C}$ with the graph of $P$ as its only basic relation.
For example, one can show as an exercise that $x+2y$ and $x-y$ can be defined
in terms of $x+y$, neither of $x+y$ and $x\times y$ can be defined in terms of
the other, and both $x+y$ and $x\times y$ can be defined in terms of
$x^{2}+y^{2}$. Notice that since $+$ and $\times$ generate all polynomials,
this implies that every polynomial can be defined in terms of $x^{2}+y^{2}$
alone.

One might wonder, for arbitrary $P$ and $Q$, under what conditions can $Q$ be
defined in terms of $P$? In this paper, we give a surprising answer: almost
always. In particular, outside a short list of exceptions, every polynomial is
by itself capable of defining $+$ and $\times$ (Theorem 8.5). The list of
exceptions roughly contains unary polynomials, linear polynomials, and
‘monomials up to a translation.’ We will see that linear polynomials are
interdefinable with vector spaces, and monomials are interdefinable with
multiplication. Thus, in a sense, the negative example given above (addition
and multiplication cannot define each other) is canonical: for polynomials in
two or more variables, every counterexample essentially arises from either
addition or multiplication. Despite the concrete nature of the result, our
proof is almost entirely abstract, and relies on deep results in model theory
and combinatorics.

Let us give more details. A reduct of a structure $\mathscr{M}=(M;\ldots)$ is,
in this paper, another structure $\mathscr{M}^{\prime}$ with the same
underlying set $M$, such that every set definable (with parameters) in
$\mathscr{M}^{\prime}$ is also definable in $\mathscr{M}$. If
$\mathscr{M}^{\prime}$ is a reduct of $\mathscr{M}$, we also say $\mathscr{M}$
is an expansion of $\mathscr{M}^{\prime}$. Two structures are definably
equivalent if each one is a reduct of the other. We are more generally
interested in the following problems:

  1. (1)

Determine when a reduct of the field $(\mathbb{C};+,\times)$ of complex
numbers defines $+$ and $\times$ (in other words, the reduct is definably
equivalent to $(\mathbb{C};+,\times)$ itself).

  2. (2)

Classify the reducts of $(\mathbb{C};+,\times)$ up to definable equivalence.

In this paper, we consider the polynomial reducts of $\mathbb{C}$ – structures
of the form $(\mathbb{C};\mathcal{P})$ where $\mathcal{P}$ is a collection of
polynomial maps $P:\mathbb{C}^{n_{P}}\rightarrow\mathbb{C}$. Notice that this
setting encompasses the definability question for single polynomials described
at the beginning of the introduction. In particular, the earlier ‘$P$ and $Q$
can be defined in terms of each other’, more precisely, means $(\mathbb{C};P)$
is definably equivalent to $(\mathbb{C};Q)$.

Our main theorem below answers Question (1) completely for polynomial reducts,
and answers Question (2) for polynomial reducts not definable exclusively by
unary polynomials. For instance, as hinted at above, our theorem implies
immediately that the reduct $(\mathbb{C};(x,y)\mapsto
x^{17}+x^{6}y^{8}-y^{3})$ defines both $+$ and $\times$.

######  Theorem 1.1.

Suppose $\mathcal{P}$ is a collection of complex polynomial maps, and let
$\mathscr{M}=(\mathbb{C};\mathcal{P})$. Then $\mathscr{M}$ is interdefinable
with exactly one of following:

  1. (i)

$(\mathbb{C};\mathcal{U})$, where $\mathcal{U}$ is a collection of unary
polynomials.

  2. (ii)

$(\mathbb{C};+,(\lambda\cdot)_{\lambda\in F})$ where $F$ is a subfield of
$\mathbb{C}$ (that is, the $F$-vector space structure on $\mathbb{C}$).

  3. (iii)

$(\mathbb{C};\times_{r})$ for some (unique) $r\in\mathbb{C}$, where
$\times_{r}$ is the operation $x\times_{r}y=(x-r)(y-r)+r$.

  4. (iv)

The full reduct $(\mathbb{C};+,\times)$.

Moreover, (i) happens if and only if each $P\in\mathcal{P}$ involves at most
one variable. (ii) happens if and only if (i) fails and each $P\in\mathcal{P}$
is linear, and in this case $F$ is generated by all coefficients on all
variables appearing in the $P\in\mathcal{P}$. (iii) happens if and only if (i)
fails and each $P\in\mathcal{P}$ is a monomial in the group operation
$\times_{r}$. And (iv) happens in every other case.

Questions (1) and (2) above were first studied by Martin [Mar88], who
classified expansions of $(\mathbb{C};+)$ and $(\mathbb{C};\times)$ by a
single rational function. Subsequent results on expansions of $(\mathbb{C};+)$
were obtained by Zilber and Rabinovich [RZ88], and Marker and Pillay [MP90].
In particular, Marker and Pillay proved that every reduct of
$(\mathbb{C};+,\times)$ expanding $(\mathbb{C};+)$ is definably equivalent to
either the full field $(\mathbb{C};+,\times)$, or to the vector space
structure $(\mathbb{C};+,(\lambda\cdot)_{\lambda\in F})$ over a subfield $F$
of $\mathbb{C}$. Recently, Hasson and the first author [CH22] gave a complete
answer to (1) in the form of an abstract condition called very ampleness. As
one concrete application, they gave a unified proof of the Marker–Pillay
result and its multiplicative counterpart, showing that a reduct of
$(\mathbb{C};+,\times)$ containing $\times$ must be definably equivalent to
either $(\mathbb{C};\times)$ or $(\mathbb{C};+,\times)$.

Meanwhile, if one does not assume $+$ or $\times$ is contained in the reduct,
concrete classifications have only been attained ‘up to finite covers’ (see
the next two paragraphs, and also [Lov04]). As reducts generated by
polynomials functions are quite general, Theorem 1.1 can be seen as a step
toward classifying all reducts up to definable equivalence. In fact, we see it
as evidence that the reducts of $(\mathbb{C};+,\times)$ can be classified ‘up
to a unary function and trivial noise’– namely there is a short list of
structures on $\mathbb{C}$ such that for any reduct $\mathscr{M}$ of
$(\mathbb{C};+,\times)$, there is a function
$f:\mathbb{C}\rightarrow\mathbb{C}$ such that the image of $\mathscr{M}$ under
$f$ is in the given list, and $\mathscr{M}$ is determined up to definably
equivalence by $f$, its image under $f$, and some extra noise coming from
binary relations (the analog of our unary polynomials).

Question (1) is closely related to Zilber’s Restricted Trichotomy Conjecture
for $(\mathbb{C};+,\times)$ (see Section 2 for the precise statement); indeed,
this conjecture will play a crucial role in our proof. Restricted to our
setting, Zilber’s conjecture predicts that non-local modularity (an abstract
model-theoretic condition) is necessary and sufficient for a reduct of
$(\mathbb{C};+,\times)$ to define an isomorphic copy of
$(\mathbb{C};+,\times)$. For $(\mathbb{C};+,\times)$, this was recently
verified in full generality by the first author ([Cas22]). In our setting
(reducts of the field itself), the conjecture is also implied by previous
partial cases due to Rabinovich [Rab93] and Hasson-Sustretov [HS21].

In general, if a reduct $\mathscr{M}$ of $(\mathbb{C};+\times)$ defines an
isomorphic copy of $(\mathbb{C};+,\times)$ as above, it must define $+$ and
$\times$ ‘up to a unary function’ – that is, there is a non-constant rational
function $f:\mathbb{C}\rightarrow\mathbb{C}$ such that $\mathscr{M}$ defines
the relations $f(x)+f(y)=f(z)$ and $f(x)f(y)=f(x)$. However, in some cases the
function $f$ cannot be eliminated. From this angle, Question (1) is the next
step in the same direction – we want to classify structures that interpret not
just any copy of $(\mathbb{C},+,\times)$, but the copy (i.e. the original
field).

###  1.2. Outline of proofs and organization of the paper

The proof of Theorem 1.1 is organized according to the Zilber trichotomy for
strongly minimal structures.

An infinite definable set is strongly minimal if all of its definable subsets,
even in elementary extensions, are finite or cofinite. A structure is strongly
minimal if its underlying set is a strongly minimal set. The complex field is
strongly minimal, and thus so are all of its reducts.

Zilber proposed a ‘coarse’ classification of strongly minimal structures ‘up
to finite correspondence.’ This is a generalization of ‘up to definable
bijection’: two definable sets $X$ and $Y$ are in finite correspondence if
there is a definable set $C\subseteq X\times Y$ such that both projections
$C\rightarrow X$ and $C\rightarrow Y$ are finite-to-one with cofinite image.

Zilber considered three possibilities for a strongly minimal structure
$\mathscr{M}$. We will not give precise definitions yet.
Trivial case: Every definable relation $X\subseteq M^{n}$ is, roughly,
determined by relations in $M$ and $M^{2}$.
Group case: Up to a finite correspondence, $\mathscr{M}$ is an abelian group
equipped only with certain subgroups of powers of $M$.
Field case: Up to a finite correspondence, $\mathscr{M}$ is an algebraically
closed field equipped only with the field operations.

######  Remark 1.2.

The union of the trivial and group cases is known in model theory as local
modularity. The group case is often called ‘non-trivial locally modular.’ The
structures $(\mathbb{C};+)$ and $(\mathbb{C};\times)$ are non-trivial locally
modular. The structure $(\mathbb{C};+,\times)$ is not locally modular.

The content of Zilber’s Restricted Trichotomy Conjecture, in our setting, is
that every reduct of $(\mathbb{C};+,\times)$ belongs to one of these three
cases. Our theorem then says that, for polynomial reducts, we can get rid of
the finite correspondences. The result is a more precise classification of
each case. In particular, we will show that the trivial case is equivalent to
(i) in Theorem 1.1; the group case is equivalent to the union of (ii) and
(iii); and the field case is equivalent to (iv). We prove each of these
statements by a separate argument.

The trivial case can be analyzed quite easily. Any polynomial depending on at
least two variables immediately violates triviality. Meanwhile, it is a short
exercise to show the converse – that if $\mathscr{M}$ is a reduct given by
unary polynomials, $\mathscr{M}$ is trivial. This analysis is done in Section
3. Note that we do not obtain a precise classification of the
interdefinability relation between trivial polynomial reducts. However, a full
classification is obtained in Corollary 3.4 for reducts defined by a single
unary polynomial.

The group case is treated in Sections 4, 5, and 6. Section 4 verifies that,
assuming non-triviality, linear polynomials are interdefinable with vector
spaces, and monomials are interdefinable with multiplication. This is again a
straightforward exercise. Conversely, Sections 5 and 6 show that polynomial
reducts in the group case always arise from linear polynomials and ‘twisted
monomials’ (monomials in the group $\times_{r}$ from Theorem 1.1). The
highlight of these sections is the use of arithmetic combinatorics to
‘eliminate’ the finite correspondence with an abelian group, thereby gaining
precise information about which polynomials can appear in the reduct.

Let us elaborate. We are given a polynomial reduct
$\mathscr{M}=(\mathbb{C};\mathcal{P})$, an abelian group $G$, and a definable
finite correspondence $C\subseteq\mathbb{C}\times G$. We want to show that all
$P\in\mathcal{P}$ are linear or monomials in some $\times_{r}$. We know that
all definable sets in all $G^{n}$ are ‘group-like’ (formally, they are Boolean
combinations of cosets of subgroups). Thus, we would like to transfer group-
like data ‘through $C$’, to recover a group operation on $\mathbb{C}$. We do
this using asymptotic finite combinatorics. The idea is that when passing a
finite set through the correspondence $C$ (say from $G$ to $\mathbb{C}$), its
cardinality will only be scaled by a constant factor; so $C$ does not change
any asymptotic growth rates, and thus any ‘group-like’ combinatorial phenomena
in $G$ should also occur in $\mathbb{C}$.

It will be easy to reduce to the case that $\mathcal{P}$ consists of a single
binary polynomial, say $P(x,y)$. In this case, the image of the graph of $P$,
through $C$, is some definable set $Y\subseteq G^{3}$, which must be ‘group-
like’. For simplicity, let us assume here that $Y$ is the graph of a
homomorphism $\phi:G^{2}\rightarrow G$ (the general case is only slightly more
complicated). Then, using known techniques, for all $\epsilon$ one can find
arbitrary large finite $B\subseteq G$ (essentially a geometric progression)
such that

$|\phi(B\times B)|<|B|^{1+\epsilon}.$

In our terminology, we say that $\phi$ has symmetric small expansion.

Setting $A$ to be the preimage of $B$ in $\mathbb{C}$ (and with a little more
work), one then obtains

$|P(A\times A)|<|A|^{1+\epsilon}$

– thus $P$ also has symmetric small expansion. Then, by Elekes–Ronyai-type
theorems from arithmetic combinatorics, we get strong information about $P$:
there are unary polynomials $f$ and $u$ such that either
$P(x,y)=f(c_{1}u(x)+c_{2}u(y))$ for some $c_{1},c_{2}\in\mathbb{C}$ or
$P(x,y)=f(u^{m}(x)u^{n}(y))$ for some integers $m,n$. With a bit more work, we
deduce all possibilities for $f$ and $u$, concluding that $P$ has one of the
desired forms.

Section 7 treats the most interesting field case. For this, we actually prove
a more general statement (Theorem 7.3): Suppose $\mathscr{M}=(M;...)$ is a
strongly minimal reduct of a curve over an algebraically closed field, which
is not locally modular (i.e. belongs to the ‘field case’). Then $\mathscr{M}$
recovers all constructible subset of powers of $M$, provided it defines
(roughly) both a binary function $M^{2}\rightarrow M$ and a generically
$d$-to-one function $M\rightarrow M$ for some $d\geq 2$. It is straightforward
to build such functions in a polynomial reduct (Theorem 8.1).

Let us describe the proof of Theorem 7.3. We are given a reduct
$\mathscr{M}=(M;...)$ of a curve, and a finite correspondence with a definable
algebraically closed field $(F,\oplus,\otimes)$. By elimination of imaginaries
in $F$, there is even a finite-to-one definable function $f:M\rightarrow
F^{n}$ for some $n$. We think of $f$ as realizing $M$ as a finite cover of a
curve over $F$. It is well known (and proven explicitly in [CH22]) that if $f$
is injective (i.e. the cover is trivial), $\mathscr{M}$ defines $+$ and
$\times$. So our goal is to show that some such $f$ is injective.

We use a generalization of an argument due to Hrushovski, for analyzing (and
reducing) finite covers of definable sets. Past iterations of Hrushovski’s
technique assume the ‘cover’ (our $M$) is a definable group. The idea is to
choose a cover of minimal index, and show by minimality that the fibers are
cosets of a finite subgroup. If the group is divisible, one can then construct
a trivial cover, thereby concluding the minimal cover was always trivial.

In our case, we have no definable group operation on $M$, so we need a more
elaborate analysis of the fibers in the minimal cover $f$. We define the
equivalence relation $\sim$ on $M$ by $x\sim y$ if $f(x)=f(y)$. We then use
minimality to show that $\sim$ is respected (in a precise sense) by all
definable finite correspondences between $M$ and itself (Lemma 7.9). Finally,
we conclude that $f$ is injective via a counting argument, applying the above
to various correspondences built from the provided binary and unary functions.

###  1.3. Notation and convention

We work throughout with strongly minimal structures $\mathscr{M}=(M;...)$ in a
language $\mathcal{L}$. If there is ambiguity in the choice of structure, we
may add to all terms defined below the prefix $\mathscr{M}$ (creating
$\mathscr{M}$-definable, $\mathscr{M}$-generic, etc.). However, unless
otherwise stated, the reader should assume all model-theoretic terms refer to
the reducts of algebraic curves we consider (not to the full field structure).

By a definable set, we refer to definability over parameters in the expansion
$\mathscr{M}^{\textrm{eq}}$, obtained by closing the class of definable sets
under quotients by definable equivalence relations.

We denote sets of parameters from $M$ by $A$, $B$, …. Unless otherwise stated,
sets of parameters are assumed to small, i.e. of smaller cardinality than
$|M|$.

##  2\. Preliminaries

###  2.1. Some model theory

Recall that $\mathscr{M}$ is saturated if whenever $\phi(\bar{x},A)$ is a
consistent set of formulas in the variable $\bar{x}$ with parameters from the
small set $A$, then $M$ contains a realization of $\phi$. Every strongly
minimal structure has a saturated elementary extension; every uncountable
strongly minimal structure in a countable language is already saturated. We
will almost always be able to assume $\mathscr{M}$ is saturated, but will make
this explicit as we go.

Recall that Morley rank gives a well-behaved dimension theory for definable
sets in $\mathscr{M}$ (for reducts of $(\mathbb{C};+,\times)$, Morley rank
agrees with the usual dimension theory of varieties). For example,
$\dim(M^{n})=n$ for each $n$, and the 0-dimensional sets are precisely the
non-empty finite sets.

A definable set of dimension $n$ is called stationary if it is not the
disjoint union of two definable sets of dimension $n$. Every definable set $X$
is a finite disjoint union of stationary sets (of the same dimension as $X$).
These sets are unique up to almost equality (defined below), and are called
the stationary components of $X$.

For a tuple $a$ and set $A$, the notation $\dim(a/A)$ denotes the smallest
dimension of a set containing $a$ and definable with parameters from $A$. One
then says that $a_{1},...,a_{n}$ are independent over $A$ if
$\dim(a_{1},...,a_{n}/A=\dim(a_{1}/A)+...+\dim(a_{n}/A)$. If $\dim(a/A)=0$ we
say that $a$ is algebraic over $A$, denoted $a\in\operatorname{acl}(A)$.

If $X\subseteq Y$ are definable, we say that $X$ is generic in $Y$ if
$\dim(X)=\dim(Y)$, small $Y$ if $\dim(X)<\dim(Y)$, and large in $Y$ if
$\dim(Y-X)<\dim Y$. One can similarly notions of almost containment and almost
equality of definable sets. We then say a definable function $f:X\rightarrow
Y$ is almost surjective if $\operatorname{Im}(f)$ is large in $Y$, and almost
finite-to-one if the union of all infinite fibers of $f$ is small in $X$.

If $X$ is definable with parameters from $A$, and $a\in X$, we say that $a$ is
generic in $X$ over $A$ if $\dim(a/A)=\dim(X)$. Generic points always exist if
$\mathscr{M}$ is saturated and $A$ is small.

###  2.2. The Zilber Trichotomy

We now explicitly state the Zilber trichotomy.

######  Definition 2.1.

Suppose $\mathscr{M}=(M,...)$ is strongly minimal.

  * •

If $\mathscr{M}$ is saturated, we say $\mathscr{M}$ is trivial if whenever
$A\subseteq M$, $b\in M$, and $a\in\operatorname{acl}(A)$, there is some $a\in
A$ with $b\in\operatorname{acl}(a)$.

  * •

If $\mathscr{M}$ is not saturated, we say $\mathscr{M}$ is trivial if some
(equivalently any) saturated elementary extension of it is trivial.

######  Definition 2.2.

Suppose $\mathscr{M}=(M,...)$ is strongly minimal. Then $\mathscr{M}$ is not
locally modular if there is a definable 2-dimensional family of plane curves
in $\mathscr{M}$. That is, a definable set $T$ of dimension 2, and a definable
set $C\subseteq M^{2}\times T$, such that:

  * •

For each $t\in T$, $\dim(C_{t})=1$.

  * •

For any $t\neq t^{\prime}\in T$, $\dim(C_{t}\cap C_{t}^{\prime})=0$.

Otherwise, we say $\mathscr{M}$ is locally modular.

Strongly minimal structures are typically organized into three levels:
trivial, non-trivial locally modular, and non-locally modular. The idea is
that these correspond to ‘graph-like,’ ‘group-like,’ and ‘field-like.’ For the
latter two levels, the terminology is no coincidence:

######  Fact 2.3.

[Hru87],[HP87] Suppose $\mathscr{M}=(M;...)$ is strongly minimal, non-trivial,
and locally modular. Then there is a definable, strongly minimal, abelian
group $(G;+,...)$ in definable finite correspondence with $\mathscr{M}$.
Moreover, for any such $G$, every definable subset of every $G^{n}$ is a
finite Boolean combination of cosets of definable subgroups of $G^{n}$.

######  Fact 2.4 (Zilber’s Restricted Trichotomy for $\mathbb{C}$ [Cas22]).

Suppose $\mathscr{M}=(M;,...)$ is strongly minimal and not locally modular.
Suppose moreover that the underlying set $M$, and all definable subsets of all
$M^{n}$, are complex constructible sets (Boolean combinations of affine
varieties). Then there is a definable field $(F;\oplus,\otimes)$, isomorphic
to $(\mathbb{C};+,\times)$, in definable finite correspondence with $M$.

##  3\. Trivial Reducts

In this section we analyze trivial polynomial reducts. This material is quite
straightforward, so we only sketch the proofs.

######  Lemma 3.1.

Let $\mathscr{M}=(\mathbb{C};...)$ be a trivial reduct of
$(\mathbb{C},+,\times)$, and let $P\in\mathbb{C}[x_{1},...,x_{n}]$ be an
$\mathscr{M}$-definable polynomial. Then $P\in\mathbb{C}[x_{i}]$ for some $i$.

###### Proof.

Suppose $P$ is definable over $A$. Replacing $\mathscr{M}$ with
$(\mathbb{C};P)$, we may assume $\mathscr{M}$ is saturated, so there is an
element $(a_{1},...,a_{n})\in\mathbb{C}^{n}$ which is generic over $A$. Now
assume $P\notin\mathbb{C}[x_{i}]$ for any $i$. Then
$P(a_{1},...,a_{n})\in\operatorname{acl}(A,a_{1},...,a_{n})$ but
$P(a_{1},...,a_{n})\notin\operatorname{acl}(A,a_{i})$ for any $i$, which
easily contradicts triviality. ∎

By Lemma 3.1, we may restrict our attention to unary polynomials, i.e. those
of the form $P(x)\in\mathbb{C}[x]$. For such polynomials, we now show the
converse:

######  Lemma 3.2.

Let $\mathcal{P}$ be a collection of unary complex polynomials. Then
$\mathscr{M}=(\mathbb{C};\mathcal{P})$ is trivial. In fact, in $\mathscr{M}$,
algebraic closure is just the union of the constants in $\mathcal{P}$ and the
closure under $P$ and $P^{-1}$ for non-constant $P\in\mathcal{P}$.

###### Proof.

We may assume $\mathcal{P}$ is finite, so $\mathscr{M}$ is saturated. Draw an
edge-colored directed graph $G$ with vertex set $\mathbb{C}$, one color for
each $P\in\mathcal{P}$, and an edge of color $P$ between $x$ and $y$ whenever
$P(x)=y$. So any color-preserving automorphism of $G$ is an automorphism of
$\mathscr{M}$. Moreover, for $x\in\mathbb{C}$, the isomorphism type of the
connected component of $G$ containing $x$ is encoded into the type of $x$ over
$\emptyset$. It follows easily that for any three generics
$x,y,z\in\mathbb{C}$ in different components, there is an automorphism of
$\mathscr{M}$ fixing $x$ and sending $y$ to $z$; in particular, this gives
$y\notin\operatorname{acl}(x)$. It follows easily that algebraic closure in
$\mathscr{M}$ is as described in the lemma, and triviality is immediate from
this description of algebraic closure. ∎

So triviality is characterized by unary polynomials. This is all we really
need here for our main result; however, for the case of a single polynomial,
we can give the following more precise classification.

In what follows, for $P(x)\in\mathbb{C}[x]$ the notation $P^{n}$ will denote
the $n$-fold composition of $P$. In particular, if $n=0$ this means the
identity map, and if $n$ is negative this means the $-n$-fold composition of
the inverse of $P$ (when it exists).

######  Theorem 3.3.

Let $P\in\mathbb{C}[x]$, and let $\mathscr{M}=(\mathbb{C};P)$. Then the
definable unary rational functions in $\mathscr{M}$ are precisely the
following:

  1. (1)

For constant $P$, the constant and identity maps.

  2. (2)

For $\deg(P)=1$, the constant maps and those of the form $P^{n}$ for
$n\in\mathbb{Z}$.

  3. (3)

For $\deg(P)=2$, the constant maps and those of the form $P^{n}$ or $r\circ
P^{n}$ for $n\geq 0$, where $r$ is the reflection across the axis of $P$; that
is, if $P(x)=ax^{2}+bx+c$, then $r(x)=\frac{-b}{a}-x$.

  4. (4)

For $\deg(P)\geq 3$, the constant maps and those of the form $P^{n}$ for
$n\geq 0$.

###### Proof.

This is an exercise in quantifier elimination. If $P$ is constant, the
identity, or an involution $x\mapsto c-x$, then everything is clear. So assume
$P$ has degree $d\geq 1$ and is not the identity or an involution. It follows
easily that any two iterates of $P$ are distinct.

Let $A\subseteq\mathbb{C}$ be a countable algebraically closed subfield
containing the coefficients of $P$. Consider the language $\mathcal{L}$
containing constant symbols $c_{a}$ for each element $a\in A$, and a unary
function symbol $f$. Let $T$ be the theory asserting each of the following:

  1. (1)

The atomic diagram of $A$ in the structure $\mathscr{M}$.

  2. (2)

$c_{a_{i}}$ has exactly $l_{i}$ preimages under $f$ for each $i$, where
$a_{1},...,a_{n}$ are the elements of $\mathbb{C}$ which do not have exactly
$d$ preimages under $P$ (note that there are finitely many such elements, each
of which belongs to $A$), and where $a_{i}$ has exactly $l_{i}$ preimages
under $P$.

  3. (3)

Every element other than $c_{a_{1}},...,c_{a_{n}}$ has exactly $d$ preimages
under $f$.

  4. (4)

If $f^{i}(x)=f^{j}(x)$ for some $i\neq j\geq 0$, then $x=c_{a}$ for one of the
finitely many elements $a$ such that $P^{i}(a)=P^{j}(a)=0$ (here was use that
since any two iterates of $P$ are distinct, for each $i$ and $j$ there are
only finitely many such elements, each of which belongs to $A$).

It is easy to see that $T$ is complete and eliminates quantifiers. Clearly
$\mathscr{M}\models T$, so $T$ completely axiomatizes
$\operatorname{Th}(\mathscr{M})$. In particular, $T$ is strongly minimal.

In general, a model of $T$ is given by a copy of $A$ together with a disjoint
set on which $f$ induces a ‘generic’ $d$-to-1 function (i.e. an everywhere
$d$-to-1 map for which $f^{i}(x)=f^{j}(x)$ has no solutions unless $i=j$). Let
$a\in\mathbb{C}$ be generic. To determine the definable rational functions in
$\mathscr{M}$, it suffices to determine $\operatorname{dcl}(a)$ (i.e. those
elements of $\mathbb{C}$ definable in $\mathscr{M}$ over the parameter $a$).
Now as in the previous lemma, we draw a graph on $\mathbb{C}$ with an edge
between $x$ and $y$ whenever $f(x)=y$. Then, similarly, any two connected
components not containing any $c_{a}$’s are isomorphic. So
$\operatorname{dcl}(a)$ is restricted to the connected component of $a$, on
which we just have a generic $d$-to-1 function. By further analyzing
automorphisms of such functions, one can easily reduce $\operatorname{dcl}(a)$
to the four cases in the theorem. For clarity, we note that if $d=2$ the map
$r$ is respresented by the unique $a^{\prime}\neq a$ with
$f(a)=f(a^{\prime})$. ∎

Theorem 3.3 implies a classification of the interdefinability relation on
unary polynomials:

######  Corollary 3.4.

Let $P,Q\in\mathbb{C}[x]$. Then $(\mathbb{C};P)$ is interdefinable with
$(\mathbb{C};Q)$ if and only if one of the following holds:

  1. (1)

$P$ and $Q$ are both among the identity and constant maps.

  2. (2)

$P$ and $Q$ are inverse linear maps.

##  4\. Non-trivial Locally Modular Reducts

Over the next three sections, we analyze non-trivial locally modular reducts.
Due to Theorem 3.2, we will always assume at least one of the polynomials in
the signature depends on at least two variables. The current section
introduces those non-trivial locally modular which ultimately turn out to be
the only ones.

To state our results, it is convenient to introduce the following:

######  Definition 4.1.

Given $r\in\mathbb{C}$, we define the operation of multiplication twisted by
$r$ on $\mathbb{C}$ by $x\times_{r}y=(x-r)(y-r)+r$.

Note that the equation $x\times_{r}y=z$ is equivalent to $(x-r)(y-r)=z-r$.
Thus it is clear that $\times_{r}$ defines a group operation on
$\mathbb{C}-\\{r\\}$, which is isomorphic to the multiplicative group of
$\mathbb{C}$.

######  Definition 4.2.

Let $r\in\mathbb{C}$. Then by a monomial twisted by $r$, we mean a polynomial
of the form $a\times_{r}{x_{1}}...\times_{r}{x_{n}}$, where $a\in\mathbb{C}$
and the $x_{i}$’s are variables which are allowed to repeat. In general we say
that a polynomial is a twisted monomial if it is a monomial twisted by $r$ for
some $r\in\mathbb{C}$.

The main result of this section is that linear polynomials and twisted
monomials are interdefinable, respectively, with vector spaces and twisted
multiplication:

######  Proposition 4.3.

Let $\mathcal{P}$ be a collection of complex polynomials, at least one of
which depends on at least two variables. Let
$\mathscr{M}=(\mathbb{C};\mathcal{P})$.

  1. (1)

Suppose each $P\in\mathcal{P}$ is linear, so has the form

$P(x_{1},...,x_{n_{P}})=b_{P}+\sum_{i=1}^{n_{P}}a_{i}^{P}x_{i}.$

for some constants $b_{P}$ and $a_{i}^{P}$. Then $\mathscr{M}$ is
interdefinable with the vector space structure on $\mathbb{C}$ over the field
$F=\mathbb{Q}(\\{a_{i}^{P}\\}_{P\in\mathcal{P},i\leq n_{P}})$.

  2. (2)

Suppose there is some $r\in\mathbb{C}$ such that each $P\in\mathcal{P}$ is a
monomial twisted by $r$. Then $\mathscr{M}$ is interdefinable with
$(\mathbb{C},\times_{r})$.

In particular, if either (1) or (2) holds, then $\mathscr{M}$ is locally
modular.

###### Proof.

In each case, it is clear that $\mathscr{M}$ is definable from the desired
structure. We show the converse.

  1. (1)

Let us fix some $P\in\mathcal{P}$ which depends on at least two variables, and
write it as $P(x_{1},...,x_{n})=a_{1}x_{1}+...+a_{n}x_{n}+b$. Without loss of
generality we assume $a_{1},a_{2}\neq 0$. Now since $a_{2}\neq 0$, we can find
a tuple $x_{2},...,x_{n}$ with $a_{2}x_{2}+...+a_{n}x_{n}+b=0$. Specializing
to this tuple, we obtain the map $x\mapsto a_{1}x$, and therefore also
$x\mapsto\frac{x}{a_{1}}$ by inverting. By a similar argument, we obtain
$x\mapsto a_{j}$ for each $j$, and therefore also $x\mapsto\frac{x}{a_{j}}$
for all non-zero $a_{j}$. Now precomposing these inverted scalings with $P$,
we get the map which replaces each non-zero $a_{j}$ with 1. Assume this map
has the form $x_{1}+...+x_{k}+b$, where $2\leq k\leq n$. Specializing all but
one $x_{j}$ to 0, we obtain $x\mapsto x+b$, and therefore by inverting we get
$x\mapsto x-b$. Now replacing $x_{k}$ with $x_{k}-b$, we obtain
$x_{1}+...+x_{k}$; and thus specializing all but two variables to 0, we get
addition. Then we can now subtract $b_{i}$ from each $P_{i}$, and assume each
$P_{i}$ has no constant term. Then specializing all but one variable to 0 in
each $P_{i}$, we get the scaling by each $a_{j}^{i}$. Combined with addition,
this is enough to recover the $F$-vector space structure.

  2. (2)

Without loss of generality we assume $r=0$, so each $P\in\mathcal{P}$ is a
monomial. Let us again distinguish some $P\in\mathcal{P}$ which depends on at
least two variables, and write it as
$P(x_{1},...,x_{n})=bx_{1}^{a_{1}}...x_{n}^{a_{n}}$, where $b\neq 0$ each
$a_{j}$ is a non-negative integer. Without loss of generality we assume
$a_{1},a_{2}\geq 1$. Then by specializing all other variables to 1, we obtain
the map $(x_{1},x_{2})\mapsto bx_{1}^{a_{1}}x_{2}^{a_{2}}$. Without loss of
generality let us identify this map as $P$; our task is now to show that
multiplication is definable from $P$ alone. Now by making appropriate
specializations as in (1), we recover the power maps $x^{a_{1}}$ and
$x^{a_{2}}$. Then the map $(x_{1},x_{2})\mapsto bx_{1}x_{2}$ is definable,
because $y=bx_{1}x_{2}$ holds if and only if there are
$x_{1}^{\prime},x_{2}^{\prime}$ with $(x_{1}^{\prime})^{a_{1}}=x_{1}$,
$(x_{2}^{\prime})^{a_{2}}=x_{2}$, and
$y=b(x_{1}^{\prime})^{a_{1}}(x_{2}^{\prime})^{a_{2}}$. Finally, setting
$x_{2}=1$ in $y=bx_{1}x_{2}$ gets scaling by $b$, and therefore scaling by
$\frac{1}{b}$. Then replacing $x_{2}$ with $\frac{x_{2}}{b}$ in
$y=bx_{1}x_{2}$ gets multiplication.

∎

For completeness, we also point out:

######  Lemma 4.4.

Let $\mathcal{L}$ be the family of languages on $\mathbb{C}$ containing each
of the following:

  1. (1)

For each subfield $F\subseteq\mathbb{C}$, the $F$-vector space language
$L_{F}$ (i.e. addition, and scaling maps for each element of $F$).

  2. (2)

For each $r\in\mathbb{C}$, the language $L(r)$ of $r$-twisted multiplication
(i.e. with the sole binary operation $\times_{r}$).

Then no two languages in $\mathcal{L}$ are interdefinable.

###### Proof.

It is well known that the $L$-structure on $\mathbb{C}$ eliminates quantifiers
for each $L\in\mathcal{L}$, making the lemma obvious. ∎

##  5\. Small expansion in strongly minimal sets

Our next goal is to show that every non-trivial locally modular polynomial
reduct is interdefinable with either a vector space or a twisted
multiplication. In this section, we develop analogs for strongly minimal sets
of some tools from additive combinatorics. In the next section, we apply these
tools to ‘recognize’ addition and multiplication abstractly from the
asymptotic combinatorics provided by local modularity. The material in this
section is technical but straightforward. The idea is to generalize the
machinery of small expansion from functions to multivalued functions.

Recall that a polynomial $P(x,y)$ over $\mathbb{R}$ or $\mathbb{C}$ is said to
have small expansion if for all $\epsilon>0$ one can find arbitrarily large
positive integers $N$ and finite sets $A$ and $B$ with $|A|=|B|=N$ and
$|P(A,B)|<N^{1+\epsilon}$. One also has a similar notion of symmetric small
expansion, obtained by further imposing that $A=B$. Polynomials with symmetric
small expansion were classified in [JRT22], essentially as those ‘coming from’
either addition or multiplication in a precise sense. Our goal is to show that
an analog of symmetric small expansion holds in certain strongly minimal
groups, and is moreover preserved through finite correspondences with these
groups.

######  Assumption 5.1.

Throughout Section 5, we work in a sufficiently saturated model $\mathscr{M}$
of a complete theory $T$. In particular, a strongly minimal set is a set $D$
definable in $\mathscr{M}$ such that $D$ is strongly minimal. One can further
assume $\mathscr{M}$ is strongly minimal too, but does not need to.

######  Definition 5.2.

Let $D$ and $E$ be strongly minimal sets, and let $X\subseteq D^{2}\times E$
be definable. We say that $X$ is a quasi-function from $D^{2}$ to $E$ if
$\dim(X)=2$, $X$ is stationary, and each of the projections $X\rightarrow
D^{2}$ and $X\rightarrow D\times E$ (for both copies of $D$) is almost finite-
to-one.

Quasi-functions should be thought of as abstractions of binary polynomials on
$D$. We now introduce symmetric small expansion for quasi-functions.

######  Notation 5.3.

Let $D$ and $E$ be strongly minimal, and let $X$ be a quasi-function from
$D^{2}$ to $E$. If $A,B\subseteq D$, we let $X(A,B)$ be the set of $z\in E$
such that for some $x\in A$ and $y\in B$ we have $(x,y,z)\in X$.

######  Definition 5.4.

Let $D$ and $E$ be strongly minimal, and let $X$ be a quasi-function from
$D^{2}$ to $E$. We say that $X$ has symmetric small expansion if for all
$\epsilon>0$, for arbitrarily large natural numbers $N$, we can find
$A\subseteq D$ with $|A|=N$ and $|X(A,A)|<N^{1+\epsilon}$.

######  Definition 5.5.

Let $D$ be strongly minimal. We say that $D$ has universal symmetric small
expansion if every quasi-function from $D^{2}$ to $D$ has symmetric small
expansion.

The following is a restatement of Definition 5.4:

######  Lemma 5.6.

Let $D$ and $E$ be strongly minimal, and let $X$ be a quasi-function from
$D^{2}$ to $E$. Then $X$ has symmetric small expansion if and only if one can
find a sequence of natural numbers $\\{N_{k}\\}\rightarrow\infty$, and sets
$A_{k}\subseteq D$ for each $k$ with $|A_{k}|=N_{k}$, such that
$\lim_{k\rightarrow\infty}\frac{\log{|X(A_{k},A_{k})}|}{N_{k}}\leq 1$.

We will call the sequence $\\{A_{k}\\}$ from Definition 5.6 a witness family
of the symmetric small expansion of $X$.

The following will also be useful:

######  Definition 5.7.

Let $D$ and $E$ be strongly minimal, let $X$ be a quasi-function from $D^{2}$
to $E$, and let $\sim$ be an equivalence relation on $D$. We say that $X$ has
symmetric small expansion respecting $\sim$ if one can find a witness family
$\\{A_{k}\\}$ to symmetric small expansion of $X$, such that each $A_{k}$ is a
union of $\sim$-classes.

We proceed to note several preservation properties of symmetric small
expansion, which are easy to see using the definitions. Lemma 5.8 gives
preservation under almost equality. Lemma 5.10, Lemma 5.11, and Corollary 5.12
discuss behavior under finite-to-one functions.

######  Lemma 5.8.

Let $D_{1}$, $D_{2}$, $E_{1}$, and $E_{2}$ be strongly minimal. Assume that
$D_{1}$ and $D_{2}$ are almost equal, and $E_{1}$ and $E_{2}$ are almost equal

  1. (1)

If $X_{1}$ is a quasi-function from $(D_{1})^{2}$ to $E_{1}$, then there is a
quasi-function $X_{2}$ from $(D_{2})^{2}$ to $E_{2}$ that is almost equal to
$X_{1}$.

  2. (2)

If $X_{1},X_{2}\subseteq(D_{1})^{2}\times E_{1}$ are definable and almost
equal, then $X_{1}$ is a quasi-function from $(D_{1})^{2}$ to $E_{1}$ if and
only if $X_{2}$ is.

  3. (3)

If $X_{1}$ is a quasi-function from $(D_{1})^{2}$ to $E_{1}$, $X_{2}$ is a
quasi-function from $(D_{2})^{2}$ to $E_{2}$, and $X_{1}$ and $X_{2}$ are
almost equal, then $X_{1}$ has symmetric small expansion if and only if
$X_{2}$ does.

###### Proof.

  1. (1)

It suffices to note that the restriction of $X_{1}$ to $(D_{1}\cap
D_{2})^{2}\times(E_{1}\cap E_{2})$ has Morley rank 2, which is easy to see
using the definition of quasi-functions.

  2. (2)

Clear from the definitions.

  3. (3)

Assume $X_{1}$ has symmetric small expansion, and let $\\{(N_{k},A_{k})\\}$ be
a witness family. It suffices to observe that, after deleting finite subsets
of each $A_{k}$ of bounded size, we obtain sets $A^{\prime}_{k}\subseteq
D_{2}$ with
$|X_{2}(A^{\prime}_{k},A^{\prime}_{k})|\leq|X_{1}(A_{k},A_{k})|+l\cdot N_{k}$
for some fixed integer $l$. Clearly, this implies the desired result. Now to
see that this is possible, we delete all points of the symmetric difference
$D_{1}\Delta D_{2}$, as well as all points of $D_{1}\cup D_{2}$ which have
infinite preimage under the projection of $X_{2}-X_{1}$ to the leftmost
$D_{1}\cap D_{2}$ factor. Since $\dim(X_{2}-X_{1})\leq 1$, there are only
finitely many such points, and by definition they do not depend on $k$ – thus
$|A_{k}-A^{\prime}_{k}|$ is indeed bounded.

Now the point is that, outside the removed points, the projection of
$X_{2}-X_{1}$ to the first copy of $D_{1}\cap D_{2}$ is finite-to-one with
fibers of size at most some fixed integer $l$. This easily implies that the
image of $X_{2}-X_{1}$ in $(D_{1}\cup D_{2})^{2}$ can contain at most $l\cdot
N_{k}$ points of $A^{\prime}_{k}\times A^{\prime}_{k}$. Thus
$X_{2}(A^{\prime}_{k},A^{\prime}_{k})$ has size at most
$|X_{1}(A_{k},A_{k})|+l\cdot N_{k}$, as desired.

∎

######  Remark 5.9.

It follows by Lemma 5.8(3) (by setting $X_{1}=X_{2}$) that whether a quasi-
function $X$ from $D^{2}$ to $E$ has symmetric small expansion is purely a
property of $X$, and does not depend on the choice of $D$ and $E$.

######  Lemma 5.10.

Let $D_{1}$, $D_{2}$, $E_{1}$, and $E_{2}$ be strongly minimal, and let
$f:D_{1}\rightarrow D_{2}$, $g:E_{1}\rightarrow E_{2}$ be definable, finite-
to-one, surjective functions.

  1. (1)

If $X_{1}$ is a quasi-function from $(D_{1})^{2}$ to $E_{1}$, then
$X_{2}:=\\{(f(x),f(y),g(z)):(x,y,z)\in X_{1}\\}$ is a quasi-function from
$(D_{2})^{2}$ to $E_{2}$.

  2. (2)

If $X_{2}$ is a quasi-function from $(D_{2})^{2}$ to $E_{2}$, then there is a
quasi-function $X_{1}$ from $(D_{1})^{2}$ to $E_{1}$ such that
$X_{2}=\\{(f(x),f(y),g(z)):(x,y,z)\in X_{1}\\}$.

###### Proof.

  1. (1)

Easy by the definitions.

  2. (2)

The preimage of $X_{2}$ in $(D_{1})^{2}\times E_{1}$ (under $f$, $f$, and $g$)
has dimension 2. One can then choose $X_{1}$ to be any stationary component of
this preimage, so long as the projection of $X_{1}$ to $X_{2}$ (using $f$,
$f$, and $g$) is surjective. This is possible because each of $f$ and $g$ is
surjective.

∎

######  Lemma 5.11.

Let $D_{1}$, $D_{2}$, $E_{1}$, and $E_{2}$ be strongly minimal. Let
$f:D_{1}\rightarrow D_{2}$, $g:E_{1}\rightarrow E_{2}$ be definable finite-to-
one surjective functions. Let $X_{1}$ be a quasi-function from $(D_{1})^{2}$
to $E_{1}$, and let $X_{2}=\\{(f(x),f(y),g(z)):(x,y,z)\in X_{1}\\}$.

  1. (1)

$X_{2}$ has symmetric small expansion if and only if $X_{1}$ has symmetric
small expansion respecting the equivalence relation $f(x)=f(y)$ on $D_{1}$.

  2. (2)

In particular, if $X_{2}$ has symmetric small expansion, then so does $X_{1}$.

  3. (3)

If $D_{1}=D_{2}$ and $f$ is the identity map, then any witness family for
symmetric small expansion of $X_{1}$ is also a witness family for $X_{2}$, and
vice versa. In particular, for any equivalence relation $\sim$ on $D_{1}$,
$X_{1}$ has symmetric small expansion respecting $\sim$ if and only if $X_{2}$
does.

###### Proof.

  1. (1)

Assume $X_{1}$ has symmetric small expansion respecting the relation
$f(x)=f(y)$. Let $\\{(N_{k},A_{k})\\}$ be a witness. Then
$|f(A_{k})|=O(N_{k})$. Moreover, it is easy to see (using that the $A_{k}$ are
closed under the relation $f(x)=f(y)$) that
$X_{2}(f(A_{k}),f(A_{k}))=g(X_{1}(A_{k},A_{k}))$, and thus that
$|X_{2}(f(A_{k}),f(A_{k}))|\leq l\cdot|X_{1}(A_{k},A_{k})|$ for some fixed
integer $l$. This shows that $\\{f(A_{k})\\}$ is a witness family for $X_{2}$.

Now assume $X_{2}$ has symmetric small expansion, and let $\\{A_{k}\\}$ be a
witness family. Then by similar reasoning, $\\{f^{-1}(A_{k})\\}$ is a witness
family for $X_{1}$.

  2. (2)

By (1).

  3. (3)

By the proof of (1).

∎

######  Corollary 5.12.

Let $D_{1}$ and $D_{2}$ be strongly minimal, and let $f:D_{1}\rightarrow
D_{2}$ be a definable finite-to-one surjective function. If $D_{2}$ has
universal symmetric small expansion, then so does $D_{1}$.

###### Proof.

By Lemmas 5.10(1) (setting $E_{1}=D_{1}$, $E_{2}=D_{2}$, $g=f$) and 5.11(2). ∎

We turn now toward verifying symmetric small expansion in certain ‘group-like’
environments. Our main tool is the following lemma.

######  Notation 5.13.

If $R$ is a ring, $S$ and $T$ are subsets of $R$, and $a,b\in R$, then by
$a\cdot S+b\cdot T$ we mean the set of all elements of $R$ of the form $a\cdot
s+b\cdot t$ for $s\in S$ and $t\in T$.

######  Lemma 5.14.

Let $R$ be an integral domain of characteristic zero, and let
$\tau_{1},...,\tau_{n}\in R$. Then one can find finite sets $S_{k}\subseteq R$
with $\lim_{k\rightarrow\infty}|S_{k}|=\infty$, such that

$\lim_{k\rightarrow\infty}\frac{\log{|Z_{k}|}}{\log{|S_{k}|}}\leq 1,$

where

$Z_{k}=\bigcup_{i,j\leq n}\tau_{i}\cdot S_{k}+\tau_{j}\cdot S_{k}.$

###### Proof.

The proof is essentially identical to the argument in [JRT22] (Lemma 3.1 and
Proposition 3.2); we give a sketch. First, note that sets $S_{k}$ with the
desired properties are closed under scaling: that is, if $\\{S_{k}\\}$ is any
such sequence, then so is $\\{c_{k}S_{k}\\}$ for any nonzero $c_{k}\in R$. It
follows that we may assume $R$ is a field.

Now let us rewrite our given elements as $\sigma_{1},...,\sigma_{l}$,
$\tau_{1},...,\tau_{m}$, where the $\sigma_{i}$ are algebraically independent
transcendentals, and each $\tau_{j}$ has degree $d_{j}$ over
$\mathbb{Q}(\sigma_{1},...,\sigma_{l},\tau_{1},...,\tau_{j-1})$. For each
$d>0$ Let $Q_{d}$ be the set of monomials in
$(x_{1},...,x_{l},y_{1},...,y_{m})$ of degree $<d$ in each variable, and let
$P_{d}$ be the set of $q\in Q_{d}$ of degree $<d_{j}$ in each $y_{j}$. Then
for each $d,r>0$ let $B_{r}^{d}$ be the set of all linear combinations
$\sum_{q\in Q_{d}}a_{q}\cdot q(\bar{\sigma},\bar{\tau})$, where each
$a_{q}\in\\{0,...,r-1\\}$.

We claim that, if we fix $d>\max\\{d_{1},...,d_{m}\\}$ and consider
$|B_{r}^{d}|$ as a function of $r$, we obtain $|B_{r}^{d}|=O(r^{D})$, where
$D=d^{l}d_{1}...d_{m}$. To see this, note first that the elements
$q(\bar{\sigma}\bar{\tau})$ for $q\in P_{d}$ are linearly independent over
$\mathbb{Q}$, by definition of the $d_{j}$. Thus $|B_{r}^{d}|\geq r^{D}$. On
the other hand, using the minimal polynomials of each $\tau_{j}$, one sees
that for fixed $d$, all $q(\bar{\sigma}\bar{\tau})$ for $q\in Q_{d}$ can be
expressed in the form

$\frac{\sum_{q\in P_{d}}a_{q}\cdot q(\bar{\sigma},\bar{\tau})}{b},$

where $b\in R$ is fixed, and the $a_{q}$ are integers. It follows easily that
$|B_{r}^{d}|$ is bounded by a constant multiple of $r^{D}$, as desired.

Next note that, for
$\alpha,\beta\in\\{\sigma_{1},...,\sigma_{l},\tau_{1},...,\tau_{m}\\}$, we
have $\alpha\cdot B_{r}^{d}+\beta\cdot B_{r}^{d}\subseteq B_{2r}^{d+1}$.
Define $Z_{r}^{d}$ as the union of all such sets $\alpha\cdot
B_{r}^{d}+\beta\cdot B_{r}^{d}$ for $\alpha,\beta$ among our $n$ given
elements. Then we obtain

$\lim_{r\rightarrow\infty}\frac{\log{|Z_{r}^{d}|}}{\log{|B_{r}^{d}|}}\leq\lim_{r\rightarrow\infty}\frac{\log{|B_{2r}^{d+1}|}}{\log{|B_{r}^{d}|}}=\lim_{r\rightarrow\infty}\frac{\log{r^{(d+1)^{l}d_{1}...d_{m}}}}{\log{r^{d^{l}d_{1}...d_{m}}}}=\left(\frac{d+1}{d}\right)^{l}.$

Thus, letting also $d\rightarrow\infty$, we can make our desired limit go to
1, and therefore extract a sequence $S_{k}$ as in the lemma. ∎

We now turn toward small expansion in strongly minimal groups. Recall that a
definable endomorphism of a strongly minimal group is either trivial or
surjective; in particular, the ring of definable endomorphisms of a strongly
minimal group has no zero divisors.

Now the first case of our desired result is the following proposition.

######  Definition 5.15.

Let $G$ be a definable set, and let $D\subseteq G^{n}$ be strongly minimal. We
say that $D$ is irredundant in $G$ if each of the $n$ projections
$D\rightarrow G$ is finite-to-one.

######  Proposition 5.16.

Let $G$ be a divisible strongly minimal group, whose ring of definable
endomorphisms is commutative. Let $D,E\subseteq G^{n}$ be strongly minimal and
irredundant in $G$, and let $X$ be a quasi-function from $D^{2}$ to $E$.
Assume that each of $D$, $E$, and $X$ is a coset of definable subgroup of the
appropriate power of $G$, and let $\sim$ be the equivalence relation on $D$
induced by the permutations of the coordinates $\\{1,...,n\\}$. Then $X$ has
symmetric small expansion respecting $\sim$.

###### Proof.

For each $i,j\leq n$, let $\pi_{ij1}(X)$ be the image of $X$ in $G^{3}$, given
by the applying the $ith$, $jth$, and 1st coordinate projections to the three
factors $D$, $D$, and $E$, respectively. Since $D$ and $E$ are irredundant in
$G$, it follows that each $\pi_{ij1}(X)$ is a quasi-function from $G^{2}$ to
$G$. Moreover, since $X$ is a coset, so is $\pi_{ij1}(X)$.

######  Claim 5.17.

We may assume that each $\pi_{ij1}$ is the graph of a function
$G^{2}\rightarrow G$.

###### Proof.

Consider the leftmost projection $\pi_{ij1}(X)\rightarrow G^{2}$. Since
$\pi_{ij1}$ is both a quasi-function and a coset, one concludes easily that
the fibers of $\pi_{ij1}(X)\rightarrow G^{2}$ (viewed as subsets of the third
$G$ factor) are cosets of a fixed finite subgroup $H_{ij}\leq G$. Let $m$ be a
common multiple of all of the $|H_{ij}|$. Then the map $x\mapsto m\cdot x$ is
constant on each fiber. So if we replace $E$ with its image under scaling the
first coordinate by $m$, the $\pi_{ij1}$ will be replaced by the graph of a
function.

Formally, let $E^{\prime}$ be the image of $E$ under scaling the first
coordinate by $m$, and let $g:E\rightarrow E^{\prime}$ be the map that scales
the first coordinate by $m$. Then $E^{\prime}$ is also a coset, and is also
irredundant in $G$. Let $X^{\prime}$ be the image of $X$ in $D^{2}\times
E^{\prime}$. By Lemma 5.11(3), $X$ has small expansion respecting $\sim$ if
and only if $X^{\prime}$ does. So we may replace $E$ with $E^{\prime}$ and $X$
with $X^{\prime}$. Finally, it follows from above that each
$\pi_{ij1}(X^{\prime})$ is indeed the graph of a function $G^{2}\rightarrow
G$. ∎

Now assume each $\pi_{ij1}(X)$ is the graph of a function $G^{2}\rightarrow
G$. Since $\pi_{ij1}(X)$ is a coset, this function has the form
$(x,y)\mapsto\sigma_{ij}(x)+\tau_{ij}(y)+a_{ij}$ for some $a_{ij}\in G$ and
some definable endomorphisms $\sigma_{ij},\tau_{ij}$ of $G$.

Now by assumption, the ring of definable endomorphisms of $G$ is commutative,
thus an integral domain. Let $\\{S_{k}\\}$ and $\\{Z_{k}\\}$ be as in Lemma
5.14 for all of the $\sigma_{ij}$ and $\tau_{ij}$. So the $S_{k}$ and $W_{k}$
are sets of definable endomorphisms of $G$. Fix $\hat{g}\in G$ generic over
all relevant data, and for each $k$ let $T_{k}$ and $W_{k}$ be the set of
images $\hat{g}$ under all endomorphisms in $S_{k}$ and $Z_{k}$, respectively.
Note that since $\hat{g}$ is generic we have $|S_{k}|=|T_{k}|$ and
$|T_{k}|=|W_{k}|$. Now for each $k$ let $A_{k}$ be the set of
$(x_{1},...,x_{n})\in D$ such that $x_{i}\in T_{k}$ for some $i$, and note by
definition that each $A_{k}$ is closed under $\sim$.

We will prove the proposition by showing that $\\{A_{k}\\}$ is a witness
family for symmetric small expansion on $X$. It is enough to show the
following two claims:

######  Claim 5.18.

$|A_{k}|=O(|S_{k}|)$.

###### Proof.

Since each $|S_{k}|=|T_{k}|$, it is equivalent to show that
$|A_{k}|=O(|T_{k}|)$. For each $i$, the set of $(x_{1},...,x_{n})\in D$ with
$x_{i}\in T_{k}$ is the preimage of $T_{k}$ in $D$ under the $ith$ projection
$\pi_{i}$. Since $D$ is irredundant in $G$, $\pi_{i}:D\rightarrow G$ is
finite-to-one. Since $D$ is a coset, $\pi_{i}$ is moreover surjective and
$l_{i}$-to-1 for some positive integer $l_{i}$. Thus $\pi_{i}^{-1}(T_{k})\cap
D$ has size $l_{i}\cdot|T_{k}|$. Since $A_{k}$ is the union of the
$\pi_{i}^{-1}(T_{k})\cap D$, the claim follows. ∎

######  Claim 5.19.

$|X(A_{k},A_{k})|=O(|Z_{k}|)$.

###### Proof.

For each $k$ let $V_{k}$ be the union of all translates $W_{k}+a_{ij}$ for
$i,j\leq n$, and let $U_{k}$ be the preimage of $V_{k}$ in $E$ under
$\pi_{1}$. Clearly $|U_{k}|=O(|Z_{k}|)$. We will show that
$X(A_{k},A_{k})\subseteq U_{k}$, which proves the claim.

To see this, let $\bar{z}=(z_{1},...,z_{n})\in X(A_{k},A_{k})$. Then there are
$\bar{x}=(x_{1},...,x_{n}),\bar{y}=(y_{1},...,y_{n})\in A_{k}$ such that
$(\bar{x},\bar{y},\bar{z})\in X$. By definition, there are $i,j\leq n$ with
$x_{i},y_{j}\in T_{k}$. Then
$z_{1}=\sigma_{ij}(x_{i})+\tau_{ij}(y_{j})+a_{ij}\in W_{k}+a_{ij}\subseteq
V_{k}$, and thus $\bar{z}\in U_{k}$. ∎

Finally, by Lemma 5.14 we have $\lim_{k\rightarrow\infty}|S_{k}|=\infty$ and
$\lim_{k\rightarrow\infty}\frac{\log{|Z_{k}|}}{\log{|S_{k}|}}\leq 1$. Combined
with the above two claims, the same assertions hold of $A_{k}$ and
$X(A_{k},A_{k})$, which proves the proposition. ∎

We now use Proposition 5.16 to prove Theorem 5.22, the main result of this
section.

######  Notation 5.20.

If $X$ is a set, then by $X^{(n)}$ we mean the $n$th symmetric power of $X$,
i.e. the quotient of $X^{n}$ by coordinate permutations.

######  Definition 5.21.

If $G$ is strongly minimal, and $D\subseteq G^{(n)}$ is strongly minimal for
some $n$, then $D$ is irredundant in $G$ if each $g\in G$ belongs to only
finitely many of the sets in $D$.

######  Theorem 5.22.

Let $G$ be a locally modular, divisible, strongly minimal group, whose ring of
definable endomorphisms is commutative. Then every strongly minimal set in
finite correspondence with $G$ has universal symmetric small expansion.

###### Proof.

Let $D$ be strongly minimal and in finite correspondence with $G$. We may view
the finite correspondence as a definable, finite-to-one function $D\rightarrow
G^{(n)}$ for some $n$. By choosing the smallest possible $n$, one easily
ensures that the image of $D$ in $G^{(n)}$ is irredundant in $G$. So by
Corollary 5.12, we may without loss of generality replace $D$ with this image.
Thus, from now on, we assume $D$ is itself an irredundant subset of $G^{(n)}$.
Let us also fix a strongly minimal set $D^{\prime}\subseteq G^{n}$ projecting
to $D$.

Now let $X$ be a quasi-function from $D^{2}$ to $D$. By Lemma 5.10, there is a
quasi-function $X^{\prime}$ from $(D^{\prime})^{2}$ to $D^{\prime}$ that
projects exactly to $X$.

By the classification of definable sets in 1-based groups ([HP87]),
equivalently Fact 2.3), there are cosets $D_{1}$ and $X_{1}$ which are almost
equal to $D^{\prime}$ and $X^{\prime}$, respectively. It follows easily that
$X_{1}\subseteq(D_{1})^{3}$, and thus $X_{1}$ is a quasi-function from
$(D_{1})^{2}$ to $D_{1}$ (by repeated applications of Lemma 5.8(2) and (3)).
Let $D_{2}$ and $X_{2}$ be the images of $D_{1}$ and $X_{1}$ under
$G^{n}\rightarrow G^{(n)}$. So $D_{2}$ is almost equal to $D$, $X_{2}$ is
almost equal to $X$, and $X_{2}$ is a quasi-function from $(D_{2})^{2}$ to
$D_{2}$.

Since $D$ is irredundant in $G$, so is $D^{\prime}$, and thus so is $D_{1}$.
Then by Proposition 5.16, $X_{1}$ has symmetric small expansion respecting
coordinate permutations. Then by Lemma 5.11, $X_{2}$ has symmetric small
expansion. So by Lemma 5.8, so does $X$. ∎

##  6\. Characterizing Locally Modularity

Throughout Section 6, we let $\mathscr{M}$ be a non-trivial locally modular
reduct of $(\mathbb{C},+,\times)$. We will show that every
$\mathscr{M}$-definable polynomial
$P(x,y)\in\mathbb{C}[x,y]-\mathbb{C}[x]-\mathbb{C}[y]$ is either linear or a
twisted monomial. We conclude that if $\mathscr{M}$ is a polynomial reduct, it
must be interdefinable with a vector space or a twisted multiplication.

Lemma 6.1 gives the necessary input from Section 5:

######  Lemma 6.1.

Let $P(x,y)\in\mathbb{C}[x,y]-\mathbb{C}[x]-\mathbb{C}[y]$ be
$\mathscr{M}$-definable. Then $P$ has symmetric small expansion.

###### Proof.

By Fact 2.3, there is a locally modular, strongly minimal, abelian group $G$
in definable finite correspondence with $M$. The lemma then follows from
Theorem 5.22, provided one can show $G$ is divisible with commutative
definable endomorphism ring.

Now by [vdD90] (Theorem 3), we may assume $G$ is an algebraic group, which (by
the finite correspondence with $\mathbb{C}$) has dimension 1. So the connected
component of the identity in $G$ is isomorphic (as an algebraic group) to
either $(\mathbb{C},+)$, $(\mathbb{C}^{\times},\times)$, or an elliptic curve.
In any case, $G$ has unbounded exponent. By strong minimality (in the
structure $\mathscr{M}$), it follows easily that $G$ is divisible.

Now as a divisible algebraic group, $G$ is irreducible. Thus $G$ itself is a
copy of the additive group, the multiplicative group, or an elliptic curve. In
any case, the definable endomorphism ring is then commutative. Indeed, every
definable endomorphism is an algebraic endomorphism, and the algebraic
endomorphism rings of these groups are commutative (they are all isomorphic to
either $\mathbb{C}$, $\mathbb{Z}$, or $\mathbb{Z}^{2}$). ∎

We now apply tools from additive combinatorics to get more information about
$\mathscr{M}$-definable polynomials. The following ad hoc terminology will be
useful.

######  Definition 6.2.

Let $P(x,y)\in\mathbb{C}[x,y]-\mathbb{C}[x]-\mathbb{C}[y]$.

  1. (1)

$P$ is weakly additive if there are non-constant unary polynomials $f,u,v$
such that $P(x,y)=f(u(x)+v(y))$.

  2. (2)

$P$ is weakly multiplicative if there are non-constant unary polynomials
$f,u,v$ such that $P(x,y)=f(u(x)v(y))$.

  3. (3)

$P$ is strongly additive if there are non-constant unary polynomials $f,u$,
and constants $c_{1},c_{2}$, such that $P(x,y)=f(c_{1}u(x)+c_{2}u(y))$.

  4. (4)

$P$ is strongly multiplicative if there are non-constant unary polynomials
$f,u$, and integers $m,n\geq 0$, such that $P(x,y)=f(u^{m}(x)u^{n}(y))$.

By Lemma 6.1 and the main result of [JRT22], we have:

######  Fact 6.3.

Let $P(x,y)\in\mathbb{C}[x,y]-\mathbb{C}[x]-\mathbb{C}[y]$. Then $P$ is either
strongly additive or strongly multiplicative.

Given an $\mathscr{M}$-definable polynomial $P(x,y)$, our remaining task is to
determine the possibilities for the polynomials $f$ and $u$ in the definition
of strongly additive (resp. multiplicative). We do this by defining new
polynomials in terms of $P$, and using that they also satisfy Fact 6.3. Lemma
6.4 and Corollary 6.5 will tell us that these new polynomials remain in the
same case as $P$ (additive or multiplicative).

######  Lemma 6.4.

Suppose $P(x,y)\in\mathbb{C}[x,y]-\mathbb{C}[x]-\mathbb{C}[y]$.

  1. (1)

If $P(x,y)=f(u(x)+v(y))$ for some unary $f,u,v$ then $P$ is non-constant on
every horizontal and vertical line.

  2. (2)

If $P(x,y)=f(u(x)v(y))$ for some unary $f,u,v$ then $P$ is constant on the
line $x=a$ if and only if $u(a)=0$, and is constant on the line $y=b$ if and
only if $v(b)=0$. In particular, $P$ is constant on at least one horizontal
line and at least one vertical line.

  3. (3)

$P$ is not both weakly additive and weakly multiplicative.

###### Proof.

(1) and (2) are obvious, and (3) is immediate from (1) and (2). ∎

######  Corollary 6.5.

Let $P(x,y)\in\mathbb{C}[x,y]-\mathbb{C}[x]-\mathbb{C}[y]$ be
$\mathscr{M}$-definable. Then:

  1. (1)

$P$ is strongly additive if and only if it is weakly additive, and is strongly
multiplicative if and only if it is weakly multiplicative.

  2. (2)

If $P$ is weakly additive then $\deg_{x}(P)=\deg_{y}(P)$.

  3. (3)

If $P$ has the form $g(v(x)w(y))$ for some unary $g,v,w$, then $v$ and $w$
have the same roots.

###### Proof.

(1) follows from Fact 6.3 and Lemma 6.4. (2) follows from (1) and the
definition of strongly additive. For (3), it follows from Lemma 6.4(2) that
the roots of $v$ and $w$ are intrinsic to $P$, and do not depend on the choice
of $g$, $v$, and $w$. But by (1), $P$ is strongly multiplicative. So we may
assume that $v=u^{m}$ and $w=u^{n}$ for some $u$, making the conclusion
obvious. ∎

We can now classify the additive case:

######  Proposition 6.6.

Suppose $P(x,y)\in\mathbb{C}[x,y]-\mathbb{C}[x]-\mathbb{C}[y]$ is
$\mathscr{M}$-definable and strongly additive. Then $P$ is linear.

###### Proof.

Write $P(x,y)=f(c_{1}u(x)+c_{2}u(y))$ as in Definition 6.2.

######  Claim 6.7.

Every $\mathscr{M}$-definable unary polynomial is linear.

###### Proof.

Let $g$ be $\mathscr{M}$-definable and unary. We may assume $g$ is not
constant. Then

$Q(x,y)=P(x,g(y))=f(c_{1}u(x)+c_{2}u(g(y)))$

is $\mathscr{M}$-definable and weakly additive. By Corollary 6.5,
$\deg_{x}(Q)=\deg_{y}(Q)$. So $\deg(u)=\deg(u\circ g)$, thus $\deg(g)=1$, thus
$g$ is linear. ∎

Now to prove Proposition 6.6, note that $g(x)=f(c_{1}u(x))$ is
$\mathscr{M}$-definable, by choosing $y\in\mathbb{C}$ with $u(y)=0$. By the
claim, $f(c_{1}u(x))$ is linear. This implies that $f$ and $u$ are linear, and
thus so is $P$. ∎

The multiplicative case is similar but trickier. We will need the following:

######  Lemma 6.8.

Suppose $f$ and $g$ are non-constant unary polynomials over $\mathbb{C}$, and
$f$ and $f\circ g$ have the same roots.

  1. (1)

If $g$ is not linear, then $f$ has only one root.

  2. (2)

If $r$ is the unique root of $f$, then $g$ is a monomial twisted by $r$.

###### Proof.

Let $R$ be the (finite) set of roots of $f$. A restatement of the assumption
on $f$ and $g$ is that $g(R),g^{-1}(R)\subseteq R$. Using that $g$ is
surjective and $R$ is finite, one concludes easily that $g(R)=g^{-1}(R)=R$. So
$g$ restricts to a permutation of $R$.

Now for each $r\in R$, we have $s=g(r)\in R$. Since $g^{-1}(R)=R$, $r$ is the
unique solution of $g(x)=s$. Let $d=\deg(g)$. Then $g-s$ has a root of
multiplicity $d$ at $x=r$, so $g(x)=a(x-r)^{d}+s$ for some constant $a$. We
now show (1) and (2):

  1. (1)

Let $r\in R$ be arbitrary, and write $g(x)=a(x-r)^{d}+s$ as above, where we
have $d\geq 2$ by assumption. Then $g^{\prime}(x)=ad(x-r)^{d-1}$ is non-
constant, so has $r$ as its unique root. Thus $r$ is determined by $g$, which
shows (1).

  2. (2)

Let $r$ be the unique root, and write $g(x)=a(x-r)^{d}+s$ as above. The
uniqueness of $r$ implies $s=r$, so $g(x)=a(x-r)^{d}+r$, as desired.

∎

We now prove the multiplicative case, using the same idea as in the additive
case:

######  Proposition 6.9.

Suppose $P(x,y)\in\mathbb{C}[x,y]-\mathbb{C}[x]-\mathbb{C}[y]$ is
$\mathscr{M}$-definable and strongly multiplicative. Then $P$ is a twisted
monomial.

###### Proof.

Write $P(x,y)=f(u^{m}(x)u^{n}(y))$ as in Definition 6.2.

######  Claim 6.10.

$u$ has only one root, say $r$, and every $\mathscr{M}$-definable unary
polynomial is a monomial twisted by $r$.

###### Proof.

If $g(x)$ is any $\mathscr{M}$-definable non-constant unary polynomial, then

$Q(x,y)=P(x,g(y))=f(u^{m}(x)u^{n}(g(y)))$

is $\mathscr{M}$-definable and weakly multiplicative. So by Lemma 6.5, $u^{m}$
and $u^{n}\circ g$ have the same roots, thus $u$ and $u\circ g$ have the same
roots.

Now since we are in the multiplicative case, $f(x,x)$ is not linear. So by
Lemma 6.8(1) with $g(x)=f(x,x)$, $u$ has a unique root. Now apply Lemma
6.8(2). ∎

Without loss of generality, we will assume the unique root of $u$ is $0$. Then
$u$ is a monomial, and we want to prove that $P$ is a monomial.

To do this, note that $g(x)=f(u^{m}(x))$ is $\mathscr{M}$-definable, by
choosing $y\in\mathbb{C}$ with $u(y)=1$. By the claim, $f(u^{m}(x))$ is a
monomial. But now $u$ and $f\circ u^{m}$ are both monomials, which clearly
implies that $f$ is a monomial. Then $P(x,y)=f(u^{m}(x)u^{n}(x))$ where $f$
and $u$ are monomials, which shows that $P$ is a monomial. ∎

Finally, we complete our analysis of the locally modular case:

######  Theorem 6.11.

Let $\mathcal{P}$ be a collection of complex polynomials, and let
$\mathscr{M}=(\mathbb{C};\mathcal{P})$. Assume that $\mathscr{M}$ is non-
trivial and locally modular. Then either every $P\in\mathcal{P}$ is linear, or
every $P\in\mathcal{P}$ is a monomial twisted by the same $r$. In particular,
$\mathscr{M}$ is interdefinable with a vector space or a twisted
multiplication.

###### Proof.

By Lemma 3.2, there is $P\in\mathcal{P}$ involving at least two variables.
Specializing all other variables generically, we may assume $P$ is binary. By
Fact 6.3, Proposition 6.6, and Proposition 6.9, $P$ is either linear or a
twisted monomial. Then by Proposition 4.3, one of the group operations $+$ or
$\times_{r}$ (for some $r$) is $\mathscr{M}$-definable. Let $\otimes$ be this
group operation. Then by the classification of definable sets in 1-based
groups ([HP87]), every $\mathscr{M}$-definable polynomial is of the form
$\bar{x}\mapsto\bar{a}\otimes f(\bar{x})$, where $f$ is a
$\otimes$-homomorphism. If $\otimes=+$, all such polynomials are linear. If
$\otimes=\times_{r}$, all such polynomials are monomials twisted by $r$. Now
apply Proposition 4.3. ∎

##  7\. The Non-Locally Modular Case

Suppose $\mathscr{M}=(M;...)$ is a non-locally modular strongly minimal
structure interpreted in an algebraically closed field. In this section, we
give a new condition implying that $\mathscr{M}$ defines all constructible
subsets of all powers of $M$. It is straightforward to see that polynomial
reducts satisfy this condition, so that non-locally modular polynomial reducts
are interdefinable with the full field structure. The proof of this
implication will be given in Section 8. We assume knowledge of standard
definitions and results from algebraic geometry; see [Sha13], for example.

######  Definition 7.1.

Let $K$ be an algebraically closed field, let $V$ and $W$ be smooth, pure-
dimensional varieties over $K$, and let $f:V\rightarrow W$ be a definable
partial function.

  * •

We say that $f$ is generically defined if its domain is dense in $V$.

  * •

We say that $f$ is almost regular at $x\in V$ if there is a neighborhood
$V^{\prime}\subseteq V$ of $x$ such that $f$ is defined on $V^{\prime}$, and
the graph of $f$ restricts to a relatively closed, pure-dimensional subset of
$V^{\prime}\times W$.

######  Remark 7.2.

Note that a generically defined map is almost regular at every generic point
(where genericity is in the sense of the structure $K$). Moreover, it follows
easily from the smoothness of the variety involved that almost regularity is
closed under composition: if $f:V\rightarrow W$ is almost regular at $x\in V$,
and $g:W\rightarrow Z$ is almost regular at $f(x)\in W$, then $g\circ f$ if
almost regular at $x$. We need almost regularity instead of regularity to
allow for the inverse Frobenius map in positive characteristic.

######  Theorem 7.3.

Let $(K,+,\times)$ be an algebraically closed field. Let $M$ be a one-
dimensional constructible set over $K$, and let $\mathscr{M}=(M,...)$ be a
non-locally modular strongly minimal reduct of the full $K$-induced structure
on $M$. Assume there are $\mathscr{M}$-definable generically defined partial
functions $P:M^{2}\rightarrow M$ and $U:M\rightarrow M$ such that:

  1. (1)

$P$ generically depends on both variables (i.e. $P$ does not, on any generic
subset of $M^{2}$, factor through either projection to $M$).

  2. (2)

$U$ is finite-to-one and not generically injective.

Then $\mathscr{M}$ defines every constructible subset of every power of $M$.

######  Remark 7.4.

From the proof, one can weaken condition (2) to a more technical restriction
on $U$: namely that whenever $a,b\in M$ are generic and $n$ is a positive
integer, the $n$-fold iterate of $x\mapsto P(P(U(x),a),b)$ is not the identity
infinitely often. We did not do this because condition (2) above is more
natural.

######  Remark 7.5.

Condition (2) is necessary. For example, in positive characteristic, Marker
and Pillay [MP90] give an example of a non-locally modular expansion of the
additive group that does not define multiplication (so here one takes
$P(x,y)=x+y$). We expect that (2) can be dropped in characteristic zero.

###### Proof.

We may assume $K$ and $\mathscr{M}$ are saturated, and $P$ and $U$ are
$\emptyset$-definable in $\mathscr{M}$. By editing finitely many points, we
may also assume that $M$ is the set of $K$-points of a smooth curve over $K$.
By Zilber’s Restricted Trichotomy (see [HS21],[Cas22]), there is an
$\mathscr{M}$-definable algebraically closed field $(F,\oplus,\otimes)$ in
$\mathscr{M}$-definable finite correspondence with $M$. By well-known
techniques, it will suffice to find an $\mathscr{M}$-definable injection
$M\rightarrow F^{n}$ for some $n$ (this is proven explicitly in [CH22],
Theorem 4.12).

By elimination of imaginaries in $F$, we can view the given finite
correspondence as a finite-to-one $\mathscr{M}$-definable function
$f:M\rightarrow F^{n}$ for some $n$. Among all such functions (over all $n$),
fix $f$ and $n$ so that the generic fiber size of $f$ is minimized. Call this
fiber size $k$. So all but finitely many fibers of $f$ have size $k$.

We will prove Theorem 7.3 by showing that $k=1$. To do this, we use the
provided functions $P$ and $U$ to try to build another function with smaller
fibers. Assuming this doesn’t work, we will collect increasingly restrictive
information about the definable sets in $\mathscr{M}$.

######  Definition 7.6.

For $x,y\in\mathbb{C}$, we define $x\sim y$ if and only if $f(x)=f(y)$.

So $\sim$ is an $\mathscr{M}$-definable equivalence relation on $M$, and
cofinitely many of the $\sim$-classes have size $k$.

Our first goal is Lemma 7.9, which says that definable finite correspondences
on $M$ must respect $\sim$ in a strong way. We prove the lemma by successively
generalizing restricted versions.

######  Lemma 7.7.

Let $g:M\rightarrow F^{m}$ be any finite-to-one $\mathscr{M}$-definable
function. Then $g$ generically respects $\sim$: for cofinitely many $x\in M$,
all $x^{\prime}\sim x$ satisfy $g(x)=g(x^{\prime})$.

###### Proof.

Consider the concatenation map $(f,g):M\rightarrow F^{n+m}$ given by
$x\mapsto(f(x),g(x))$. The fibers of $(f,g)$ are intersections of fibers of
$f$ and $g$. Fix $x\in M$ generic over the parameters defining these two maps.
Then the fiber containing $x$ in $(f,g)$ is a subset of $f^{-1}(f(x))$. By the
minimality of $k$, these two fibers are equal. Thus, every $x^{\prime}\sim x$
satisfies $g(x)=g(x^{\prime})$. Since $x$ is generic, this property then holds
on a cofinite subset of $M$. ∎

Lemma 7.8 generalizes Lemma 7.7 from functions to finite correspondences:

######  Lemma 7.8.

Let $C\subseteq M^{2}$ be any $\mathscr{M}$-definable finite correspondence.
Then for cofinitely many $(x,y)\in C$, for every $x^{\prime}\sim x$, there is
some $y^{\prime}\sim y$ with $(x^{\prime},y^{\prime})\in C$.

###### Proof.

We view $C$ as a multivalued function from $M$ to $M$, say $x\mapsto
C(x)\subseteq M$. The composition $f\circ C$ is a multivalued function from
$M$ to $F$. Let $X=f\circ C(M)$. By elimination of imaginaries in $F$, there
is an $\mathscr{M}$-definable injection $g:X\rightarrow F^{m}$ for some $m$.

We now apply Lemma 7.7 to the finite-to-one $\mathscr{M}$-definable function
$h=g\circ f\circ C:M\rightarrow F^{m}$. Namely, let $(x,y)\in C$ be generic
over all relevant parameters, and let $x^{\prime}\sim x$. Note that $x$ is
generic in $M$ over the same parameters, since $C$ is a finite correspondence.
Now by Lemma 7.7, we have $g\circ f\circ C(x)=g\circ f\circ C(x^{\prime})$.
Since $g$ is injective, $f\circ C(x)=f\circ C(x^{\prime})$. But $y\in C(x)$,
so $f(y)\in f\circ C(x)$, thus $f(y)\in f\circ C(x^{\prime})$. So there is
some $y^{\prime}\in C(x^{\prime})$ such that $f(y^{\prime})=f(y)$, i.e.
$y^{\prime}\sim y$.

So the desired property holds for all generic $(x,y)\in C$ over the relevant
parameters, and thus holds on a cofinite subset of $C$. ∎

We now arrive at Lemma 7.9. Essentially, it says that the finite exceptional
set of Lemma 7.8 can be traced to certain geometric abnormalities in the set
$C$.

######  Lemma 7.9.

There is a fixed finite set $T\subseteq M$ with the following property:
suppose $C\subseteq M^{2}$ is $\mathscr{M}$-definable and one-dimensional. Let
$X$ and $Y$ be $\sim$-classes, and let $(x_{0},y_{0})\in C\cap(X\times Y)$
with $x_{0},y_{0}\notin T$. Moreover assume:

  1. (1)

$C$ is relatively closed and pure-dimensional in a neighborhood of $X\times
Y$.

  2. (2)

The sets $\\{y:(x_{0},y)\in C\\}$ and $\\{x:(x,y_{0})\in C\\}$ are finite.

Then the projections $C\cap(X\times Y)\rightarrow X$ and $C\cap(X\times
Y)\rightarrow Y$ are surjective – that is, for any $x\sim x_{0}$, there is
$y\sim y_{0}$ with $(x,y)\in C$, and vice versa.

###### Proof.

By (1) and (2) above, we may assume $C$ is pure-dimensional and relatively
closed in $M^{2}$, and that both projections $C\rightarrow M^{2}$ are finite-
to-one. Indeed, one can $\mathscr{M}$-definably (over parameters) delete all
isolated points of $C$, as well as all infinite fibers in each projection
$C\rightarrow M^{2}$, then take the closure of the remaining set. It follows
from (1), (2), and (3) that the resulting set is still one-dimensional,
$(x_{0},y_{0})$ was not deleted, and some neighborhood of $X\times Y$ had no
new points added.

So assume $C$ is a closed, pure-dimensional finite correspondence on $M$. Our
strategy is to ‘projectivize’ the entire problem. We show (Claim 7.13) that
the projective version ‘completes’ Lemma 7.8, making the desired property hold
at all points of $C$ instead of most points. One then needs to check where
exactly any ‘projective’ points were used. We will see that, because $C$ is
closed and pure-dimensional in $M^{2}$, projective points only arise because
of $\sim$, and have nothing to do with $C$.

Let us continue with the proof. Let $\hat{M}$ be a smooth projective
completion of $M$. Let $E\subseteq M^{2}$ be the graph of $\sim$. So
$\dim(E)=1$. Let $\hat{C},\hat{E}\subseteq\hat{M}^{2}$ be pure-dimensional
projective curves differing from $C$ and $E$ in finitely many points,
respectively. So $\hat{C}$ and $\hat{E}$ are finite correspondences on
$\hat{M}$. The assumption that $C$ is closed and pure-dimensional in $M^{2}$,
equivalently stated, gives that $C=\hat{C}\cap M^{2}$ and $\hat{C}$ is pure-
dimensional.

######  Definition 7.10.

Let $T$ be the set of all elements of $M$ which appear as coordinates of
elements of the symmetric difference $E\Delta\hat{E}$.

Clearly $T$ is finite and independent of $C$. We will show that $T$ satisfies
the lemma.

######  Definition 7.11.

Define the sets $V\subseteq\hat{M}^{4}$ and $W\subseteq\hat{M}^{3}$ as
follows:

  * •

$(x,y,x^{\prime},y^{\prime})\in V$ if
$(x,y),(x^{\prime},y^{\prime})\in\hat{C}$ and
$(x,x^{\prime}),(y,y^{\prime})\in\hat{E}$.

  * •

$(x,y,x^{\prime})\in W$ if $(x,y)\in\hat{C}$ and $(x,x^{\prime})\in\hat{E}$.

So $V$ and $W$ are projective varieties, and there are natural projections
$V\rightarrow W$, $W\rightarrow\hat{C}$, and $W\rightarrow\hat{E}$, all of
which are finite-to-one (because $\hat{C}$ and $\hat{E}$ are finite
correspondences). We need the following two additional claims about $V$ and
$W$:

######  Claim 7.12.

$W$ is a pure-dimensional curve.

###### Proof.

Since $W\rightarrow\hat{C}$ is finite-to-one, $\dim(W)\leq 1$. On the other
hand, one checks easily that every generic $(x,y)\in C$ is in the image of
$W\rightarrow\hat{C}$, which forces $\dim(W)=1$. Pure-dimensionality then
follows because $W$ has codimension 2 in $\hat{M}^{3}$ and is defined by the
two pure codimension 1 conditions $(x,y)\in\hat{C}$ and
$(x,x^{\prime})\in\hat{E}$ (here we use that $\hat{M}$ is smooth and $C$ is
pure-dimensional). ∎

######  Claim 7.13.

$V\rightarrow W$ is surjective.

###### Proof.

By projectivity and Claim 7.12, it suffices to show that $V$ is almost
surjective, i.e. the image of $V$ is almost equal to $W$. To do this, let
$(x,y,x^{\prime})\in W$ be generic over all relevant parameters. It follows
easily from Claim 7.12 that $(x,y)$ and $(x,x^{\prime})$ are generics in
$\hat{C}$ and $\hat{E}$, respectively, and thus are also generics in $C$ and
$E$, respectively. By Lemma 7.8, there is some $y$ with $(y,y^{\prime})\in E$
and $(x^{\prime},y^{\prime})\in C$. Note that $(y,y^{\prime})$ is also generic
in $E$, thus $(y,y^{\prime})\in\hat{E}$. So $(x,y,x^{\prime},y^{\prime})\in
V$, as desired. ∎

We now prove Lemma 7.9. Recall that we are given $\sim$-classes $X$ and $Y$,
and some $(x_{0},y_{0})\in C\cap(X\times Y)$ with $x_{0},y_{0}\notin T$. We
want to show that $C\cap(X\times Y)\rightarrow X$ and $C\cap(X\times
Y)\rightarrow Y$ are surjective. By symmetry, it is enough to consider the
projection to $X$. So, let $x\in X$ be arbitrary. So $x_{0}\sim x$. Since
$x_{0}\notin T$, we get $(x_{0},x)\in\hat{E}$, thus $(x_{0},y_{0},x)\in W$. By
Claim 7.13, there is $y\in\hat{M}$ with $(x,y)\in\hat{C}$ and
$(y_{0},y)\in\hat{E}$. Since $y_{0}\notin T$, we get $(y_{0},y)\in E$. In
particular, $y\in M$. Then since $C=\hat{C}\cap M^{2}$, we get $(x,y)\in C$. ∎

Fix $T$ as in Lemma 7.9. Adding finitely many points if necessary, we assume
that $T$ is a union of $\sim$-classes, including all classes of size $\neq k$.

######  Definition 7.14.

We refer to $\sim$-classes outside $T$ as good.

######  Convention 7.15.

For the rest of the proof of Theorem 7.3, we will absorb $\sim$ and $T$ into
the language of $\mathscr{M}$, thereby assuming they are
$\emptyset$-definable.

Corollary 7.16 strengthens Lemma 7.9 in the case of graphs of functions.

######  Corollary 7.16.

Let $q:M\rightarrow M$ be a finite-to-one, generically defined,
$\mathscr{M}$-definable partial function. Let $X$ and $Y$ be good
$\sim$-classes, and assume $q$ is almost regular at all points of $X$. If
$q(x)\in Y$ for some $x\in X$, then $q$ defines a bijection from $X$ to $Y$.

###### Proof.

Let $C$ be the graph of $q$. By assumption, the restriction of $C$ to some
neighborhood of $X\times Y$ is a closed, pure dimensional finite
correspondence (see Remark 7.2). So Lemma 7.9 applies. We conclude that
$C\cap(X\times Y)\rightarrow Y$ is surjective, and thus $q(X)\supseteq Y$. But
$|X|=|Y|=k$ by goodness, so in fact $q$ must biject $X$ and $Y$. ∎

######  Remark 7.17.

Let $q:M\rightarrow M$ be as in the setup of Corollary 7.16. Suppose $X\in
M/\sim$ is generic in the sense of the structure $(K,+,\times)$, over the
parameters defining $M$, $\sim$, $T$, and $q$. Then $q$ is automatically
almost regular at all points of $X$. In particular, Corollary 7.16 implies
that $q$ bijects $X$ with some (necessarily also $K$-generic) $Y\in M/\sim$.

In light of Remark 7.17, we make the following harmless convention, which will
only matter for applications of Corollary 7.16 in the proof of Lemma 7.21:

######  Convention 7.18.

For the rest of the proof of Theorem 7.3, the term generic, applied to
elements of $\mathscr{M}$-definable sets, always refers to genericity in the
structure $(K,+,\times)$, where we absorb the parameters defining $M$ and all
basic relations of $\mathscr{M}$.

Recall that to prove Theorem 7.3, we want to show $k=1$. We will do this by
combining several iterated applications of Lemma 7.9 and Corollary 7.16.

Our first task is to construct a generically defined partial function
$Q:M^{2}\rightarrow M$ such that the unary maps $x\mapsto Q_{y}(x)=Q(x,y)$
admit a certain configuration of ‘fixed points’ (i.e. solutions to
$Q(x,y)=x$). We then conclude by applying Lemma 7.9 to the set defined by
$Q(x,y)=x$. The function $Q$ will be built from $P$ by replacing $x$ with a
composition of several unary maps, and applying Corollary 7.16 at each step in
the composition. It is here that we use our unary function $U$ from the
statement of Theorem 7.3.

######  Notation 7.19.

Given $y\in M$, by $P_{y}$ we mean the map $x\mapsto P(x,y)$.

######  Remark 7.20.

Since $P$ generically depends on both variables, it follows easily that
$P_{y}$ is finite-to-one for generic $y\in M$. It also follows that if
$(x,y)\in M^{2}$ is generic, then so are $(x,P(x,y))$ and $(y,P(x,y))$. Since
$M^{2}$ is stationary, it then follows that if $(x,z)\in M^{2}$ is generic,
there is some generic $y\in M$ such that $P(x,y)=z$ and each of $(x,y)$ and
$(y,z)$ is generic in $M^{2}$.

######  Lemma 7.21.

There are generic $\sim$-classes $X$ and $Y$, an element $y_{0}\in Y$, and an
$\mathscr{M}$-definable generically defined partial map $Q:M^{2}\rightarrow
M$, generically depending on both variables, with the following properties:

  1. (1)

$Q$ is almost regular at all points of $X\times Y$. In particular, the graph
of $Q$ is relatively closed and pure two-dimensional in a neighborhood of
$X\times Y\times M$.

  2. (2)

For each $x\in X$, map $y\mapsto Q(x,y)$ is finite-to-one and almost regular
at all points of $Y$.

  3. (3)

The map $x\mapsto Q(x,y_{0})$ is the identity on $X$, but is not generically
the identity on $M$.

###### Proof.

Let $X_{0}$ be any generic $\sim$-class, and let $x_{0}\in X_{0}$. By
Corollary 7.16, the map $U$ bijects $X_{0}$ with a generic $\sim$-class
$X_{1}$. Since $U$ is $\emptyset$-definable in $\mathscr{M}$, $X_{0}$ and
$X_{1}$ are interalgerbraic.

Now let $a\in M$ be generic over $(X_{0},X_{1})$. By Corollary 7.16 and Remark
7.20, the map $P_{a}$ bijects $X_{1}$ with a generic $\sim$-class $X_{2}$ that
is independent from $X_{1}$, and thus also from $X_{0}$. Let $x_{2}=P_{a}\circ
U(x_{0})\in X_{1}$. Then $(x_{0},x_{2})$ is generic in $M^{2}$, and thus so is
$(x_{2},x_{0})$. So by Remark 7.20 again, there is a generic $b\in M$ such
that $P_{b}(x_{2})=x_{0}$ and $x_{2}$ is generic in $M$ over $b$. By Corollary
7.16 again, $P_{b}$ bijects $X_{2}$ to $X_{0}$.

Now by construction, the map $P_{b}\circ P_{a}\circ U$ defines a permutation
$\tau$ of the finite set $X_{0}$. Let $m$ be the order of $\tau$. Then the
$m$-fold composition $(P_{b}\circ P_{a}\circ U)^{m}$ induces the identity on
$X_{0}$.

We then satisfy the lemma by setting $X=X_{0}$, $y_{0}=b$ (so $Y$ is the class
of $b$), and $Q(x,y)=P_{y}\circ r(x)=P(r(x),y)$, where

$r(x)=P_{a}\circ U\circ(P_{b}\circ P_{a}\circ U)^{m-1}.$

Let us verify (1)-(3) in the lemma statement:

  1. (1)

By the construction of $Q$, and since almost regularity is closed under
composition, it is enough to see that $U$ is almost regular at all points of
$X_{0}$, $P_{a}$ is almost regular at all points of $X_{1}$, $P_{b}$ is almost
regular at all points of $X_{2}$, and $P$ is almost regular at all points of
$X_{2}\times Y$. In each case, this follows since the relevant points are
generic in the domain over the parameters defining the function.

  2. (2)

Let $x\in X$, and let $Q^{x}$ be the map $y\mapsto Q(x,y)$. It follows from
(1) that $Q^{x}$ is almost regular at all points of $y$. We show that $Q^{x}$
is finite-to-one. For this, note that $r(x)\in X_{2}$ by construction. So
$r(x)$ is generic in $M$, which shows that $Q^{x}$ (which is $y\mapsto
P(r(x),y)$) is a generic specialization of $P$. Now use that $P$ generically
depends on both variables (see Remark 7.20).

  3. (3)

The map $y_{0}\mapsto Q(x,y_{0})$ is the composition $(P_{b}\circ P_{a}\circ
U)^{m}$, which is the identity on $X_{0}$ by construction. Now suppose toward
a contradiction that $y_{0}\mapsto Q(x,y_{0})$ is the identity. Then
$y_{0}\mapsto Q(x,y_{0})$ is generically injective, and thus each of $P_{b}$,
$P_{a}$, and $U$ is generically injective. But $U$ is non-generically
injective by assumption, contradiction.

∎

######  Notation 7.22.

For the rest of the proof of Theorem 7.3, fix $X$, $Y$, $y_{0}$, and $Q$ as in
Lemma 7.21. We also define $C=\\{(x,y)\in M^{2}:Q(x,y)=x\\}$.

By construction, we have $(x,y_{0})\in C$ for each $x\in X$. We want to apply
Lemma 7.9 to $C$ at these points, but we first need to check that the
hypotheses are satisfied:

######  Lemma 7.23.

For each $x\in X$, the hypotheses of Lemma 7.9 apply to $C$ at $(x,y_{0})$.
That is:

  1. (1)

$\dim(C)=1$.

  2. (2)

$C$ is relatively closed and pure-dimensional in some neighborhood of $X\times
Y$.

  3. (3)

The set $\\{y^{\prime}:(x,y^{\prime})\in C\\}$ is finite.

  4. (4)

The set $\\{x^{\prime}:(x^{\prime},y_{0})\in C\\}$ is finite.

###### Proof.

(3) follows from Lemma 7.21(2), and (4) follows from Lemma 7.21(3). That $C$
is closed in a neighborhood of $X\times Y$ follows from Lemma 7.21(1).

Next, note that $C$ can be thought of as the intersection of the graph of $Q$
with the hypersurface $\\{(x,y,z):x=z\\}$. By Lemma 7.21(1), the graph of $Q$
is pure 2-dimensional in a neighborhood of $X\times Y\times M$, so its
intersection with a hypersurface has no isolated points. In particular, since
$(x,y_{0})\in C$, we get $\dim(C)\geq 1$.

Now if $\dim(C)=2$, then $C$ is dense in $M^{2}$. But since $C$ is closed in a
neighborhood of $X\times Y$, this would imply that $C$ contains a neighborhood
of $X\times Y$, contradicting (3). So $\dim(C)=1$, and we get (1). Finally,
since $C$ is closed and has no isolated points in a neighborhood of $X\times
Y$, we also get (2). ∎

We are now fully set up. The next three claims constitute the final part of
the argument.

######  Claim 7.24.

The projection $C(X\times Y)\rightarrow Y$ is surjective.

###### Proof.

By Lemma 7.9 at the point $(x,y_{0})$, for any $x\in X$. ∎

######  Claim 7.25.

The projection $C(X\times Y)\rightarrow X$ is injective.

###### Proof.

Fix $x\in X$. By Lemma 7.21(2), the map $Q^{x}$, given by $y\mapsto Q(x,y)$,
is non-constant and almost regular at all points of $Y$. By Lemma 7.21(3), we
have $Q^{x}(y_{0})=x$. So by Corollary 7.16, $Q^{x}$ bijects $Y$ to $X$. In
particular, there can be only one $y\in Y$ with $Q(x,y)=x$. ∎

######  Claim 7.26.

$C$ defines a bijection between $X$ and $Y$.

###### Proof.

By construction, $X$ and $Y$ are good, so $|X|=|Y|=k$. Now apply the previous
two claims. ∎

We now finish the proof of Theorem 7.3. By Claim 7.26, the map $x\mapsto
Q(x,y_{0})$ has only one fixed point in $X$. By Lemma 7.21, the same map has
$k$ fixed points in $X$. We conclude that $k=1$. In particular, our
$\mathscr{M}$-definable map $f:M\rightarrow F^{n}$ is generically injective.
After editing finitely many points, we may assume $f$ is injective. The
theorem now follows from Theorem 4.12 of [CH22]. ∎

##  8\. Proof of the Main Theorem

In the final section, we collect the results up to this point and prove
Theorem 1.1. First we show that Theorem 7.3 applies to polynomial reducts:

######  Theorem 8.1.

Let $\mathcal{P}$ be a set of complex polynomials, and let
$\mathscr{M}=(\mathbb{C},\mathcal{P})$. If $\mathscr{M}$ is not locally
modular, then $\mathscr{M}$ defines $+$ and $\times$.

###### Proof.

By Theorem 7.3 and Remark 7.4, it suffices to recover (1) a binary polynomial
that depends on both variables, and (2) a non-linear unary polynomial. We
build these in the following two claims.

######  Claim 8.2.

There is an $\mathscr{M}$-definable binary polynomial depending on both
variables.

###### Proof.

By non-triviality and Lemma 3.2, there is an $\mathscr{M}$-definable
polynomial depending on at least two variables. Now specialize all but two
variables generically. ∎

######  Claim 8.3.

There is an $\mathscr{M}$-definable non-linear unary polynomial.

###### Proof.

Suppose there is no such polynomial. By non-local modularity, $\mathscr{M}$ is
not interdefinable with a vector space, so there must be some
$P\in\mathcal{P}$ which is not linear. By assumption, $P$ uses at least two
variables.

Now by non-linearity, there is a monomial $m$ in $P$ with total degree at
least 2. Clearly, then, there are two variables $(x,y)$ that each appear in
$P$ and such that $m$ has total degree at least 2 in $x$ and $y$: indeed, if
two variables appear in $m$ this is trivial, and if not then let $x$ be the
unique variable in $m$ and let $y$ be any other variable appearing in $P$.

Let $x$ and $y$ be variables as above. Then, specializing all other variables
generically, we recover a binary non-linear polynomial, say $Q(x,y)$.
Considering further specializations of $x$ and $y$, our assumption forces that
$Q$ is linear in each of $x$ and $y$. That is, $Q(x,y)=axy+bx+cy+d$ for some
constants $a,b,c,d$. By non-linearity, $a\neq 0$. But then $Q(x,x)$ is
$\mathscr{M}$-definable and non-linear, a contradiction. ∎

∎

We now deduce the main theorem:

######  Theorem 8.4 (Theorem 1.1).

Suppose $\mathcal{P}$ is a collection of complex polynomial maps, and let
$\mathscr{M}=(\mathbb{C};\mathcal{P})$. Then $\mathscr{M}$ is interdefinable
with exactly one of following:

  1. (i)

$(\mathbb{C};\mathcal{U})$, where $\mathcal{U}$ is a collection of unary
polynomials.

  2. (ii)

The $F$-vector space structure on $\mathbb{C}$ for some subfield
$F\leq\mathbb{C}$.

  3. (iii)

$(\mathbb{C};\times_{b})$ for some (unique) $b\in\mathbb{C}$.

  4. (iv)

The full reduct $(\mathbb{C};+,\times)$.

Moreover, (1) happens if and only if each $P\in\mathcal{P}$ involves at most
one variable. (2) happens if and only if (1) fails and each $P\in\mathcal{P}$
is linear, and in this case $F$ is generated by all coefficients on all
variables appearing in the $P\in\mathcal{P}$. (3) happens if and only if (4)
fails and each $P\in\mathcal{P}$ is a monomial in the group operation
$\times_{b}$. And (4) happens in every other case.

###### Proof.

If $\mathscr{M}$ is trivial, then by Lemma 3.1, $\mathscr{M}$ is
interdefinable with a structure as in (1). If $\mathscr{M}$ is non-trivial
locally modular, then by Theorem 6.11, $\mathscr{M}$ is interdefinable with a
structure as in (2) or (3). If $\mathscr{M}$ is not locally modular, then by
Theorem 8.1, $\mathscr{M}$ is interdefinable with the structure in (4). In
particular, (1) holds if and only if $\mathscr{M}$ is trivial; (2) or (3)
holds if and only if $\mathscr{M}$ is non-trivial locally modular; and (4)
holds if and only if $\mathscr{M}$ is not locally modular.

The ‘moreover’ clause follows for (1) by Lemmas 3.1 and 3.2. For (2) and (3),
quantifier elimination implies that all $P\in\mathcal{P}$ are (respectively)
linear or monomials twisted by $r$, while the converse is Lemma 4.3. Then (4)
must happen in all other cases. ∎

As a result, we conclude that ‘almost all’ complex polynomials in several
variables are capable of defining each other:

######  Theorem 8.5.

Let $P\in\mathbb{C}[x_{1},...,x_{m}]$ and $Q\in\mathbb{C}[x_{1},...,x_{n}]$ be
polynomials. Assume that at least two variables appear in each of $P$ and $Q$,
and neither of $P$ and $Q$ is linear or a twisted monomial. Then there is a
first-order definition of $Q$ in terms of $P$, and vice versa.

###### Proof.

By Theorem 1.1, the structure $(\mathbb{C},P)$ defines addition and
multiplication, and therefore defines $Q$. The other direction holds
symmetrically. ∎

We ends with a remark on the possibility of further generalization.

######  Remark 8.6.

Analogues of Theorem 8.1 are expected in many other settings (e.g., with
polynomials replaced by rational function). Except for the material of Section
6, our analysis goes through in these settings. The only missing ingredient is
a suitable counterpart of Fact 6.3 in such setting. It is expected that this
can be obtained from the recent result in [CPS23] together with an analysis as
in [JRT22].

## References

  * [Cas22] Benjamin Castle, _Restricted trichotomy in characteristic zero_ , 2022.
  * [CH22] Benjamin Castle and Assaf Hasson, _Very ampleness in strongly minimal sets_ , 2022.
  * [CPS23] Artem Chernikov, Ya’acov Peterzil, and Sergei Starchenko, _Model-theoretic elekes-szabó for stable and o-minimal hypergraphs_ , 2023.
  * [HP87] U. Hrushovski and A. Pillay, _Weakly normal groups_ , Logic colloquium ’85 (Orsay, 1985), Stud. Logic Found. Math., vol. 122, North-Holland, Amsterdam, 1987, pp. 233–244. MR 895647
  * [Hru87] Ehud Hrushovski, _Locally modular regular types_ , Classification theory (Chicago, IL, 1985), Lecture Notes in Math., vol. 1292, Springer, Berlin, 1987, pp. 132–164.
  * [HS21] Assaf Hasson and Dmitry Sustretov, _Incidence systems on cartesian powers of algebraic curves_ , 2021.
  * [JRT22] Yifan Jing, Souktik Roy, and Chieu-Minh Tran, _Semialgebraic methods and generalized sum-product phenomena_ , Discrete Anal. (2022), Paper No. 18, 23. MR 4527758
  * [Lov04] James Loveys, _Linear reducts of the complex field_ , Notre Dame J. Formal Logic 45 (2004), no. 3, 161–190. MR 2130784
  * [Mar88] Gary A. Martin, _Definability in reducts of algebraically closed fields_ , J. Symbolic Logic 53 (1988), no. 1, 188–199. MR 929384
  * [MP90] D. Marker and A. Pillay, _Reducts of $({\bf C},+,\cdot)$ which contain $+$_, J. Symbolic Logic 55 (1990), no. 3, 1243–1251. MR 1071326
  * [Rab93] E. D. Rabinovich, _Definability of a field in sufficiently rich incidence systems_ , QMW Maths Notes, vol. 14, Queen Mary and Westfield College, School of Mathematical Sciences, London, 1993, With an introduction by Wilfrid Hodges. MR 1213456
  * [RZ88] Eugenia Rabinovich and Boris Zilber, _Additive reducts of algebraically closed fields_ , manuscript, 1988.
  * [Sha13] Igor R. Shafarevich, _Basic algebraic geometry. 1_ , third ed., Springer, Heidelberg, 2013, Varieties in projective space. MR 3100243
  * [vdD90] L. P. D. van den Dries, _Weil’s group chunk theorem: a topological setting_ , Illinois J. Math. 34 (1990), no. 1, 127–139. MR 1031890