2.1 Multiplicity operators and Weierstrass polydiscs

In this section we denote by $B\subset {\mathbb C}^{n}$ the unit ball. The norm $\left \lVert \cdot \right \rVert $ always denotes the maximum norm. We will need the following basic lemma on multiplicity operators:

Lemma 8. Set $F_{1},\dotsc ,F_{n}:B\to D(1)$. Suppose that $s=\left \lvert M^{\smash {(k)}}_{0}F \right \rvert \neq 0$ for some multiplicity operator $M^{\smash {(k)}}$. Let $\ell \in ({\mathbb C}^{n})^{*}$ have unit norm and set $0<\rho <s$. Then there is a ball $B^{\prime }$ around the origin of radius at least $s/\operatorname {poly}_{n}(k)$ and a union of at most k discs $U_{\rho }$ of total radius at most $\operatorname {poly}_{n}(k)\cdot \rho $, such that

(21)$$ \begin{align} {\mathbf z}\in B^{\prime}\setminus\ell^{-1}\left(U_{\rho}\right) \implies \log \left\lVert F({\mathbf z}) \right\rVert \ge (k+1)\log\rho-\operatorname{poly}_{n}(k). \end{align} $$

We now state a result relating the multiplicity operators to the construction of a Weierstrass polydisc for a curve:

Lemma 9. Set $F_{1},\dotsc ,F_{n-1}:B\to D(1)$. Suppose that $s=\left \lvert M^{\smash {(k)}}_{0}F \right \rvert \neq 0$ for some $(n-1)$-dimensional multiplicity operator $M^{\smash {(k)}}$ with respect to the variables ${\mathbf w}=z_{2},\dotsc ,z_{n}$. Then there exists a Weierstrass polydisc for the set $\{F=0\}$ in the standard coordinates $\Delta =D(r_{1})\times \dotsb \times D(r_{n})$ with all the radii satisfying

(22)$$ \begin{align} \log r_{i} \ge \operatorname{poly}_{n}(k) \log s. \end{align} $$

Proof. We claim that one can find a polydisc $\Delta _{w}=D(r_{2})\times \dotsb \times D(r_{n})$ such that

(23)$$ \begin{align} \log \left\lVert F(0,{\mathbf w}) \right\rVert\ge (k+1)\log s-\operatorname{poly}_{n}(k) \text{ for every }{\mathbf w}\in\partial\Delta_{w}, \end{align} $$

and moreover,

(24)$$ \begin{align} \log r_{i} \ge \operatorname{poly}_{n}(k)\log s \text{ for } i=2,\dotsc,n. \end{align} $$

To prove this, apply Lemma 8 to $F(0,{\mathbf w})$, with $\ell $ given by each of the ${\mathbf z}_{2},\dotsc ,{\mathbf z}_{n}$-coordinates with a suitable choice $\rho =s/\operatorname {poly}_{n}(k)$, and then choose $\Delta _{w}$ to be a polydisc inside the balls $B^{\prime }$ and with each $\partial D\left (r_{j}\right )$ disjoint from the set $U_{\rho }$ obtained for $\ell ={\mathbf z}_{j}$.

Since $F_{1},\dotsc ,F_{n-1}$ have unit maximum norms, their derivatives are bounded by $O(1)$ in $B^{2}$ by the Cauchy estimate. It follows that $F(z,{\mathbf w})$ cannot vanish on $\partial \Delta _{w}$ for $z\in D(r_{1})$, where

(25)$$ \begin{align} \log r_{1}\sim (k+1)\log s-\operatorname{poly}_{n}(k), \end{align} $$

so $D(r_{1})\times \Delta _{w}$ indeed gives a Weierstrass polydisc satisfying the final condition $\log r_{1}\ge \operatorname {poly}_{n}(k) \log s$.

Suppose that $\Gamma \subset {\mathbb C}^{n}$ is an analytic curve, $\Delta =D_{z}\times \Delta _{w}$ is a Weierstrass polydisc for $\Gamma $ and $G:\Delta \to {\mathbb C}$ is holomorphic.

Definition 10. We define the analytic resultant of G with respect to $\Delta $ to be the holomorphic function ${\mathcal R}_{\Delta ,\Gamma }(G):D_{z}\to {\mathbb C}$ given by

(26)$$ \begin{align} \left[{\mathcal R}_{\Delta,\Gamma}(G)\right](z) = \prod_{w:(z,w)\in\Gamma\cap\Delta} G(z,w). \end{align} $$

Our second result concerns a lower estimate for analytic resultants in terms of multiplicity operators.

Lemma 11. Let $F_{1},\dotsc ,F_{n}:B\to D(1)$ be holomorphic. Set $\Gamma =\{F_{1}=\dotsb =F_{n-1}=0\}$ and suppose that $\Delta =D(r)\times \Delta _{w}\subset B$ is a Weierstrass polydisc in the standard coordinates for $\Gamma $ with multiplicity $\mu $. Suppose that $s=\left \lvert M^{\smash {(k)}}_{0}(F)\right \rvert \neq 0$ for some multiplicity operator $M^{\smash {(k)}}$. Set $0<\rho <s$. Then for z in a ball of radius $\Omega _{n}(s)$ around the origin and outside a union of balls of radius $O_{n}(\rho )$, we have

(27)$$ \begin{align} \log \lvert R(z)\rvert> \mu\cdot((k+1)\log\rho-\operatorname{poly}_{n}(k)), \quad R:={\mathcal R}_{\Delta,\Gamma}(F_{n}):D(r)\to{\mathbb C}. \end{align} $$

2.2 Multiplicity operators along ${\mathcal F}$

When $P=(P_{1},\dotsc ,P_{n})\in {\mathcal O}({\mathbb M})^{n}$, we may apply the multiplicity operator $M^{\smash {(k)}}$ to P by evaluating the derivatives along ${\boldsymbol \xi }_{1},\dotsc ,{\boldsymbol \xi }_{n}$. This amounts to computing, for each point $p\in {\mathbb M}$, the multiplicity operator of $P{\vert _{{\mathcal L}}}_{p}$ in the ${\boldsymbol \xi }$-chart.

Lemma 12. For any multiplicity operator $M^{\smash {(k)}}$, we have

(28)$$ \begin{align} \delta\left(M^{\smash{(k)}} P\right) = \operatorname{poly}\left(\delta_{P},\delta_{\boldsymbol\xi},k\right). \end{align} $$

We will require the following result of Gabrielov and Khovanskii [Reference Gabrielov and Khovanskii26]:

Theorem 5. With P as before and $p\in {\mathbb M}\setminus \Sigma _{V(P)}$,

(29)$$ \begin{align} \operatorname{mult}_{p} P < \operatorname{poly}(\deg{\boldsymbol\xi}, \deg P). \end{align} $$

As a consequence, we have the following:

Proposition 13. Let $V\subset {\mathbb M}$ be a complete intersection $V=V(P_{1},\dotsc ,P_{m})$ with $m\le n$. Then

(30)$$ \begin{align} \delta(\Sigma_{V}) = \operatorname{poly}\left(\delta_{\boldsymbol\xi},\delta_{V}\right). \end{align} $$

Moreover, if $m=n$, then $\Sigma _{V}$ is set-theoretically cut out by the functions $\left \{M^{\smash {(k)}}(P)\right \}$, where $M^{\smash {(k)}}$ varies over all multiplicity operators of order $k=\operatorname {poly}(\deg {\boldsymbol \xi },\deg P)$.

Proof. We have $p\in \Sigma _{V}$ if and only if $p\in \Sigma _{\mathcal F}$ or $\dim \left ({\mathcal L}_{p}\cap V\right )>n-m$. Since clearly $\delta (\Sigma _{\mathcal F})=\operatorname {poly}\left (\delta _{\boldsymbol \xi }\right )$, we only have to write equations for the latter condition. This is equivalent to the statement that for every ${\boldsymbol \xi }$-linear subspace of ${\mathcal L}_{p}$ of dimension m, the intersection $V\cap L$ is nonisolated – that is, has infinite multiplicity. We express this using multiplicity operators as follows.

Let ${\mathbf c}^{1},\dotsc ,{\mathbf c}^{m}$ be n-tuples of indeterminate coefficients and let

(31)$$ \begin{align} {\boldsymbol\xi}_{\mathbf c}=\left({\mathbf c}^{1}\cdot{\boldsymbol\xi},\dotsc,{\mathbf c}^{m}\cdot{\boldsymbol\xi}\right) \end{align} $$

denote the subfoliation of ${\boldsymbol \xi }$ generated by the corresponding linear combinations. Then for every $p\in {\mathbb M}\setminus \Sigma _{\mathcal F}$, we obtain a linear subspace ${\mathcal L}_{p,{\mathbf c}}\subset {\mathcal L}_{p}$ and we seek to express the condition that ${\mathcal L}_{p,{\mathbf c}}\cap V$ is an intersection of infinite multiplicity for every ${\mathbf c}$. By Theorem 5, if the multiplicity of the intersection is finite, then it is bounded by $k=\operatorname {poly}(\deg {\boldsymbol \xi },\deg P)$. It is enough to express the condition that the multiplicity exceeds this number for every ${\mathbf c}$. According to Proposition 7, for every fixed value of ${\mathbf c}$ this condition can be expressed by considering all multiplicity operators $M^{\smash {(k)}}(P)$ with respect to ${\boldsymbol \xi }_{\mathbf c}$. Expanding these expressions with respect to the variables ${\mathbf c}$ and taking the ideal generated by all the coefficients, we obtain equations for the vanishing for every ${\mathbf c}$. The estimates on the degrees and heights of these equations follow easily from Lemma 12.

We record a useful corollary of Proposition 13:

Corollary 14. Let $V=V(P_{1},\dotsc ,P_{n})$ be a complete intersection and set $p\in {\mathbb B}$. There exists a multiplicity operator $M^{\smash {(k)}}$ of order $k=\operatorname {poly}(\deg {\boldsymbol \xi },\deg V)$ such that

(32)$$ \begin{align} \log\left\lvert M^{\smash{(k)}}_{p}(P)\right\rvert \ge \operatorname{poly}\left(\delta_{\boldsymbol\xi},\delta_{V}\right)\cdot \log\operatorname{dist}(p,\Sigma_{V}). \end{align} $$

4.1 Parametrising linear subfoliations

Set $k\le n$ and let $A(n,k)$ denote the affine variety of full rank matrices $({\boldsymbol \alpha }_{1},\dotsc ,{\boldsymbol \alpha }_{k})\in \operatorname {Mat}_{n\times k}$. Define $L_{k}{\mathbb M}:=A(n,k)\times {\mathbb M}$ and consider the vector fields

(41)$$ \begin{align} L_{k}({\boldsymbol\xi})_{i} = {\boldsymbol\alpha}_{i}\cdot{\boldsymbol\xi}, \quad i=1,\dotsc,k. \end{align} $$

The leaves of $L_{k}{\mathbb M}$ with $L_{k}({\boldsymbol \xi })$ correspond to the leaves obtained by choosing a k-dimensional subspace of $\langle {\boldsymbol \xi }_{1},\dotsc ,{\boldsymbol \xi }_{n}\rangle $ and using it to span a k-dimensional subfoliation of ${\mathcal F}$.

4.2 Main statement

Our goal is to construct an affine variety $\tilde {\mathbb M}:=N\times {\mathbb M}$ depending only on ${\mathbb M}$, and vector fields $\tilde {\boldsymbol \xi }$ depending on ${\mathbb M},V$, with the following properties:

1. 1. If we denote by $\pi _{\mathbb M}:\tilde {\mathbb M}\to {\mathbb M}$ the projection and by $\phi _{p},\tilde \phi _{a,p}$ the ${\boldsymbol \xi },\tilde {\boldsymbol \xi }$ charts, respectively, then for any $(a,p)\in \tilde {\mathbb M}$ we have $\pi _{\mathbb M}\circ \tilde \phi _{a,p}=\phi _{p}\circ \Phi _{a,p}$, where $\Phi _{a,p}$ is the germ of a self map of $({\mathbb C}^{n},0)$. In particular, ${\mathcal L}_{p}=\pi _{\mathbb M}\left ({\mathcal L}_{a,p}\right )$.

2. 2. Whenever $\phi _{p}$ extends to the unit ball, the germ $\Phi _{a,p}$ extends to $B^{2} $ and

   (42)$$ \begin{align} \left\lVert \Phi_{a,p}-\operatorname{id} \right\rVert_{B^{2}}<0.1. \end{align} $$
   In other words, the reparametrisation is close to the identity.

3. 3. We have effective estimates

   (43)$$ \begin{align} \deg\tilde{\boldsymbol\xi}=\deg{\boldsymbol\xi}+O(1), \qquad h\left(\tilde{\boldsymbol\xi}\right)=\operatorname{poly}\left(\delta_{\boldsymbol\xi},\deg V\right). \end{align} $$

4. 4. For $k=\operatorname {codim} V$, if we set $\tilde {\mathbb M}_{k}:=L_{k}\tilde {\mathbb M}$ and denote by $\tilde V$ the natural pullback to $\tilde {\mathbb M}_{k}$, then

   (44)$$ \begin{align} \pi_{\mathbb M}\left(\Sigma_{\tilde V}\right)\subset \Sigma_{V}. \end{align} $$
   In other words, no ‘new’ unlikely intersections are formed when considering linear subfoliations of $\tilde {\mathbb M},\tilde {\boldsymbol \xi }$.

We also remark that one can similarly achieve general position with respect to any $O(1)$ different varieties $V_{i}\subset {\mathbb M}$, by the same argument.

4.3 Polynomial time reparametrisation

Fix $D\in {\mathbb N}$ and let ${\mathcal M}_{D}$ denote the space of polynomial maps $\Phi :{\mathbb C}^{n}\to {\mathbb C}^{n}$ with coordinate-wise degree at most D. Let $P_{D}({\mathbb M})$ denote the affine variety obtained from ${\mathcal M}_{D}\times {\mathbb C}^{n}_{s}\times {\mathbb M}$ by imposing the condition $\det \tfrac {\partial \Phi ({\mathbf s})}{\partial {\mathbf s}}\neq 0$, where we use $\Phi $ for the coordinate on ${\mathcal M}_{D}$ and ${\mathbf s}$ for the coordinate on ${\mathbb C}^{n}$. Consider the vector fields

(45)$$ \begin{align} P_{D}(\xi_{i}) = \tfrac{\partial}{\partial{\mathbf s}_{i}} + \tfrac{\partial\Phi({\mathbf s})}{\partial{\mathbf s}}\cdot{\boldsymbol\xi}. \end{align} $$

Then the local $P_{D}({\boldsymbol \xi })$-chart at a point $(\Phi ,a,p)$ is given by

(46)$$ \begin{align} \phi_{\Phi,a,p}({\mathbf x}) = \left(\Phi,a+{\mathbf x},\phi_{p}(\Phi(a+{\mathbf x})-\Phi(a))\right). \end{align} $$

In particular, the projection of the leaf $P_{D}({\mathcal L})_{\Phi ,a,p}$ to ${\mathbb M}$ is the germ ${\mathcal L}_{p}$, but the time parametrisation is adjusted according to $\Phi $ around a.

4.4 Codimension of unlikely intersection

Set

(47)$$ \begin{align} \tilde {\mathbb M} = L_{k}(P_{D}{\mathbb M})= A(n,k)\times{\mathcal M}_{D}\times{\mathbb C}^{n}_{s}\times{\mathbb M}. \end{align} $$

Denote by $\tilde V$ the pullback of V to $\tilde {\mathbb M}$.

Lemma 19. Let $p\in {\mathbb M}\setminus \Sigma _{V}$, $A\in A(n,k)$ and $a\in {\mathbb C}^{n}_{s}$. Then the set

(48)$$ \begin{align} \left\{\Phi\in{\mathcal M}_{D} : (A,\Phi,a,p)\in\Sigma_{\tilde V} \right\}\subset{\mathcal M}_{D} \end{align} $$

is algebraic of codimension at least D.

Proof. Algebraicity follows from Proposition 13. Replacing a by $0$ and $\Phi ({\mathbf x})$ by $\Phi (a+{\mathbf x})-\Phi (a)$, we may assume without loss of generality that $a=0$. Similarly, replacing A by $({\boldsymbol \xi }_{1},\dotsc ,{\boldsymbol \xi }_{k})$ and $\Phi ({\mathbf x})$ by its appropriate linear change of variable, we may assume without loss of generality that $A=({\boldsymbol \xi }_{1},\dotsc ,{\boldsymbol \xi }_{k})$.

Denote $\Phi ^{\prime }_{j}=\Phi _{j}-\Phi _{j}(0)$. Then the leaf at $(A,\Phi ,0,p)$ is defined by

(49)$$ \begin{align} (A,\Phi,0)\times{\mathcal L}_{p}^{\prime}, \quad {\mathcal L}_{p}^{\prime}={\mathcal L}_{p}\cap\left\{\Phi^{\prime}_{k+1}=\dotsb=\Phi^{\prime}_{n}=0\right\}. \end{align} $$

We must check when the intersection of ${\mathcal L}_{p}^{\prime }$ and V is a complete intersection. It is enough to bound the codimension of the condition that $\Phi ^{\prime }_{k+1}$ vanishes identically on (a component of) ${\mathcal L}_{p}\cap V$, that $\Phi ^{\prime }_{k+2}$ vanishes identically on (a component of) ${\mathcal L}_{p}\cap \left \{\Phi ^{\prime }_{k+1}=0\right \}\cap V$ and so on.

From the foregoing we conclude that it is enough to prove the following simple claim: let $\gamma \subset ({\mathbb C}^{n},0)$ be the germ of an analytic curve. Then the set of polynomials of degree at most D without a free term vanishing identically on $\gamma $ has codimension at least D. Note that this set is linear. Choose t to be a (linear) coordinate on ${\mathbb C}^{n}$ which is nonconstant on $\gamma $. Then clearly $t,\dotsc ,t^{D}$ are linearly independent on $\gamma $, and the claim follows.

Now choose $D=\dim A(n,k)+n+\dim {\mathbb M}+1$. Denote by $\pi _{\Phi }:\tilde {\mathbb M}\to {\mathcal M}_{D}$ the projection. Then by a dimension-counting argument using Lemma 19, the codimension of $\pi _{\Phi }\left (\Sigma _{\tilde V}\right )$ is positive. By Proposition 13 the degree of the Zariski closure $Z:=\operatorname {Clo}\pi _{\Phi }\left (\Sigma _{\tilde V}\right )$ is bounded by $\operatorname {poly}(\deg V,\deg {\boldsymbol \xi })$. If we choose any $\Phi _{0}\not \in Z$ and restrict $\tilde {\mathbb M}$ to $\Phi =\Phi _{0}$, then the final condition in Section 4.2 is satisfied by definition. It remains only to show that $\Phi _{0}$ can be chosen close to the identity map and with appropriately bounded height. This follows immediately from the following general statement:

4.5 Generic choice of a complete intersection

Let $V\subset {\mathbb M}$ be a variety defined over ${\mathbb K}$. In this section we show that one can choose a complete intersection W containing V with $\Sigma _{W}$ being ‘as small as possible’ and with effective control over $\delta _{W}$. We will need the following elementary lemma:

The following is our main result for this section:

Proposition 22. Let $0\le m\le \dim {\mathbb M}$ be an integer. There exists a complete intersection W of pure codimension m that contains V and satisfies $\delta _{W}=\operatorname {poly}\left (\delta _{\boldsymbol \xi },\delta _{V}\right )$ and

(50)$$ \begin{align} \Sigma_{W} = \left\{p\in{\mathbb M} : \dim\left(V\cap{\mathcal L}_{p}\right)>n-m\right\}. \end{align} $$

Proof. We remark that the inclusion $\subset $ in equation (50) is trivial. Suppose that we have already constructed a complete intersection $W_{k}$ of pure codimension $k<m$ satisfying the conditions. We will show how to choose a polynomial equation P vanishing on V, with $\delta _{P}$ bounded, such that $W_{k+1}=W\cap V(P)$ satisfies

(51)$$ \begin{align} \Sigma_{W_{k+1}} = \left\{p\in{\mathbb M} : \dim\left(V\cap{\mathcal L}_{p}\right)>n-k-1\right\}. \end{align} $$

The claim then follows by induction on k.

Let $D=\dim {\mathbb M}+1$ and let ${\mathcal P}_{D}$ denote the space of polynomials in the ambient space of ${\mathbb M}$ of degree at most D. Consider $\tilde {\mathbb M}:={\mathcal P}_{D}^{S}\times {\mathbb M}$ and let $\tilde P\in {\mathcal O}\left (\tilde {\mathbb M}\right )$ be given by

(52)$$ \begin{align} \tilde P(Q_{1},\dotsc,Q_{S},\cdot) = Q_{1}P_{1}+\dotsb+Q_{S}P_{S}, \quad (Q_{1},\dotsc,Q_{S})\in{\mathcal P}_{D}^{S}. \end{align} $$

Set $\tilde W_{k+1}=\tilde W_{k}\cap V\left (\tilde P\right )$, where $\tilde W_{k}:={\mathcal P}_{D}^{S}\times W_{k}.$

Let p satisfy $\dim \left (V\cap {\mathcal L}_{p}\right )<n-k$. By assumption, $p\not \in \Sigma _{W_{k}}$. We claim that the codimension in ${\mathcal P}_{D}^{S}$ of the set

(53)$$ \begin{align} \left\{ Q\in{\mathcal P}_{D}^{S} : (Q,p)\in\Sigma_{\tilde W_{k+1}}\right\} \end{align} $$

is at least D. Indeed, the condition is equivalent to the fact that $\tilde P$ does not vanish identically on any of the irreducible components of ${\mathcal L}_{p}\cap W_{k}$. It is enough to check the codimension for each component C separately. Since V is set-theoretically cut out by $P_{1},\dotsc ,P_{S}$ and $\dim \left (V\cap {\mathcal L}_{p}\right )<n-k$, one of the polynomials $P_{j}$ – say, without loss of generality, $P_{1}$ – does not vanish identically on C. Then for any fixed value of $Q_{2},\dotsc ,Q_{S}$, at most one value of $Q_{1}{\vert _{C}}$ can give $\tilde P{\vert _{C}}\equiv 0$, and we have already seen in the proof of Lemma 19 that the codimension of this affine linear condition is at least D.

We now finish as in Section 4.4. Namely, by Proposition 13 we see that $\Sigma _{\tilde W_{k+1}}$ is algebraic and

(54)$$ \begin{align} \deg \Sigma_{\tilde W_{k+1}}=\operatorname{poly}(\deg({\boldsymbol\xi}),\deg(V)). \end{align} $$

Set $Z=\operatorname {Clo} \pi \left (\Sigma _{\tilde W_{k+1}}\right )$, where $\pi :\tilde M\to {\mathcal P}_{D}^{S}$, and note that by a dimension-counting argument Z has positive codimension. Choosing a point $Q\not \in Z$ using Lemma 20 and setting $P=\tilde P(Q,\cdot )$ finishes the proof.

5.1 Proof of Theorem 1

We prove Theorem 1 in dimension n assuming that it holds for dimension at most $n-1$ and that Theorem 2 holds for dimension at most n.

Set $V^{\prime }:=V(P_{1},\dotsc ,P_{n-1})$. Note that $\Sigma _{V^{\prime }}\subset \Sigma _{V}$. We start by passing to general position with respect to V and $V^{\prime }$ as in Section 4.2. This has the effect of slightly reparametrising the time variables, and in the new parametrisation the original balls ${\mathcal B},{\mathcal B}^{2}$ are contained in balls of radius slightly larger than $1,1/2$. However, dividing these balls into $O(1)$ balls and rescaling time (i.e., rescaling ${\boldsymbol \xi }$), we see that it is enough to prove Theorem 1 for ${\mathcal B},{\mathcal B}^{2}$ in the new parametrisation.

Applying Theorem 2, we construct a collection of Weierstrass polydiscs $\{\Delta _{\alpha }\subset {\mathcal B}\}$ for $V^{\prime }$ such that the union of the $\Delta ^{2}_{\alpha }$ covers ${\mathcal B}^{2}$. Since $\#\{\Delta _{\alpha }\}$ is appropriately bounded, it will suffice to count the zeros of $P_{n}$ on $V^{\prime }$ inside each $\Delta ^{2}_{\alpha }$ separately. Fix one such polydisc $\Delta :=\Delta _{\alpha }$ and set $\Delta =D_{z}\times \Delta _{w}$ (in some unitary system of coordinates). We also have that $\mu :=e(V^{\prime }\cap {\mathcal B},\Delta )$ is appropriately bounded.

Recall the analytic resultant defined in equation (26). The zeros of $P_{n}$ on $V^{\prime }$ inside $\Delta $ correspond (with multiplicities) to the zeros of ${\mathcal R}_{\Delta ,{\mathcal B}\cap V^{\prime }}(P_{n})$ in $D_{z}$. We want to count those zeros contained in $D_{z}^{2}$. Recall the following consequence of Jensen’s formula [Reference Il’yashenko and Yakovenko30]:

Proposition 23. Let $f:\bar D\to {\mathbb C}$ be holomorphic. Denote by M (resp., m) the maximum of $\lvert f(z)\rvert $ on $\bar D$ (resp., $\bar D^{2}$). Then there exists a constant C such that

(55)$$ \begin{align} \#\left\{z\in D^{2} : f(z)=0 \right\} \le C\cdot\log\frac M m. \end{align} $$

We apply this proposition to ${\mathcal R}_{\Delta ,{\mathcal B}\cap V^{\prime }}(P_{n})$ in $D_{z}$. We first note that M is a product of $\lvert P_{n}\rvert $ evaluated at $\mu $ points $p_{1},\dotsc ,p_{\mu }\in {\mathbb B}$. It is clear that $\log \left \lvert P_{n}\left (p_{j}\right )\right \rvert \le \operatorname {poly}(\delta (P_{n}))$, so $\log M$ is appropriately bounded.

It remains to show that $\log (1/m)$ is appropriately bounded. Let $p_{1},\dotsc ,p_{\mu }$ denote the points of $V^{\prime }$ lying over the origin in $\Delta ^{\prime }$. Consider the multiplicity operators $M^{\smash {(k)}}(P_{1},\dotsc ,P_{n-1})$ with respect to the direction of the ${\mathbf w}$-coordinates (which we think of as a leaf of the foliated space $L_{n-1}{\mathbb M}$). By Corollary 14, at every point $p_{j}$ there is such a multiplicity operator with $\log \left \lvert 1/M^{\smash {(k)}}_{p_{j}}(P_{1},\dotsc ,P_{n-1})\right \rvert $ appropriately bounded in absolute value (here we use the fact that we perturbed to general position). According to Lemma 9, each point $p_{j}$ is the centre of a Weierstrass polydisc $\Delta _{j}$ in the same coordinate system, and with the logarithms of all radii appropriately bounded in absolute value.

Denote ${\mathcal R}_{j}:={\mathcal R}_{\Delta _{j},{\mathcal B}\cap V^{\prime }}(P_{n})$. The domains of all these functions (and of ${\mathcal R}$ itself) contain a disc D of radius r, with $\log (1/r)$ appropriately bounded. Note that

(56)$$ \begin{align} \lvert{\mathcal R}(z)\rvert \ge \frac{\prod_{j=1,\dotsc,\mu} \left\lvert{\mathcal R}_{j}(z)\right\rvert}{e^{\operatorname{poly}\left(\delta_{\boldsymbol\xi},\delta_{V}\right)}}, \end{align} $$

since the numerator contains the value of $P_{n}$ evaluated at every point of ${\mathcal B}\cap V^{\prime }$ over z (possibly more than once), and these evaluations are always bounded from above by $e^{\operatorname {poly}\left (\delta \left (P_{n}\right )\right )}$, as we have already seen. It will therefore suffice to find a point in D where $\log \left (1/\left \lvert {\mathcal R}_{j}\right \rvert \right )$ is appropriately bounded for every j. For this we use Lemma 11. Namely, the lemma shows that $\log \left (1/\left \lvert {\mathcal R}_{j}\right \rvert \right )$ is appropriately bounded outside a union of balls of total radius smaller than $r/\mu $, and taking the union over $j=1,\dotsc ,\mu $, one can find a point where this happens simultaneously for every j. This shows that $\log (1/m)$ is appropriately bounded and concludes the proof of Theorem 1.

5.2 Proof of Corollary 2

This follows immediately by applying Proposition 22 with $m=n$ and applying Theorem 1 to the W that one obtains.

5.3 Proof of Theorem 2

We will prove Theorem 2 in dimension n assuming that Theorem 1 holds in smaller dimensions. It will be enough to find a Weierstrass polydisc $\Delta \subset {\mathcal B}$ around the origin containing a ball of radius r such that $1/r$ and $e(V\cap {\mathcal B},\Delta )$ are appropriately bounded. Indeed, if we can do this, then by a simple rescaling and covering argument we can find a collection of polydiscs covering ${\mathcal B}^{2}$.

According to Corollary 18, it will be enough to show that $\operatorname {vol}({\mathcal B}\cap V)$ is appropriately bounded. This volume can be estimated using complex integral geometry in the spirit of Crofton’s formula. Namely, according to [Reference Chirka and Hoksbergen19, Proposition 14.6.3] we have

(57)$$ \begin{align} \operatorname{vol}(V\cap{\mathcal B}) = \operatorname{const}(n) \int_{G\left(n,\operatorname{codim} V\right)} \#(V\cap{\mathcal B}\cap L) \,\mathrm d L, \end{align} $$

where $G(n,k)$ denotes the space of all k-dimensional linear subspaces of ${\mathbb C}^{n}$ with the standard measure.

We now pass to general position with respect to V, as in Section 4.2. Since our reparametrising map can be assumed to be close to the identity, this does not change the volume by a factor of more than (say) $2$. Hence it is enough to estimate the volume in the new coordinates, and by equation (57) it will suffice to show that $\#(V\cap {\mathcal B}\cap L)$ is appropriately bounded for every ${\boldsymbol \xi }$-linear subspace of dimension $k=\operatorname {codim} V$. Since the ${\mathcal B}\cap L$ are all unit balls in leaves of $L_{k}{\mathbb M}$, the result now follows by the inductive application of Theorem 1 (using the fact that $L_{k}{\mathbb M}$ has no new unlikely intersections with V).

6.1 Interpolating algebraic points

Set $n\in {\mathbb N}$. The asymptotic constants in this section will depend only on n. Let $\Delta =\Delta _{\mathbf x}\times \Delta _{\mathbf w}\subset {\mathbb C}^{n}$ be a Weierstrass polydisc for an analytic set $X\subset {\mathbb C}^{n}$ of pure dimension m. Set $F\in {\mathcal O}\left (\bar \Delta \right )$. Let ${\mathcal M}\subset {\mathbb N}^{n}$ be the set

(58)$$ \begin{align} {\mathcal M} := {\mathbb N}^{m}\times\{0,\dotsc,e(X,\Delta)-1\}^{n-m}. \end{align} $$

We also set $E:=e(X,\Delta )^{n-m}$. Recall the following result combining [Reference Binyamini and Novikov12, Theorem 3] and [Reference Binyamini and Novikov11, Proposition 8]:

Proposition 24. On $\Delta ^{2}$ there is a decomposition

(59)$$ \begin{align} F = \sum_{\alpha\in{\mathcal M}} c_{\alpha} {\mathbf x}^{\alpha} + Q, \quad Q\in{\mathcal O}\left(\Delta^{2}\right), \end{align} $$

where Q vanishes on $\Delta ^{2}\cap X$ and

(60)$$ \begin{align} \left\lVert c_{\alpha} {\mathbf x}^{\alpha} \right\rVert_{\Delta^{2}} = O\left(2^{-\lvert\alpha\rvert} \cdot \left\lVert F \right\rVert_{\Delta}\right). \end{align} $$

We now fix $\Phi \in {\mathcal O}(\Delta )^{m+1}$. In [Reference Binyamini and Novikov12], Proposition 24 was used in combination with the interpolation determinant method of Bombieri and Pila [Reference Bombieri and Pila17] to produce an algebraic hypersurface interpolating the points of $X\cap \Delta ^{2}$ where $\Phi $ takes algebraic values of a given height in a fixed number field. However, this method does not produce good bounds when one considers the more general $\left [X\cap \Delta ^{2}\right ](g,h;\Phi )$, where the number field may vary. Instead, we will use an alternative approach proposed by Wilkie [Reference Wilkie53], which is based on the following variant of the Thue–Siegel lemma. This idea was used in a slightly different context by Habegger in [Reference Habegger29].

Lemma 25 [Reference Waldschmidt51, Lemma 4.11]

Set $A\in \operatorname {Mat}_{\mu \times \nu }({\mathbb R})$. For any $N\in {\mathbb N}$, there exists a vector ${\mathbf v}\in {\mathbb Z}^{\nu }\setminus \{0\}$ satisfying

(61)$$ \begin{align} \begin{aligned} \left\lVert {\mathbf v} \right\rVert_{\infty}\le N+1, \qquad \left\lVert A{\mathbf v} \right\rVert_{\infty} \le N^{\frac{\mu-\nu}\mu} \left\lVert A \right\rVert_{\infty}. \end{aligned} \end{align} $$

By combining Proposition 24 and Lemma 25, we obtain the following:

Lemma 26. Set $d,N\in {\mathbb N}$. There exists a polynomial $P\in {\mathbb Z}[y_{1},\dotsc ,y_{m+1}]\setminus \{0\}$ with $\deg P\le d$ and all coefficients bounded in absolute value by N, such that

(62)$$ \begin{align} \left\lVert (P\circ\Phi){\vert_{X\cap\Delta^{2}}} \right\rVert \le \operatorname{poly}(d) E \left\lVert \Phi \right\rVert_{\Delta}^{d} (N \log N) 2^{-d\left(E^{-1}\log N\right)^{\frac1{m+1}}}. \end{align} $$

Proof. Let $\Phi ^{\alpha }$ for $\alpha \in {\mathbb N}^{m+1}$ denote the monomial in the $\Phi $ variables with the usual multiindex notation. Note that $\left \lVert \Phi ^{\alpha } \right \rVert \le \left \lVert \Phi \right \rVert ^{\lvert \alpha \rvert }$. For each $\lvert \alpha \rvert \le d$, apply Proposition 24 to $\Phi ^{\alpha }$ to get

(63)$$ \begin{align} \Phi^{\alpha} = \sum_{\beta\in{\mathcal M}} c_{\alpha,\beta} {\mathbf x}^{\beta} + Q, \quad Q\in{\mathcal O}\left(\Delta^{2}\right), \end{align} $$

where Q vanishes on $\Delta ^{2}\cap X$ and

(64)$$ \begin{align} \left\lVert c_{\alpha,\beta} {\mathbf x}^{\beta} \right\rVert_{\Delta^{2}} = O\left(2^{-\lvert\beta\rvert} \cdot \left\lVert \Phi \right\rVert_{\Delta}^{d}\right). \end{align} $$

Fix $k\in {\mathbb N}$ to be chosen later. Using Lemma 25, we find a linear combination $\sum _{\lvert \alpha \rvert \le d}v_{\alpha } \Phi ^{\alpha }$ with $v_{\alpha }$ integers and $\lvert v_{\alpha }\rvert <N$, not all zero, such that for every $\lvert \beta \rvert \le k$ we have

(65)$$ \begin{align} \left\lVert \sum_{\lvert\alpha\rvert\le d} v_{\alpha} c_{\alpha,\beta}x^{\beta} \right\rVert_{\Delta^{2}} = O\left(\left\lVert \Phi \right\rVert_{\Delta}^{d}\right)\cdot N^{\frac{\mu-\nu}\mu}, \quad \mu \sim E k^{m}, \ \nu \sim d^{m+1}. \end{align} $$

We now write

(66)$$ \begin{align} \sum_{\lvert\alpha\rvert\le d} v_{\alpha} \Phi^{\alpha} = \sum_{\lvert\beta\rvert\le k}\sum_{\lvert\alpha\rvert\le d} v_{\alpha} c_{\alpha,\beta}x^{\beta} +\sum_{\lvert\beta\rvert>k} \sum_{\lvert\alpha\rvert\le d} v_{\alpha} c_{\alpha,\beta}x^{\beta} =A+B. \end{align} $$

For A we have by equation (65) the estimate

(67)$$ \begin{align} A \le O\left(E k^{m} \left\lVert \Phi \right\rVert_{\Delta}^{d}\right)\cdot N^{\frac{\mu-\nu}\mu}, \end{align} $$

and for B we have by equation (64) the estimate

(68)$$ \begin{align} B \le O\left(N d^{m+1} \left\lVert \Phi \right\rVert_{\Delta}^{d} \sum_{\lvert\beta\rvert>k} 2^{-\beta}\right) = O\left(N d^{m+1}\left\lVert \Phi \right\rVert_{\Delta}^{d}2^{-k}\right). \end{align} $$

Choosing $k=d\left (E^{-1}\log N\right )^{1/(m+1)}$ proves the lemma.

We will compare the upper bound of Lemma 26 with the following elementary lower bound at points where $\Phi $ takes algebraic values of bounded height and degree:

Lemma 27 [Reference Habegger29, Lemma 14]

Let $P\in {\mathbb Z}[y_{1},\dotsc ,y_{m+1}]$ be a polynomial of degree d and all coefficients bounded in absolute value by N. Suppose that ${\mathbf y}\in \left ({\mathbb Q}^{\mathrm {alg}}\right )^{m+1}$ and $P({\mathbf y})\neq 0$. Then

(69)$$ \begin{align} \lvert P({\mathbf y})\rvert \ge \left(d^{m+1} N H({\mathbf y})^{d(m+1)}\right)^{-[{\mathbb Q}({\mathbf y}):{\mathbb Q}]}. \end{align} $$

For a subset $A\subset {\mathbb C}^{n}$, we denote

(70)$$ \begin{align} A(g,h;\Phi) := \{ p\in A : [{\mathbb Q}(\Phi(p)):{\mathbb Q}] \le g \text{ and } h(\Phi(p))\le h\}. \end{align} $$

We now come to our interpolation result.

Proposition 28. The set $\left [\Delta ^{2}\cap X\right ](g,h;\Phi )$ is contained in the zero locus of $P\circ \Phi $, where $P\in {\mathbb Z}[y_{1},\dotsc ,y_{m+1}]\setminus \{0\}$ and

(71)$$ \begin{align} \deg P\sim g\cdot E\cdot \left(gh+\log\left\lVert \Phi \right\rVert_{\Delta}\right)^{m}, \qquad h(P) \sim E\cdot\left(gh+\log\left\lVert \Phi \right\rVert_{\Delta}\right)^{m+1}. \end{align} $$

Proof. Set $d,N\in {\mathbb N}$ and construct the polynomial P as in Lemma 26. At any point ${\mathbf x}\in \left [\Delta ^{2}\cap X\right ](g,h;\Phi )$, if $P\circ \Phi (x)\neq 0$, then

(72)$$ \begin{align} \left[\operatorname{poly}(d)\cdot N \cdot 2^{(m+1)hd}\right]^{-g} \le \lvert P\circ\Phi({\mathbf x})\rvert \le \operatorname{poly}(d) E \left\lVert \Phi \right\rVert_{\Delta}^{d} (N \log N) 2^{-d\left(E^{-1}\log N\right)^{\frac1{m+1}}}, \end{align} $$

and hence

(73)$$ \begin{align} 2^{-O\left(\frac{g \log N}d + gh +\log\left\lVert \Phi \right\rVert_{\Delta}+\frac{\log E}d\right)} \le 2^{-\left(E^{-1} \log N\right)^{\frac1{m+1}}}. \end{align} $$

Now choose

(74)$$ \begin{align} d&=C^{m+1}E \left(gh+\log\left\lVert \Phi \right\rVert_{\Delta}\right)^{m}\cdot g, \end{align} $$

(75)$$ \begin{align} \log N&=C^{m+1}E\left(gh+\log\left\lVert \Phi \right\rVert_{\Delta}\right)^{m+1}. \end{align} $$

Then formula (73) becomes

(76)$$ \begin{align} 2^{-O\left(gh +\log\left\lVert \Phi \right\rVert_{\Delta}\right)} \le 2^{-C\left(gh +\log\left\lVert \Phi \right\rVert_{\Delta}\right)}, \end{align} $$

which is impossible for a sufficiently large constant $C=C(m)$, and we deduce that $P\circ \Phi $ vanishes on $\left [\Delta ^{2}\cap X\right ](g,h;\Phi )$, as claimed.

6.2 Finishing the proof of Theorem 3

Theorem 3 follows immediately from the following inductive step, where we start with ${\mathcal W}={\mathbb C}^{\ell }$ and proceed until $\operatorname {dist}({\mathcal B},\Sigma (V,{\mathcal W}))<\varepsilon $:

Proposition 29. Let ${\mathcal W}\subset {\mathbb C}^{\ell }$ be an irreducible ${\mathbb Q}$-variety of positive dimension. Suppose that $\varepsilon :=\operatorname {dist}({\mathcal B},\Sigma (V,W))$ is positive. Then there exists a collection of irreducible ${\mathbb Q}$-subvarieties $\{{\mathcal W}_{\alpha }\subset {\mathcal W}\}$ of codimension $1$ such that

(77)$$ \begin{align} {\mathcal W}\cap A(g,h) \subset \cup_{\alpha} {\mathcal W}_{\alpha} \end{align} $$

and

(78)$$ \begin{align} \#\{{\mathcal W}_{\alpha}\}, \max_{\alpha} \delta_{{\mathcal W}_{\alpha}} = \operatorname{poly}\left(\delta_{\boldsymbol\xi},\delta_{V},\delta_{\mathcal W},\delta_{\Phi},g,h,\log \varepsilon^{-1}\right). \end{align} $$

Proof of Proposition 29

Define $m:=\dim {\mathcal W}$. Set $V^{\prime }=V\cap \Phi ^{-1}({\mathcal W})$. We claim that

(79)$$ \begin{align} \left\{ p: \dim\left(V^{\prime}\cap{\mathcal L}_{p}\right)\ge m \right\} \subset \Sigma(V,{\mathcal W};\Phi). \end{align} $$

Indeed, if $p\not \in \Sigma (V,{\mathcal W};\Phi )$, then $\Phi _{V\cap {\mathcal L}_{p}}$ is finite. If $V^{\prime }\cap {\mathcal L}_{p}$ has a component C of dimension at least m, then $\dim \Phi (C)\ge m$ and $C\subset {\mathcal W}$, so $\Phi (C)$ is a component of ${\mathcal W}$ – contradicting the definition of $\Sigma (V,{\mathcal W};\Phi )$.

Using Proposition 22, we find a complete intersection W of codimension $n-m+1$ containing $V^{\prime }$ and satisfying $\Sigma _{W}\subset \Sigma (V,{\mathcal W};\Phi )$ with appropriate control over $\delta _{W}$. Using Theorem 2, we cover ${\mathcal B}^{2}$ by sets $\Delta _{\beta }^{2}$, where $\Delta _{\beta }$ is a Weierstrass polydisc for ${\mathcal B}\cap W$ and $\#\left \{\Delta _{\beta }\right \}$ and $e\left ({\mathcal B}\cap W,\Delta _{\beta }\right )$ are bounded as in equation (78).

Choose a set of m coordinates $S\subset \{1,\dotsc ,\ell \}$ such that the projection of ${\mathcal W}$ to these coordinates is dominant. Using Proposition 28 we construct a polynomial $P_{\beta }\in {\mathbb Z}[y_{1},\dotsc ,y_{\ell }]\setminus \{0\}$ depending only on the variables $\{y_{s}\}_{s\in S}$ such that $\delta (P_{S})=\operatorname {poly}(g,h,\delta _{\Phi })$ and $P_{\beta }\circ \Phi $ vanishes identically on $\left [\Delta _{\beta }^{2}\cap W\right ](g,h;\Phi )$. Finally, taking $\{{\mathcal W}_{\alpha }\}$ to be the union of the collection of irreducible components of ${\mathcal W}\cap \left \{P_{\beta }=0\right \}$ for every $\beta $ proves the claim.

7.1 Simultaneous torsion points

Let $T\subset {\mathbb C}^{4}\times ({\mathbb C}\setminus \{0,1\})$ denote the fibred product of two copies of the Legendre family,

(80)$$ \begin{align} T = \left\{(x_{1},y_{1},x_{2},y_{2},\lambda) : y_{j}^{2}=x_{j}\left(x_{j}-1\right)\left(x_{j}-\lambda\right), j=1,2\right\}. \end{align} $$

The fibre of T over $\lambda $ is an elliptic square $E_{\lambda }\times E_{\lambda }$, and we use the additive notation for the group law on this scheme. We will also write $P=(x_{1},y_{1})$ and $Q=(x_{2},y_{2})$.

The proof is given in Section 8. This theorem implies finiteness of the set of simultaneous torsion points, which is the main statement of [Reference Masser and Zannier38]. It also implies that the set of simultaneous torsion points is effectively computable in polynomial time: for each possible torsion order k up to the bound provided in the theorem, one can compute the algebraic equations $\left (P^{k},Q^{k},c\right )=(\infty ,\infty ,c)$ using the group law on T, intersect with the equation defining $C\subset T$ and use elimination theory or Gröbner-basis algorithms to compute the sets of solutions c.

We remark that numerous variations on the theme of Theorem 6 have been studied by Masser and Zannier [Reference Masser and Zannier41, Reference Masser and Zannier37, Reference Masser and Zannier39, Reference Masser and Zannier40], Barroero and Capuano [Reference Barroero and Capuano4, Reference Barroero and Capuano2, Reference Barroero and Capuano3] and Schmidt [Reference Schmidt49]. These include very interesting applications to the solvability of Pell’s equation in polynomials and to integrability in elementary terms. Effective bounds for these contexts, analogous to Theorem 6, should in principle provide the last step toward effective solvability of these classical problems. While we do not address these generalisations directly in this paper, they do appear to be similarly amenable to our methods. We have developed some of the material (most specifically the growth estimates in Appendix A) with an eye to treating the more general types of period maps arising in these applications.

7.2 André–Oort for modular curves

We refer the reader to [Reference Pila45] for the general terminology related to the André–Oort conjecture in the context of ${\mathbb C}^{n}$. We will prove the following:

The proof is given in Section 9. Note that this is the only point in this paper where the implied asymptotic constant is not effectively computable in principle. The constants depend on Siegel’s asymptotic lower bound for class numbers, and obtaining an effective form of this bound is a well-known and deep problem. The effectivity of this universal constant notwithstanding, Theorem 7 still establishes the polynomial-time decidability of the André–Oort conjecture in ${\mathbb C}^{n}$ for fixed n. We also note that the constants do depend effectively on n, so the result also establishes the decidability of the conjecture for ${\mathbb C}^{n}$ with n considered as a variable. We remark that the André–Oort conjecture for more general products of modular curves can be proved by reduction to the ${\mathbb C}^{n}$ case, and this certainly preserves effectivity, but we do not pursue the details of this here.

7.3 A Galois-orbit lower bound for torsion points

We will prove the following:

Theorem 8. Let E be an elliptic curve defined over a number field ${\mathbb K}$, and $p\in A$ a torsion point of order n. Then

(81)$$ \begin{align} n=\operatorname{poly}_{g}([{\mathbb K}:{\mathbb Q}],h_{\mathrm{Fal}}(E),[{\mathbb K}(p):{\mathbb K}]). \end{align} $$

The proof is given in Section 10. This theorem is not new: it follows (with more precise dependence on the parameters) from the work of David [Reference David22]. It has also been generalised to abelian varieties of arbitrary genus under some mild conditions [Reference David21] (see also [Reference Masser and Zannier41] for the general case). The proof presented here is different, replacing the use of transcendence methods by point counting using an idea of Schmidt.

We restrict our formal presentation to the elliptic case, as the general case requires some additional technical tools that we do not treat in this paper. However, we sketch in Section 10.4 how the proof extends to arbitrary genus (we restrict to principally polarised abelian varieties, and do not consider the general case). We also mention further implications for Galois-orbit lower bounds in Shimura varieties in Section 10.2.

8.1 The foliation

We will construct a one-dimensional foliation encoding for each $c\in C$ a pair of lattice generators $(f,g)$ for the curve $E_{\lambda (c)}$ and a pair of elliptic logarithms $z,w$ for the points $P(c),Q(c)\in E_{\lambda (c)}$. This can be done with the help of the classical Picard–Fuchs differential operator as follows.

We will work in the space over ${\mathbb C}$ given by

(82)$$ \begin{align} {\mathbb M} := C\times G, \quad G:=(\operatorname{Mat}_{2\times2},+)\rtimes(\operatorname{GL}_{2},\cdot), \end{align} $$

where we use the matrix $M_{L}$ (resp., $M_{P}$) to denote the coordinate on the second (resp., third) factor, and more specifically write

(83)$$ \begin{align} M_{L} = \begin{pmatrix} z & w \\ \dot z & \dot w \end{pmatrix}, \qquad M_{P} = \begin{pmatrix} f & g \\ \dot f & \dot g \end{pmatrix}. \end{align} $$

We consider G as a semidirect product with respect to the left action of $\operatorname {GL}_{2}$ on $\operatorname {Mat}_{2\times 2}$ given by $M_{P}\cdot M_{L}=M_{P} M_{L}$ – that is, with the product rule

(84)$$ \begin{align} (M_{L},M_{P})\left(M_{L}^{\prime},M_{P}^{\prime}\right) = \left(M_{L}+M_{P} M_{L}^{\prime},M_{P} M_{P}^{\prime}\right). \end{align} $$

Let $\Sigma \subset {\mathbb C}_{\lambda }$ denote the set consisting of $0,1$, the critical values of $\lambda {\vert _{C}}$ and the points where $\lambda =x_{1}(\lambda )$ or $\lambda =x_{2}(\lambda )$ (compare [Reference Masser and Zannier38, p. 459], where a similar choice is made). We set $A_{\lambda }={\mathbb C}\setminus \Sigma $ and replace C by the part of C that lives over $A_{\lambda }$.

We will take our foliation ${\mathcal F}$ to be generated by a vector field

(85)$$ \begin{align} \xi := \tfrac{\partial}{\partial\lambda} + \tfrac{\partial x_{1}}{\partial\lambda}\tfrac{\partial}{\partial x_{1}}+\dotsb+\tfrac{\partial\dot w}{\partial\lambda}\tfrac{\partial}{\partial\dot w}, \end{align} $$

and will show how to express each of the $\tfrac {\partial }{\partial \lambda }$-derivatives of the coordinates as regular functions on ${\mathbb M}$.

We start with the coordinates of C. Since we assume $\lambda {\vert _{C}}$ is submersive, there is a unique lift of $\tfrac {\partial }{\partial \lambda }$, thought of as a section of $T(A_{\lambda })$, to a section $\xi _{C}$ of $T(C)$. The coordinates of this section are regular functions, and their height and degree can be readily estimated, for example, by writing out $T(C)$ explicitly as a Zariski tangent bundle. The $\tfrac {\partial }{\partial x_{j}}$- and $\tfrac {\partial }{\partial y_{j}}$-coordinates of $\xi _{C}$ give our $\tfrac {\partial x_{j}}{\partial \lambda }$ and $\tfrac {\partial y_{j}}{\partial \lambda }$.

We now turn to the equations for $(f,g)$. Recall that each elliptic period

(86)$$ \begin{align} I(\lambda) := \oint_{\delta(\lambda)}\omega, \quad \omega=\frac{\,\mathrm d x}y, \end{align} $$

where $\delta (\lambda )\in H_{1}(E_{\lambda })$ is a continuous family, satisfies the Picard–Fuchs equation

(87)$$ \begin{align} L I(\lambda)=0, \quad L=\lambda(1-\lambda)\tfrac{\partial^{2}}{\partial\lambda^{2}}+(1-2\lambda)\tfrac{\partial}{\partial\lambda}-\frac14. \end{align} $$

We encode the fact that f satisfies this second-order equation by requiring

(88)$$ \begin{align} \tfrac{\partial}{\partial\lambda} f = \dot f, \qquad \tfrac{\partial}{\partial\lambda} \dot f = \frac{(1/4)f - (1-2\lambda)\dot f}{\lambda(1-\lambda)}. \end{align} $$

Note that $\lambda (1-\lambda )$ is invertible on $A_{\lambda }$. We impose the same equations on $(g,\dot g)$.

Finally, to handle $z,w$, recall that each elliptic logarithm

(89)$$ \begin{align} \hat I(\lambda) := \int_{\infty}^{P(\lambda)} \omega \end{align} $$

satisfies an inhomogeneous Picard–Fuchs equation. More explicitly, applying the operator L to $\hat I(\lambda )$, we obtain by direct computation

(90)$$ \begin{align} L\hat I(\lambda) = B+\int_{\infty}^{P(\lambda)} L \omega =B+\int_{\infty}^{P(\lambda)} \frac12 \,\mathrm d\left(\frac{y}{(x-\lambda)^{2}}\right) = B+\frac12 \frac{y_{1}(\lambda)}{(x_{1}(\lambda)-\lambda)^{2}}, \end{align} $$

where B denotes the terms coming from the derivation of the boundary points – for example, $\omega (P(\lambda )^{\prime })$ for the first derivative. To make this computation explicitly, write $y:=\sqrt {x(x-1)(x-\lambda )}$ as a function of $x,\lambda $, express the integral as a path integral in the x-plane and use the usual derivation rules.

Denote the right-hand side of equation (90) by $R_{z}$. Then $R_{z}$ is a regular function on ${\mathbb M}$ by our definition of $A_{\lambda }$, and the explicit derivation readily shows that $\delta _{R}=\operatorname {poly}(\delta _{C})$. We may thus write the equations for z as

(91)$$ \begin{align} \tfrac{\partial}{\partial\lambda} z = \dot z, \qquad \tfrac{\partial}{\partial\lambda} \dot z = \frac{(1/4)z - (1-2\lambda)\dot z+R_{z}}{\lambda(1-\lambda)}. \end{align} $$

We impose a similar set of equations on $(w,\dot w)$, with the right-hand side $R_{w}$.

As a consequence of this construction, one leaf ${\mathcal L}_{0}$ of our foliation is given (locally) by the graph over C of $(f,g,z,w)$, where $f,g$ are taken to be the two generators of the lattice $E_{\lambda }$ and $z,w$ are taken to be elliptic logarithms of $P(\lambda ),Q(\lambda )$. As one analytically continues, this leaf ${\mathcal L}_{0}$ obtains other choices for the generators $f,g$ and the logarithms $z,w$.

We will also require a description of the remaining leaves. This is fairly simple to obtain: our equations for $f,g$ are equivalent to the Gauss–Manin linear equations $Lf=Lg=0$. For the standard leaf ${\mathcal L}_{0}$, these are taken to be two linearly independent solutions, and any other solution is obtained by replacing $M_{P}$ with $M_{P}G_{P}$ for some $G_{P}\in \operatorname {GL}_{2}({\mathbb C})$. Similarly, the equations for $z,w$ are equivalent to $Lz=R_{z},Lw=R_{w}$, and since $f,g$ form a basis of solutions of the homogeneous equations on any leaf, any other leaf with the same $f,g$ is obtained by replacing $M_{L}$ with $M_{L}+M_{P}G_{L}$ for some $G_{L}\in \operatorname {Mat}_{2\times 2}({\mathbb C})$. In other words, ${\mathcal F}$ is a flat structure of the principal G-bundle ${\mathbb M}$, where G acts on itself by multiplication on the right.

8.2 Degree and height bounds

We need two lemmas from [Reference Masser and Zannier38] on the degree and height of points $c\in C$ where either P or Q is torsion.

Lemma 30 [Reference Masser and Zannier38, Lemma 7.1]

Let $c\in C$ be such that $P(c)$ or $Q(c)$ is torsion of order n. Then

(92)$$ \begin{align} n \le \operatorname{poly}(\delta_{C},[{\mathbb Q}(\lambda(c)):{\mathbb Q}],h(\lambda(c))). \end{align} $$

Lemma 31 [Reference Masser and Zannier38, Lemma 8.1]

Let $c\in C$ be such that $P(c)$ or $Q(c)$ is torsion. Then

(93)$$ \begin{align} h(\lambda(c)) \le \operatorname{poly}(\delta_{C}). \end{align} $$

Recall that we defined $A_{\lambda }:={\mathbb C}\setminus \Sigma $ for some finite set $\Sigma $. For $\delta>0$, we define $\Lambda _{\delta }\subset A_{\lambda }$ as

(94)$$ \begin{align} \Lambda_{\delta} := \left\{ \lambda : \lvert\lambda\rvert<\delta^{-1},\ \forall\sigma\in\Sigma: \lvert\lambda-\sigma\rvert>\delta \right\}. \end{align} $$

We record a consequence of Lemma 31.

8.3 Setting up the domain for counting

Let $c\in C$ be such that $P(c),Q(c)$ are both torsion, and let n denote the maximum among their orders of torsion and $N(c):=[{\mathbb Q}(c):{\mathbb Q}]$. According to Lemma 31, we have $h(\lambda (c))=\operatorname {poly}(\delta _{C})$. Then by Lemma 32, at least half of the Galois orbit of $\lambda (c)$ lies in a set $\Lambda _{\delta }$ with some $\delta =2^{-\operatorname {poly}\left (\delta _{C}\right )}$. Moreover,

(95)$$ \begin{align} n=\operatorname{poly}(\delta_{C},N(c)) \end{align} $$

by Lemma 30.

We choose a collection of $\operatorname {poly}(\delta _{C})$ discs $D_{i}\subset A_{\lambda }$ such that

(96)$$ \begin{align} D_{i}^{1/4}\subset\Lambda_{\delta/2}, \quad \Lambda_{\delta}\subset\cup_{i} D_{i}. \end{align} $$

This is possible by elementary plane geometry using a logarithmic subdivision process. For example, it is enough to show that for each $r>0$, one can make such a choice of discs $D_{i}$ with $D_{i}^{1/4}\subset \Lambda _{r/2}$ to cover $\Lambda _{r}\setminus \Lambda _{2r}$. This is equivalent, after rescaling by r, to proving the same fact for $r=1$, and here the number of discs $D_{i}$ is easily seen to depend polynomially on the number of points in $\Sigma $.

In conclusion, we have proved the following:

8.4 Growth estimates for the leaf

We will consider the ball ${\mathcal B}$ in ${\mathcal L}_{0}$ corresponding to $D^{1/2}$ in the $\lambda $-coordinate, with the $P,Q$ coordinates corresponding to the branch of C chosen in Lemma 33. To apply Theorem 3 we must estimate the radius of the ball ${\mathbb B}_{R}$ containing this leaf. This can possibly be done by hand for the elliptic case treated in this paper, but we give a more general approach using growth estimates for differential equations which seems easier to carry out in more general settings.

We start with the coordinates $P,Q$. Since

(97)$$ \begin{align} \log \operatorname{dist}^{-1}\left(D^{1/2},\Sigma\right) = \operatorname{poly}(\delta_{C}), \end{align} $$

one can check that the coordinates $P,Q$ are bounded by $e^{\operatorname {poly}\left (\delta _{C}\right )}$. For instance, one may use the general effective bounds for semialgebraic sets proved in [Reference Basu and Roy5], though for this special case much more elementary arguments would suffice. We proceed to consider the remaining coordinates, which are given by (transcendental) elliptic integrals and require a more delicate approach.

Consider first the elliptic periods $f,g$. Fix some $\lambda _{0}\in {\mathbb C}\setminus \{0,1\}$, say $\lambda _{0}=1/2$. For some fixed choice of the integration paths staying away from $0,1,\lambda _{0},\infty $, we can directly estimate

(98)$$ \begin{align} \lvert f\rvert,\left\lvert\dot f\right\rvert,\lvert g\rvert,\left\lvert\dot g\right\rvert,\frac1{\operatorname{Im}(f/g)} < M_{0} \end{align} $$

at $\lambda =\lambda _{0}$ with $M_{0}$ an effective constant. Indeed, for such a path the integrals are nicely convergent, and one can approximate them up to any given precision effectively and find such a constant. Our goal is to deduce an effective estimate for these quantities after analytic continuation from $\lambda _{0}$ to $D^{1/2}$.

Recall that $f,g$ satisfy the Picard–Fuchs differential equation (87). Since this is a Fuchsian equation, the theorem of Fuchs [Reference Ilyashenko and Yakovenko31, Theorem 19.20] implies that $f,g$ (and their derivatives) grow polynomially as one approaches the singular locus of the operator (here $\lambda =0,1,\infty $) along geodesic lines on ${\mathbb P}^{1}$. In Appendix A we prove an effective version of this theorem. Specifically, using Theorem 9 we get for any $\lambda \in D^{1/2}$ the estimate

(99)$$ \begin{align} \lvert f\rvert,\left\lvert\dot f\right\rvert,\lvert g\rvert,\left\lvert\dot g\right\rvert < e^{\operatorname{poly}\left(\delta_{C}\right)}. \end{align} $$

Here we can and do assume, for instance, that we analytically continue the leaf from $\lambda _{0}$ to $D^{1/2}$ along some sequence of discs in ${\mathbb P}^{1}$ as explained in the comment following Theorem 9, staying at a distance $e^{-\operatorname {poly}\left (\delta _{C}\right )}$ from the singularities. We absorb $M_{0}$ in the asymptotic notation.

The estimate for $\operatorname {Im} (f/g)$ requires a different argument. The ratio of periods $f/g$ defines a map $D_{i}^{1/4}\to {\mathbb H}$, and by the Schwarz–Pick lemma we have

(100)$$ \begin{align} \operatorname{diam}_{H}\left( (f/g)\left(D^{1/2}_{i}\right) \right) \le \operatorname{diam}_{D^{1/4}_{i}} D^{1/2}_{i} = \operatorname{const}. \end{align} $$

Thus as we continue from $\lambda _{0}$ to $D^{1/2}$ along a finite sequence of discs $D_{i}$ the ratio $f/g$ varies by at most $\operatorname {poly}(\delta _{C})$ in ${\mathbb H}$. In particular, $\operatorname {Im}^{-1}(f/g)<e^{\operatorname {poly}\left (\delta _{C}\right )}$ in $D^{1/2}$.

The proof for the elliptic logarithms $z,w$ is similar to that for $f,g$. At the origin $\lambda _{1}$ of D, we choose $z,w$ to be given by an integral (89) with some standard choice of the path far from $0,1,\lambda _{1}$. Then as before, we can estimate $\lvert z\rvert ,\left \lvert \dot z\right \rvert ,\lvert w\rvert ,\left \lvert \dot w\right \rvert $ at $\lambda _{1}$ by $e^{\operatorname {poly}\left (\delta _{C}\right )}$. Our goal is to prove the same in $D^{1/2}$. Recall that $z,w$ satisfy a nonhomogeneous Picard–Fuchs equation (90). Here the right-hand side consists of the regular functions $R_{z},R_{w}$ on $A_{\lambda }$, which can be estimated from above by $\operatorname {poly}\left (\operatorname {dist}^{-1}(\lambda ,\Sigma )\right )$ in the same way as estimating the branches $P,Q$. Now using Theorem 9 again gives

(101)$$ \begin{align} \lvert z\rvert,\left\lvert\dot z\right\rvert,\lvert w\rvert,\left\lvert\dot w\right\rvert < e^{\operatorname{poly}\left(\delta_{C}\right)}. \end{align} $$

To conclude, we have the following:

Lemma 35. For any $\lambda \in D^{1/2}$, we have effective estimates

(102)$$ \begin{align} \lvert f\rvert,\left\lvert\dot f\right\rvert,\lvert g\rvert,\left\lvert\dot g\right\rvert,\lvert z\rvert,\left\lvert\dot z\right\rvert,\lvert w\rvert,\left\lvert\dot w\right\rvert,\frac1{\operatorname{Im}(f/g)} \le e^{\operatorname{poly}\left(\delta_{C}\right)}. \end{align} $$

In other words, the ball ${\mathcal B}$ constructed earlier is contained in ${\mathbb M}_{R}$ for $\log R=\operatorname {poly}(\delta _{C})$.

8.5 Setting up the counting

We will be interested in counting representations of $z,w$ as rational combinations of $f,g$. For this it will be convenient to expand our ambient space and foliation. Define

(103)$$ \begin{align} \hat{\mathbb M} := {\mathbb M} \times \operatorname{Mat}_{2\times2}({\mathbb C})_{U}, \end{align} $$

where U denotes the coordinate on the second factor in matrix form. We define the foliation $\hat {\mathcal F}$ on $\hat {\mathbb M}$ as the product of the foliation ${\mathcal F}$ on ${\mathbb M}$ with the full-dimensional foliation on the second factor (i.e., where a single leaf is the entire space). We will work with a ball $\hat {\mathcal B}$ of radius $\hat R$ contained in $\hat {\mathbb B}_{\hat R}$, where $\hat R$ will be suitably chosen later.

Consider the subvariety $V\subset \hat {\mathbb M}$ given by

(104)$$ \begin{align} V := \{ (z,w) = (f,g) U \}. \end{align} $$

Note that we do not restrict the entries of U to ${\mathbb R}$, as this would not be covered by Theorem 3. Let $\hat {\mathcal L}_{0}$ denote the lifting of the standard leaf to $\hat {\mathbb M}$. We will apply Theorem 3 with $\Phi :=U$. Let G act on $\operatorname {Mat}_{2\times 2}({\mathbb C})_{U}$ on the right by the formula

(105)$$ \begin{align} U\cdot(G_{L},G_{P}) = G_{P}^{-1}(U+G_{L}). \end{align} $$

Then the diagonal action on $\hat {\mathbb M}$ restricts to an action of G on V, and the map $\Phi $ is of course G-equivariant. We use this to deduce two functional transcendence statements for all leaves from the corresponding statements for the standard leaf.

Proof. If the map is not finite, then there is some $U_{0}$ whose fibre, that is, the set

(106)$$ \begin{align} \{\lambda\in A_{\lambda} : (z(\lambda),w(\lambda))=(f(\lambda),g(\lambda))U_{0}\}, \end{align} $$

is locally of dimension $1$. For the standard leaf $\hat {\mathcal L}_{0}$ this contradicts the functional transcendence lemma [Reference Masser and Zannier38, Lemma 5.1], as it implies that $z,w$ are algebraic over $f,g$. Since all other leaves are obtained by the G-action, and $\Phi $ is equivariant, the same follows for all other leaves.

Lemma 37. Let ${\mathcal W}\subset {\mathbb C}^{4}$ be a positive-dimensional algebraic block such that $\Sigma (V,{\mathcal W})$ meets a ball ${\mathcal B}\subset \hat {\mathcal L}_{0}$. Then ${\mathcal W}$ is contained in the affine linear space defined by

(107)$$ \begin{align} (z(\lambda_{0}),w(\lambda_{0})) = (f(\lambda_{0}),g(\lambda_{0})) U \end{align} $$

for some $\lambda _{0}\in \lambda ({\mathcal B})$.

We remark that Lemma 37 implies, in particular, that any block coming from the standard leaf can contain at most one real point: it is a product of two affine linear spaces with complex angle $(f(\lambda _{0}):g(\lambda _{0}))$. By G-equivariance, the blocks coming from other leaves are obtained as G-translates. For a sufficiently nearby leaf – that is, a G-translate sufficiently close to the origin – the angle is still complex. All such nearby blocks therefore also contain at most one real point. This will be crucial later in our application of Theorem 3.

8.6 Finishing the proof

We fix $\varepsilon =e^{-\operatorname {poly}\left (\delta _{C}\right )}$, to be suitably chosen later. Apply Theorem 3 to the ball $\hat {\mathcal B}$ with $V,\Phi $ constructed in Section 8.5. Recall that by Lemma 35 the ball ${\mathcal B}$ is contained in a ball of radius $e^{\operatorname {poly}\left (\delta _{C}\right )}$ in ${\mathbb M}$. The same lemma also shows that $\operatorname {Im}(f/g)\ge e^{-\operatorname {poly}\left (\delta _{C}\right )}$ uniformly on ${\mathcal B}$. We choose $\varepsilon $ small enough so that, by Lemma 37, any block coming from a leaf of distance $\varepsilon $ to ${\mathcal B}$ is still a product of affine spaces with complex angle (and in particular contains at most one real point). Setting $A:={\mathbb R}^{2}\cap \Phi \left (\hat {\mathcal B}^{2}\cap V\right )$, we have

(108)$$ \begin{align} \# A(1,h) = \operatorname{poly}\left(\delta_{C},\hat R,h\right). \end{align} $$

On the other hand we have the following:

Proof. Recall that $P(c),Q(c)$ are both torsion of order at most n, and the same is therefore true for each $c_{\sigma }$. In the equation

(109)$$ \begin{align} (z,w) = (f,g) U \end{align} $$

with real U, each $c_{\sigma }$ corresponds to a single value of U, with all coordinates rational and denominators not exceeding n. The claim follows once we prove that the entries of U are bounded from above by $e^{\operatorname {poly}\left (\delta _{C}\right )}$. This follows from Lemma 35. Indeed, we have, for example,

(110)$$ \begin{align} z = f u_{11} + g u_{12}, \end{align} $$

which can be interpreted as a pair of ${\mathbb R}$-linear equations on $u_{11},u_{12}$ by taking real and imaginary parts. The determinant of this system is at least $e^{-\operatorname {poly}\left (\delta _{C}\right )}$, because $\operatorname {Im}(f/g)$ is at least $e^{-\operatorname {poly}\left (\delta _{C}\right )}$, and the bounds on U follow easily.

In fact, the proof of Theorem 3 gives a bound $\operatorname {poly}(\delta _{C},h)$ not only for $\#A(1,h)$ but for the number of different points $\lambda \in D$ corresponding to points in A. A reader having forgotten the proof of Theorem 3 may instead appeal to Corollary 2, which shows that the number of different values of $\lambda $ corresponding to a single point of A is at most $\operatorname {poly}(\delta _{C},h)$. Indeed, for any fixed value $U=U_{0}$ in A, apply the corollary to the set

(111)$$ \begin{align} {\mathcal B}^{2}\cap V\cap\{U=U_{0}\}, \end{align} $$

using Lemma 36 to see that $\Sigma $ is empty in this case. It is in fact a simple exercise to remove the dependence on h in this bound, but as we do not need this, we leave it for the reader.

We are now ready to finish the proof. Recall that in Lemma 33 the number of points $c_{\sigma }$ is at least $N(c)/\operatorname {poly}(\delta _{C})$. Thus with $h=\operatorname {poly}(\delta _{C},\log n)$, we have

(112)$$ \begin{align} N(c)/\operatorname{poly}(\delta_{C}) \le \#A(1,h) \le \operatorname{poly}(\delta_{C},\log n)=\operatorname{poly}(\delta_{C},\log N(c)), \end{align} $$

where the last estimate is by equation (95). This immediately implies $N(c)=\operatorname {poly}(\delta _{C})$, as claimed.

9.1 The foliation

We follow Pila’s proof [Reference Pila45], which uses the uniformisation of modular curves by the j-function $j:\Omega \to {\mathbb C}$, where $\Omega \subset {\mathbb H}$ denotes the standard fundamental domain for the $\operatorname {SL}_{2}({\mathbb Z})$-action. To apply Theorem 3, we encode this graph as a leaf of an algebraic foliation. This could be done by replacing $j:{\mathbb H}\to {\mathbb C}$ with the $\lambda $-function $\lambda :{\mathbb H}\to {\mathbb C}$ and expressing the inverse $\tau :{\mathbb C}\to {\mathbb H}$ as the ratio of two elliptic integrals, which satisfy a Picard–Fuchs differential equation as discussed in Section 8.1. For variation here we use an alternative approach, expressing j directly as a solution of a Schwarzian-type differential equation (which was used for a similar purpose in [Reference Binyamini8]).

Recall that the Schwarzian operator is defined by

(113)$$ \begin{align} S(f) = \left(\frac{f^{\prime\prime}}{f^{\prime}}\right)^{\prime} - \frac12 \left(\frac{f^{\prime\prime}}{f^{\prime}}\right)^{2}. \end{align} $$

We introduce the differential operator

(114)$$ \begin{align} \chi(f) = S(f) + R(f) (f^{\prime})^{2}, \quad R(f) = \frac{f^{2}-1968f+2,654,208}{2f^{2}(f-1728)^{2}}, \end{align} $$

which is a third-order algebraic differential operator vanishing on Klein’s j-invariant j [Reference Masser33, p. 20]. As observed in [Reference Freitag and Scanlon24], it easy to check that the solutions of $\chi (f)=0$ are exactly the functions of the form $j_{g}(\tau ):=j\left (g^{-1}\cdot \tau \right )$ where $g\in \operatorname {PGL}_{2}({\mathbb C})$ acts on ${\mathbb C}$ in the standard manner.

The differential equation may be written in the form $f^{\prime \prime \prime }=A(f,f^{\prime },f^{\prime \prime })$, where A is a rational function. More explicitly, consider the ambient space $M:={\mathbb C}\times {\mathbb C}^{3}\setminus \Sigma $ with coordinates $(\tau ,y,\dot y,\ddot y)$, where $\Sigma $ consists of the zero loci of $y,y-1728$ and $\dot y$. In particular, we will write ${\mathbb C}_{y}:={\mathbb C}\setminus \{0,1728\}$. On M, the vector field

(115)$$ \begin{align} \xi := \tfrac{\partial}{\partial\tau} + \dot y \tfrac{\partial}{\partial y} + \ddot y\tfrac{\partial}{\partial\dot y}+A(y,\dot y,\ddot y)\tfrac{\partial}{\partial\ddot y} \end{align} $$

encodes the differential equation, in the sense that any trajectory is given by the graph of a function $j_{g}(\tau )$ and its first two derivatives.

We define our n-dimensional foliation ${\mathcal F}$ on the ambient space ${\mathbb M}:=M^{n}$ by taking an n-fold cartesian product of M with its one-dimensional foliation determined by the vector field $\xi $. We let ${\mathcal L}$ denote the standard leaf given by the product of the graphs of the j function, and note that any other leaf is obtained as a product of graphs of

(116)$$ \begin{align} (j(g_{1}\tau_{1}),\dotsc,j(g_{n}\tau_{n})), \quad (g_{1},\dotsc,g_{n})\in\operatorname{GL}_{2}({\mathbb C})^{n}. \end{align} $$

In fact, one may easily check that ${\mathcal F}$ is invariant under an appropriate algebraic action of $\operatorname {GL}_{2}({\mathbb C})^{n}$, where the action is trivial on y and is computed by the chain rule on $\dot y,\ddot y$.

9.2 Reduction to maximal special points

Denote by $V^{\mathrm {ws}}$ the weakly special locus of V – that is, the union of all weakly special subvarieties of V. In [Reference Binyamini8, Theorem 4] it is shown that one can effectively compute $V^{\mathrm {ws}}$, and in particular $\delta (V^{\mathrm {ws}})=\operatorname {poly}_{n}(\delta _{V})$. It is also shown that as a consequence of this, one can reduce the problem of computing all maximal special subvarieties to the problem of computing all special points $p\in V_{\alpha }\setminus V_{\alpha }^{\mathrm {ws}}$, for some auxiliary collection of varieties $V_{\alpha }\subset {\mathbb C}^{n_{\alpha }}$ with $n_{\alpha }\le n$ and $\sum _{\alpha } \delta (V_{\alpha })=\operatorname {poly}_{n}(\delta _{V})$.

We remark that even though in [Reference Binyamini8] only the bounds on the number and degrees of these auxiliary subvarieties are explicitly stated, the construction in fact yields an effective algorithm, as can be observed directly from the proof. We also note that the proof itself relies on differential algebraic constructions, though of a very different nature than in the present paper. In conclusion, it will suffice to prove Theorem 7 only for special points outside $V^{\mathrm {ws}}$.

9.3 A bound for maximal special points

We will use Theorem 3 to count maximal special points in V as a function of the discriminant. Toward this end, we define $\hat V:=\pi _{y}^{-1}(V)\subset {\mathbb M}$, where $\pi _{y}:{\mathbb M}\to {\mathbb C}_{y}^{n}$ is the projection to the coordinates $(y_{1},\dotsc ,y_{n})$. We let $\Phi =(\tau _{1},\dotsc ,\tau _{n})$. Note that $\Phi $ restricts to the germ of a finite map locally at every ${\mathcal L}_{p}$.

The following proposition will allow us to control the blocks coming from nearby leaves. We denote by $J:{\mathbb H}^{n}\to {\mathbb C}^{n}$ the n-fold product of the j-function.

Proposition 39. Let ${\mathcal B}$ be a $\xi $-ball in the standard leaf and ${\mathcal W}$ a positive- dimensional algebraic block coming from a nearby leaf at distance $\varepsilon $. Then

(117)$$ \begin{align} J({\mathcal W}\cap\pi_{\tau}({\mathcal B})) \subset N_{\delta}(V^{\mathrm{ws}}), \quad \delta=O_{\mathcal B}(\varepsilon), \end{align} $$

where $N_{\delta }(V^{\mathrm {ws}})$ denotes the $\delta $-neighbourhood of $V^{\mathrm {ws}}$ with respect to the Euclidean metric on ${\mathbb C}^{n}$.

Proof. If ${\mathcal W}$ comes from the standard leaf, then the modular Ax–Lindemann theorem established in [Reference Pila45] shows that ${\mathcal W}$ is contained in a pre–weakly special subvariety ${\mathcal W}^{\prime }$ with ${\mathcal W}^{\prime }\cap {\mathbb H}^{n}\subset J^{-1}(V)$. More accurately, some branch of a germ of ${\mathcal W}$ is contained in ${\mathcal W}^{\prime }$, but since ${\mathcal W}$ is irreducible, in fact ${\mathcal W}\subset {\mathcal W}^{\prime }$. Thus $J({\mathcal W}\cap {\mathbb H}^{n})\subset V^{\mathrm {ws}}$ by definition.

Recall that by formula (116), all other leaves are obtained by a $g\in \operatorname {GL}_{2}({\mathbb C})^{n}$-translate of the standard leaf. A tubular neighbourhood of ${\mathcal B}$ of radius $\varepsilon $ is thus generated by translates with $\left \lVert \operatorname {id}-g \right \rVert =O_{\mathcal B}(\varepsilon )$. If ${\mathcal W}$ comes from a leaf in this neighbourhood, then we have by the foregoing argument

(118)$$ \begin{align} J\left(g^{-1}({\mathcal W}\cap{\mathbb H}^{n})\right)\subset V^{\mathrm{ws}}. \end{align} $$

To finish, we should show that

(119)$$ \begin{align} J({\mathcal W}\cap\pi_{\tau}({\mathcal B})) \subset N_{O_{\mathcal B}(\varepsilon)}\left(J\left(g^{-1}({\mathcal W}\cap{\mathbb H}^{n})\right)\right). \end{align} $$

This follows at once because ${\mathcal B}$ is precompact. First, ${\mathcal W}\cap \pi _{\tau }({\mathcal B})$ is contained in a neighbourhood of $g^{-1}({\mathcal W}\cap {\mathbb H}^{n})$, since the derivative of the G-action is bounded in $\pi _{\tau }({\mathcal B})\subset {\mathbb H}^{n}$. And then $J({\mathcal W}\cap \pi _{\tau }({\mathcal B}))$ is contained in a neighbourhood of $J\left (g^{-1}({\mathcal W}\cap {\mathbb H}^{n})\right )$, since the derivative of J is bounded in $\pi _{\tau }({\mathcal B})$.

Let $p\in V\setminus V^{\mathrm {ws}}$ be a special point. We associate to p the complexity measure

(120)$$ \begin{align} \Delta(p) := \sum_{i=1}^{n} \lvert\operatorname{disc}(p_{i})\rvert, \end{align} $$

where $\operatorname {disc} p_{i}$ is the discriminant of the endomorphism ring of the elliptic curve corresponding to $p_{i}$. The Chowla–Selberg formula combined with standard estimates on L-functions imply

(121)$$ \begin{align} h(p) = O_{\varepsilon}(\Delta(p)^{\varepsilon}), \quad \text{for any }\varepsilon>0 \end{align} $$

(see, e.g., [Reference Habegger28, Lemma 4.1] and the estimate for the logarithmic derivative of the L-function in [Reference Tsimerman50, Corollary 3.3]).

Lemma 40. For any $\varepsilon>0$ and special point $p\in V\setminus V^{\mathrm {ws}}$,

(122)$$ \begin{align} \log \operatorname{dist}^{-1} (p_{\sigma},V^{\mathrm{ws}}) = \operatorname{poly}_{n}(\delta_{V}) O_{\varepsilon}(\Delta(p)^{\varepsilon}) \end{align} $$

holds for at least two-thirds of the Galois conjugates $p_{\sigma }$ of p.

Proof. This follows from $\delta (V^{\mathrm {ws}})=\operatorname {poly}_{n}(\delta _{W})$ and equation (121). For instance, choose a polynomial P with $h(P)=\operatorname {poly}_{n}(\delta _{V})$ vanishing on $V^{\mathrm {ws}}$ but not on p. Then $h(P(p))=\operatorname {poly}_{n}(\delta _{V},h(p))$, and in particular for two-thirds of the conjugates $p_{\sigma }$ we have

(123)$$ \begin{align} -\log \lvert p_{\sigma}\rvert=O_{\varepsilon}(\Delta(p)^{\varepsilon}), \qquad -\log \left\lvert P(p_{\sigma}) \right\rvert &= \operatorname{poly}_{n}(\delta_{V},O_{\varepsilon}(\Delta(p)^{\varepsilon})). \end{align} $$

On the other hand, for these conjugates if $d_{\sigma }:=\operatorname {dist} (p_{\sigma },V^{\mathrm {ws}})$, then by the mean value theorem (assuming, e.g., $d_{\sigma }<1$),

(124)$$ \begin{align} \left\lvert P(p_{\sigma}) \right\rvert \le \,\mathrm d_{\sigma}\cdot \max_{B_{p_{\sigma}}\left(d_{\sigma}\right)} \left\lVert \,\mathrm d P \right\rVert =e^{\operatorname{poly}_{n}\left(\delta_{V},\Delta(p)^{\varepsilon}\right)} \cdot d_{\sigma}. \end{align} $$

Taking logs and comparing the last two estimates implies equation (122) on $d_{\sigma }$.

Let $K\subset \Omega ^{n}\subset {\mathbb H}^{n}$ be a compact subset of the fundamental domain $\Omega ^{n}$ with

(125)$$ \begin{align} \operatorname{vol}(K) \ge \frac23 \operatorname{vol}(\Omega^{n}). \end{align} $$

According to Duke’s equidistribution theorem [Reference Duke23], for $\lvert \operatorname {disc}(p)\rvert \gg 1$ at least two-thirds of the conjugates $p_{\sigma }$ correspond to points in K. Thus at least one-third of the conjugates $p_{\sigma }$ both lie in K and satisfy Lemma 40. Call such conjugates good conjugates.

According to Brauer-Siegel [Reference Brauer18], the number of good conjugates is at least

(126)$$ \begin{align} \frac13 [{\mathbb Q}(p):{\mathbb Q}] \ge \Delta(p)^{c} \quad \text{for some }c>0. \end{align} $$

We also recall from [Reference Pila45] that for each $p_{\sigma }$, the corresponding preimage $\tau _{\sigma }\in \Omega ^{n}$ satisfies

(127)$$ \begin{align} [{\mathbb Q}(\tau_{\sigma}):{\mathbb Q}] \le 2n, \qquad H(\tau_{\sigma}) = \operatorname{poly}_{n}(\Delta(p)). \end{align} $$

We are now ready to finish the proof. Cover the part of ${\mathcal L}$ corresponding to K by finitely many unit balls ${\mathcal B}\subset {\mathcal L}$ and apply Theorem 3 with $\varepsilon _{0}$ to each of them. We choose

(128)$$ \begin{align} \log \varepsilon_{0}^{-1} = \operatorname{poly}_{n}(\delta_{V}) O_{\varepsilon}(\Delta(p)^{\varepsilon}) \end{align} $$

corresponding to the bound in Lemma 40, so that for any good conjugate $p_{\sigma }$ the $\varepsilon _{0}$-neighbourhood of $p_{\sigma }$ does not meet $V^{\mathrm {ws}}$. Then according to Corollary 39, none of the positive-dimensional blocks ${\mathcal W}_{\alpha }$ coming from nearby leaves at a distance $\varepsilon _{0}$ can contain the corresponding $\tau _{\sigma }$. Counting with $g=2n$ and $e^{h}=\operatorname {poly}_{n}(\Delta (p))$, we see that each good conjugate must come from a zero-dimensional ${\mathcal W}_{\alpha }$, and the number of good conjugates is therefore $\operatorname {poly}_{n}(\delta _{V},O_{\varepsilon }(\Delta (p)^{\varepsilon }))$. Choosing $\varepsilon $ sufficiently small compared to c and comparing this to formula (126), we conclude that $\Delta (p)<\operatorname {poly}_{n}(\delta _{V})$.

9.4 Computation of the maximal special points

To compute the finite list of maximal special points $p\in V\setminus V^{\mathrm {ws}}$, we start by enumerating all CM-points $p\in {\mathbb C}^{n}$ up to a given $\Delta =\delta _{V}$ (in polynomial time). For example, they are all obtained as images under $\pi $ of points $\tau $ in ${\mathbb H}^{n}$, whose coordinates are each imaginary quadratic with height $\operatorname {poly}_{n}(\Delta )$. It is simple to enumerate all such points; call them $\left \{\tau _{j}\right \}$.

For each $\tau _{j}$ and each equation $P_{k}=0$ defining V, we should check whether $P_{k}\left (\pi \left (\tau _{j}\right )\right )$ vanishes. Since $\delta _{\pi \left (\tau _{j}\right )}=\operatorname {poly}_{n}(\Delta )$, we have