1.1. The transcendental Bézout problem

The classical Bézout theorem states that the number of common zeros of n polynomials in n variables is generically bounded by the product of their degrees. The transcendental Bézout problem is concerned with the count of zeros of entire maps $\mathbb {C}^n \to \mathbb {C}^n$ . It is motivated by a number of influential mathematical ideas. The starting point is Serre’s famous G.A.G.A. [Reference Serre32], by now understood as a meta-mathematical principle stating that complex projective analytic geometry reduces to algebraic geometry. A prototypical result is a theorem of Chow [Reference Chow11], by which every closed complex submanifold of $\mathbb {C} P^n$ is necessarily algebraic (i.e., is given as the set of solutions of a system of polynomial equations). However, as the following simple example shows, Chow’s theorem fails in the affine setting.

Example 1.1. Consider an analytic function $f:\mathbb {C} \to \mathbb {C}$ given by

$$ \begin{align*} f(z)= e^z+1 = (e^x \cos y +1) +ie^x \sin y,\; z=x+iy. \end{align*} $$

Zeros of f form an infinite discrete set $\{(2k+1)\pi i, \; k \in \mathbb {Z}\}$ . It is not biholomorphically equivalent to any algebraic (and hence finite) proper subset of $\mathbb {C}$ .

In order to revive at least some parts of G.A.G.A. in the affine framework, one needs a substitute of the notion of the degree of a polynomial for entire mappings $f:\mathbb {C}^n \to \mathbb {C}^n$ . As it is put in [Reference Gol’dberg, Levin and Ostrovskii18], “A transcendental entire function that can be expanded into an infinite power series can be viewed as a “polynomial of infinite degree”, and the fact that the degree is infinite brings no additional information to the statement that an entire function is not a polynomial.” To this end, one introduces the maximum modulus

$$ \begin{align*} \mu(f,r) = \max _{z \in B_r} |f(z)|\;, \end{align*} $$

where $B_r$ stands for the closed ball of radius r. This quantity has at least two degree-like features. First, assume thatFootnote ¹

$$ \begin{align*}\limsup_{r \to \infty} \frac{\log \mu(f,r)}{\log r} < k+1.\end{align*} $$

Then, remarkably, f is a polynomial of the total degree $\leq k$ . This is a minor generalization of Liouville’s classical theorem. Thus, one can distinguish polynomials in terms of the maximum modulus.

In what follows, let $\zeta (f,r)$ denote the number of zeros of a continuous map $f: \mathbb {C}^n \to \mathbb {C}^n$ inside the ball $B_r$ .

The second feature of the maximum modulus of an entire function $f: \mathbb {C} \to \mathbb {C}$ is given by the following statement which readily follows from Jensen’s formula: if $f(0)\neq 0$ , then for every $a>1$ ,

(1.1) $$ \begin{align} \zeta(f,r) \leq C \log \mu(f,ar) \;\;\forall r>0\;, \end{align} $$

where C is a positive constant depending on a and $f(0)$ . For instance, in Example 1.1, both $\zeta $ and $\log \mu $ grow linearly in r.

These two features might have given a hope that $\log \mu (f,r)$ is an appropriate substitute of the degree for an entire map $f: \mathbb {C}^n \to \mathbb {C}^n$ (this was known as the transcendental Bézout problem). However, this analogy was overturned by Cornalba and Shiffman [Reference Cornalba and Shiffman13] who famously constructed, for $n=2$ , an entire map f with $\log \mu (f,r) \leq C_{\epsilon } r^\epsilon $ for every $\epsilon>0$ (and hence of growth order zero), with $\zeta (f,r)$ growing arbitrarily fast. As Griffiths wrote in [Reference Griffiths19] “This is the first instance known to this author when the analogue of a general result in algebraic geometry fails to hold in analytic geometry.”

1.2. Coarse zero count

One of the motivations for the present paper is to further explore the Cornalba–Shiffman example using the notion of coarse zero count introduced in [Reference Buhovsky, Payette, Polterovich, Polterovich, Shelukhin and Stojisavljević9], which is based on topological persistence. The idea, roughly speaking, is to discard the zeros corresponding to small oscillations of the map. It turns out that with such a count, we are able to get a Jensen-type estimate (1.1), albeit with a possibly non-sharp power of $\log \mu (f,r)$ in the right-hand side; see (1.2) below.

Given a continuous map $f: \mathbb {C}^n \to \mathbb {C}^n$ and positive numbers $\delta , r>0$ , we define the counting function $\zeta (f,r,\delta )$ of $\delta $ -coarse zeros of f inside a ball $B_r$ as the number of connected components of the set $f^{-1}(B_\delta )\cap B_r$ which contain zeros of f; see Figure 1.

Figure 1

Dots represent zeros of f, whereas shaded regions depict the set $f^{-1}(B_\delta )$ .

Theorem 1.2. For any analytic map $f: \mathbb {C}^n \to \mathbb {C}^n$ and any $a>1$ , $r>0$ , and $\delta \in (0, \frac {\mu (f, ar)}{2})$ , we have

(1.2) $$ \begin{align} \zeta(f,r,\delta) \leq C \left(\log\left(\frac{\mu(f, ar)}{\delta}\right)\right)^{2n-1}, \end{align} $$

where the constant C depends only on a and n.

This theorem is proved in Section 2.1. Its generalization in the framework of topological persistence is presented in Section 6.

Note that by Liouville’s theorem, unless f is constant, $\mu (f, ar)$ is unbounded. Therefore, for any given $\delta> 0,$ the condition $\delta \in (0,\mu (f,ar)/2)$ holds for all r large enough.

Remark 1.3. Consider a higher-dimensional generalization of Example 1.1: take an analytic map $f:\mathbb {C}^n \to \mathbb {C}^n$ given by

$$ \begin{align*} f(z_1,\dots,z_n) = \left(e^{z_1}+1,\ldots,e^{z_n}+1\right).\end{align*} $$

It is easy to see that $\log \mu (f,r)$ grows linearly in r and $\zeta (f,r,\delta )$ grows as $r^n$ when $r \to \infty $ , for $\delta $ sufficiently small. It would be interesting to understand whether the power of the logarithm in (1.2) is sharp or it can be improved, possibly, to n.

It follows from Theorem 1.2 that for the Cornalba-Shiffman example, the coarse count of zeros grows slower than any positive power of r; see Theorem 1.5 below for precise asymptotics.

Remark 1.4. Consider a function f of growth order $\leq \rho ;$ that is, for all $\epsilon>0$ , there exist positive constants $A_{\epsilon }, B$ , such that

$$\begin{align*}|f(z)| \leq A_{\epsilon} e^{B|z|^{\rho+\epsilon}}\end{align*}$$

everywhere. Then by (1.2), $\zeta (f,r,\delta )$ grows slower than $r^{(2n-1)\rho +\epsilon }$ for every $\epsilon>0.$ At the same time, it was shown in [Reference Carlson10, equation (1.9)] that for any $\alpha>0$ , $\zeta (f+c, r)$ grows slower than $r^{(2n-1)\rho +1 +\alpha }$ for almost all $c>0$ small enough. Although the growth rate in (1.2) is slightly sharper, it is interesting to note that the power $(2n-1)\rho $ appears in both bounds.

1.3. Cornalba–Shiffman example: a coarse perspective

Let us remind the Cornalba–Shiffman construction. Let $g:\mathbb {C} \to \mathbb {C}$ be given by

$$ \begin{align*} g(z)=\prod_{i=1}^\infty \left( 1- \frac{z}{2^i} \right).\end{align*} $$

For $k\geq 1$ an integer, let

$$ \begin{align*}g_k(z)=\frac{g(z)}{1-\frac{z}{2^k}} \end{align*} $$

be the function defined by the same product with k-th term excluded. All the infinite products converge uniformly on compact subsets of $\mathbb {C}$ , and hence, g and $g_k$ are holomorphic by Weierstrass’ theorem. For a positive integer c, we define a polynomial $P_c:\mathbb {C} \rightarrow \mathbb {C}$ as

$$ \begin{align*}P_c(w)=\prod_{j=1}^c \left( w- \frac{1}{j} \right).\end{align*} $$

Given a strictly increasing sequence of positive integers $\mathfrak {c}=\{c_i \}$ , $c_1<c_2<\ldots $ define $f:\mathbb {C}^2\to \mathbb {C}$ as

$$ \begin{align*}f(z,w)=\sum_{i=1}^\infty 2^{-c_i^2}g_i(z)P_{c_i}(w).\end{align*} $$

f converges uniformly on compact sets and is hence holomorphic by Weierstrass’ theorem in several variables. Finally, we define a map $F:\mathbb {C}^2 \rightarrow \mathbb {C}^2$ , $F(z,w)=(g(z),f(z,w)).$ As shown in [Reference Cornalba and Shiffman13], for all $\mathfrak {c}$ , F is of order zero. However, the zero set of F is given by

$$ \begin{align*}F^{-1}(0)=\left\lbrace \Big(2^i,\frac{1}{j} \Big) ~|~ i=1,2,\ldots; j=1,\ldots ,c_i \right\rbrace ,\end{align*} $$

as depicted in Figure 2. The dots represent zeros of F, and the number of zeros $\zeta (F,r)=\zeta (F,r,0)$ equals the number of dots inside the circle.

Figure 2

Classical count of zeros.

We now see that by taking $\mathfrak {c}$ , which increases sufficiently fast, $\zeta (F,r)$ can grow arbitrarily fast, which disproves the two-dimensional transcendental Bézout problem. More precisely, in [Reference Cornalba and Shiffman13], Cornalba and Shiffman made a remark that $c_i=2^{2^i}$ would suffice. Indeed, it is not difficult to check that for $\lambda>0$ , if $c_i=\lfloor 2^{\lambda i } \rfloor $ , then $\zeta (F,r)=\Theta (r^\lambda )$ (i.e., the order of growth of the number of zeros is $\lambda ),$ while for $c_i=2^{2^i}$ , $\log \zeta (F,r)= \Theta (r)$ and the order of growth of $\zeta (F,r)$ is infinite. Here and further on, we write $a(r)=\Theta (b(r))$ if $a(r)=O(b(r))$ and $b(r)=O(a(r))$ as $r \to \infty $ ; we will also write $a(r) \sim b(r)$ if $\lim _{r\to \infty } a(r)/b(r)=1$ .

Let us re-examine the same class of examples from the coarse point of view. The following result is proved in Section 3.

Theorem 1.5. Let $\mathfrak {c}$ be an arbitrary increasing sequence of positive integers. When $r\to +\infty $ it holds

$$ \begin{align*}\log \mu(F, r)=\Theta ((\log r)^2),\end{align*} $$

and for a fixed $\delta>0$

$$ \begin{align*}\zeta(F,r,\delta) \sim \log r.\end{align*} $$

Let us explain the geometric picture behind Theorem 1.5, while referring the reader to Section 3 for detailed proofs. For a fixed $\delta $ , we show that the set $\{|F|\leq \delta \}$ , while possibly being complicated for small radius r, stabilizes for large radii and can be described rather accurately. More precisely, we show that there exists $k_0$ , which depends only on $\delta $ , such that $\{|F|\leq \delta \}$ contains intervals $\{2^k \} \times [0,1]$ for all $k\geq k_0.$ Thus, for $k\geq k_0$ , the zeros on each of the intervals $\{2^k \} \times [0,1]$ are counted coarsely as one zero, and the coarse count increases at the rate $\log r.$ This implies $\zeta (F,r,\delta )=O(\log r).$ Furthermore, for $k\geq k_0$ , $\{ |F|\leq \delta \}$ will never intersect hyperplanes $H_k= \{ (w,z) ~|~ Re(z)=2^k+2^{k-1} \}.$ In other words, $\{|F|\leq \delta \}$ consists of parts contained between those hyperplanes and which contain intervals $\{2^k \} \times [0,1],$ as shown on Figure 3 (shaded regions represent the set $\{ |F|\leq \delta \}$ ). This implies that $\zeta (F,r,\delta )= \Theta (\log r),$ which we can improve to $\zeta (F,r,\delta ) \sim \log r$ as claimed by Theorem 1.5.

Figure 3

Coarse count of zeros.

Putting together Theorems 1.2 and 1.5, we come to the following conclusion. It follows from Theorem 1.2 for $n=2$ that

$$ \begin{align*}\zeta(F,r,\delta) \leq C_a (\log \mu(F, ar)-\log \delta)^3 .\end{align*} $$

On the logarithmic scale, this inequality tells us that for fixed a and $\delta $ ,

(1.3) $$ \begin{align} \log \zeta(F,r,\delta) = O \left(\log \log \mu(F,a r)\right), \end{align} $$

when $r\to +\infty .$ Theorem 1.5 implies that (1.3) is asymptotically sharp as

(1.4) $$ \begin{align} \log \zeta(F,r,\delta) = \Theta (\log \log \mu(F, ar)). \end{align} $$

In other words, Cornalba–Shiffman examples exhibit highly oscillatory behavior on small scales, which increases the count of zeros in an uncontrollable way and contradicts the transcendental Bézout problem. However, equality (1.4) shows that if we discard small oscillations, the same examples behave essentially as predicted by the coarse version of the transcendental Bézout problem.

1.4. Islands vs peninsulas

A connected component of $f^{-1}(B_\delta )\cap B_r$ is called an island if it is disjoint from $S_r=\partial B_r$ , and a peninsula otherwise. We prove the following result in Section 2.3: this is a combination of Corollary 2.11 and Corollary 2.9.

Let $\zeta ^0(f,r,\delta )$ denote the number of islands, and let $\tau (f,r,\delta )$ denote the total number of zeros of f with multiplicities contained in islands in $B_r.$ Since an island can contain more than one zero, clearly,

$$\begin{align*}\zeta^0(f,r,\delta) \leq \tau(f,r,\delta). \end{align*}$$

The following result is a consequence of Rouché’s theorem for analytic mappings; see Section 4.

Theorem 1.7. For all $a>1$ , $r>0$ and $\delta \in (0, \mu (f, ar)/2)$ ,

(1.5) $$ \begin{align} \tau(f,r,\delta) \leq C_1 \big(\log(\mu(f, ar)/\delta)\big)^{n}, \end{align} $$

where $C_1$ depends only on a and n.

Note that in view of Remark 1.3, estimate (1.5) is sharp.

In general, $\zeta (f,r,\delta )$ and $\zeta ^0(f,r,\delta )$ can behave rather differently. Indeed, this is the case for the Cornalba-Shiffman example, as we discuss below.

Let F be a Cornalba-Shiffman map defined previously. It is natural to ask what is the possible growth of the coarse count of islands $\zeta ^0(F,r,\delta ).$ We show that, as opposed to $\zeta (F,r,\delta )$ , $\zeta ^0(F,r,\delta )$ can grow arbitrarily slow, with an upper bound depending on $\mathfrak {c}.$ More precisely, in Section 4, we prove the following theorem.

Theorem 1.9. Let $\lambda , \delta>0$ , $l\geq 1$ an integer and write $\exp _2(x)=2^x.$ If $c_i=\lfloor \underbrace {\exp _2 \ldots \exp _2}_{l \text { times}} (\lambda i ) \rfloor $ , then there exists a constant $m_{l,\lambda ,\delta }$ such that for all $r\geq \underbrace {\exp _2 \ldots \exp _2}_{l+1 \text { times}}(1)$ , it holds

$$ \begin{align*} \zeta^0(F,r,\delta)\leq \frac{1}{\lambda}\underbrace{\log \ldots \log}_{l+1 \text{ times}} r + m_{l,\lambda,\delta}. \end{align*} $$

In particular, for $c_i=2^{2^i}$ as in [Reference Cornalba and Shiffman13], we have that $\zeta ^0(F,r,\delta )=O(\log \log \log r).$

From the geometric perspective, the slow growth of $\zeta ^0(F,r,\delta )$ is due to elongation of $\{|F|\leq \delta \}$ in the w-direction. Namely, as r increases, new groups of zeros of roughly the same modulus r appear, while the components of $\{|F|\leq \delta \}$ which contain these zeros grow in the w-direction faster than r (the diameter of their w-projection grows faster than r). Hence, it takes larger r for a component of $\{|F|\leq \delta \}$ to be fully contained in $B_r$ (i.e., to contribute to $\zeta ^0(F,r,\delta )).$

2.1. Proof of Theorem 1.2

In order to prove Theorem 1.2, we will approximate an analytic map by a polynomial. To this end, we first recall a version of the classical Cauchy estimates for complex analytic mappings.

Proposition 2.1. Let $f: \mathbb {C}^n \to \mathbb {C}^m$ be a complex analytic mapping, $a>1,$ and $R_k = f-p_k$ be the Taylor remainder for the approximation of f by the Taylor polynomial mapping $p_k$ at $0$ of degree $<k.$ Then for all $r>0, k \geq 0,$

$$\begin{align*}\mu(R_k,r) \leq C_{a} a^{-k} \mu(f,ar)\end{align*}$$

for the constant $C_{a} = \frac {a}{a-1}$ depending only on $a.$

We give a proof for clarity.

Proof. Let $v \in S^{2n-1} \subset \mathbb {C}^n$ and $u \in B_r(\mathbb {C}) \subset \mathbb {C}.$ Write a point z in $B_r = B_r(\mathbb {C}^n)$ as $z = u v.$ Then $R_k(z) = f(uv)-p_k(uv).$ Now take $v' \in S^{2m-1}$ , and set $g(u) = \langle f(uv),v'\rangle , q_k(u) = \langle p_k(uv), v' \rangle .$ Then $q_k$ is the Taylor polynomial of g of degree $<k$ , and $r_k = g - q_k$ is the corresponding Taylor remainder. It is enough to bound $r_k(u)$ uniformly in $v,v'$ for $u \in B_r(\mathbb {C}).$

Let $0<r<\rho .$ We use the following integral formula for the Cauchy remainder; see [Reference Ahlfors1, pages 125-126]: for $u,w$ with $|u|=r$ , $|w|=\rho $ ,

$$\begin{align*}r_k(u) = \frac{1}{2\pi i} \int_{S^1_{\rho}} g(w) \Big(\frac{u}{w}\Big)^k \frac{1}{w-u} dw. \end{align*}$$

Therefore,

$$\begin{align*}|r_k(u)| \leq \Big(\frac{r}{\rho}\Big)^k \frac{\rho}{\rho - r} \mu(g,\rho),\end{align*}$$

and picking $\rho = ar,$ we get

$$\begin{align*}|r_k(u)| \leq C_a a^{-k} \mu(g,ar) \leq C_a a^{-k} \mu(f,ar).\end{align*}$$

So taking maxima over $u, v$ and $v',$ we obtain

$$\begin{align*}\mu(R_k,r) \leq C_a a^{-k} \mu(f,ar).\\[-34pt] \end{align*}$$

Next, we highlight topological properties of polynomials needed for the proof of Theorem 1.2.

Last, we formulate a lemma which connects Proposition 2.2 to the coarse count of zeros of near-polynomial maps.

2.2. Higher-dimensional counts

A similar approach leads to a proof of the following more general statement, albeit with a slightly weaker exponent on the right-hand side. See also Section 6. Consider the invariants $\zeta _d(f,r,\delta )$ from Section 1.5.1.

Theorem 2.6. For any analytic map $f: \mathbb {C}^n \to \mathbb {C}^m$ , $m\leq n$ and any $a>1$ , $r>0$ , an integer $ 0 \leq d \leq 2n -1$ , and $\delta \in (0, \frac {\mu (f, ar)}{2})$ , we have

(2.1) $$ \begin{align} \zeta_d(f,r,\delta) \leq C \left(\log\left(\frac{\mu(f, ar)}{\delta}\right)\right)^{2n}, \end{align} $$

where the constant C depends only on a and n.

Proof. First, we notice that it is enough to prove the theorem in the case $m=n$ by the same reasoning as in Remark 2.5.

By Proposition 2.1, we can approximate $ f $ by a Taylor polynomial mapping $ p $ at $ 0 $ of degree $ <k $ such that $ |f-p| \leq C_{a} a^{-k} \mu (f,ar) $ on $ B_r $ . Here, we choose $ k $ to be the minimal positive integer such that $ C_a a^{-k} \mu (f,ar) < \delta /2 $ . Since $ ||f|-|p|| \leq |f-p| < \delta /2 $ , there exist natural maps

$$ \begin{align*}H_d(\{f=0\} \cap B_r) \xrightarrow{i_1} H_d(\{|p| \leq \delta/2\} \cap B_r) \xrightarrow{i_2} H_d(\{|f|\leq \delta \} \cap B_r).\end{align*} $$

It follows that

$$ \begin{align*} \zeta_d(f,r,\delta) &= \dim Im ( i_2 \circ i_1) \leq \dim H_d(\{|p|\leq \delta/2\} \cap B_r) \\ &= \dim H_d(\{ |p|^2 \leq \delta^2/4\} \cap B_r). \end{align*} $$

Identifying $\mathbb {C}^n=\mathbb {R}^{2n}$ , we have that $|p|^2$ is a real polynomial of degree $<2k$ and $\{ |p|^2 \leq \delta ^2/4\}\cap B_r$ is defined by two polynomial inequalities

$$ \begin{align*}\frac{\delta^2}{4}-|p|^2\geq 0 \text{ and } r-x_1^2-\ldots -x_{2n}^2\geq 0.\end{align*} $$

By [Reference Milnor27, Theorem 3], we have that $\dim H_d(\{ |p|^2 \leq \delta ^2/4\} \cap B_r)$ is bounded by $\frac {1}{2}(4k+2k)(3+2k)^{2n-1} $ , which finishes the proof.

2.3. Zeros and islands

In this section, we prove Theorem 1.6, which is a combination of Corollary 2.11 and Corollary 2.9. It is a structure theorem for islands of analytic mappings. We start with the following general result which is possibly known, but we could not locate it in the literature.

Proof. Write $g = u + iv,$ where $u, v$ are the real and imaginary part of g, respectively, and i is the imaginary unit. Then $h = |u|^2 + |v|^2.$ Now $d_q h = 2\langle u, d_q u \rangle + 2 \langle v, d_q v \rangle ,$ while $d_q g = d_q u + i d_q v.$ This shows that if q is a critical point of $h,$ then $g(q) \in \mathbb {C}^m$ is orthogonal to the image of $d_q g.$ Therefore, as $n \geq m,$ if $g(q) \neq 0,$ then the kernel $K = \ker d_q g$ is a nontrivial complex-linear subspace of $\mathbb {C}^n.$ Now, as $d_q g$ vanishes on $K,$ writing the second order Taylor approximation

$$\begin{align*}g(z) = g(q) + \frac{1}{2} d^2_q g(z-q, z-q) + o(|z-q|^2)\end{align*}$$

of g at $q,$ where $d^2_q g$ is the complex Hessian (or quadratic differential) of g at $q,$ we obtain that

$$\begin{align*}h(z) = h(q) + \mathrm{Re}\langle d^2_q g(z-q,z-q), g(q)\rangle + o(|z-q|^2)\end{align*}$$

is the second order Taylor approximation of h at $q,$ where the brackets denote the Hermitian inner product on $\mathbb {C}^m.$ Hence, the Hessian $d^2_q h$ is given on $a,b \in K$ by

$$\begin{align*}d^2_q h(a,b) = 2\, \mathrm{Re}\langle d^2_q g(a,b), g(q)\rangle.\end{align*}$$

Since $d^2_q g$ is a $\mathbb {C}^m$ -valued complex bilinear form, $\langle d^2_q g, g(q)\rangle $ is a $\mathbb {C}$ -valued complex bilinear form. Therefore, by a classical observation (see [Reference Milnor26, Assertion 1, p.39]), the quadratic form of $d^2_q h|_K$ has zero signature as the real part of a complex quadratic form. Therefore, it cannot be positive-definite. This contradicts the hypothesis that q is an interior non-degenerate minimum. Hence, $g(q) = 0.$

Proof. Denote by K the island of $\{ |g|\leq \delta \}$ with interior $V.$ First, we claim that $|g|$ is not constant on $K.$ To this end, let $q\in V$ , L a complex line passing through q parametrized by $z\in \mathbb {C}$ and denote by $u(z)=\sum _{j=1}^n |g_j(z)|^2$ the restriction of $|g|^2$ to $L.$ If $|g|$ was constant on K, u would be constant on a neighborhood of q in L and we would have that

$$ \begin{align*}0=\Delta u = 4 \sum_{j=1}^n \frac{\partial g_j}{\partial z} \frac{\partial \bar{g}_j }{\partial \bar{z}}= 4 \sum_{j=1}^n \Big{|} \frac{\partial g_j}{\partial z} \Big{|}^2.\end{align*} $$

This implies that each $g_j$ is constant on a neighborhood of q and thus constant on the whole $L\cap B_r.$ Since L was taken arbitrarily, we conclude that g is constant, which is impossible since $\{ |g| \leq \delta \}$ has an island by the assumption.

Now, assume by contradiction that $0<\min _K |g|< \max _K |g|=\delta .$ Let $0<2\varepsilon < \min \{ \min _K |g| , \max _K |g| - \min _K |g| \}$ , and assume that $g':K\to \mathbb {C}^m$ satisfies $|g-g'|<\varepsilon $ on $K.$ Let $q,q' \in K$ for which $|g(q)|=\min _K |g|$ , $|g'(q')|=\min _K |g'|.$ Since

$$ \begin{align*}|g(q')|<|g'(q')|+\varepsilon\leq |g'(q)| +\varepsilon <|g(q)| + 2 \varepsilon = \min_K |g| +2\varepsilon < \max_K |g|,\end{align*} $$

we have that $q' \in V$ because $|g|$ equals $\delta $ on $K\setminus V.$ Thus, $|g'|$ has an interior local minimum $q'$ . By means of a transversality argument, we show the following proposition at the end of this subsection.

Proposition 2.10. Assume K contains no zeros of $g.$ Then there exists an open set U, $K\subset U$ such that for every $\varepsilon>0$ , there exists a holomorphic function $g':U \to \mathbb {C}^m$ such that $|g-g'|<\varepsilon $ on U and $|g'|^2$ is Morse.

For $g'$ given by Proposition 2.10, $q'$ must be a zero by Proposition 2.8. However, since $2\varepsilon < \min _K |g|$ , we have that $0=|g'(q')|> |g(q')|-\varepsilon >\frac {1}{2} \min _K |g|>0$ , which is a contradiction.

We note that the condition $n \geq m$ is essential for Proposition 2.8 and Corollaries 2.9, 2.11, as can be seen from the mapping $g:B_r \to \mathbb {C}^2$ , $B_r \subset \mathbb {C}$ , $g(z) = (1,z).$ Similarly, so is non-degeneracy in Proposition 2.8, as shown by the example $g: B_r \to \mathbb {C}^2$ , $B_r \subset \mathbb {C}^2$ , $g(z,w) = (z,1-z).$

Proof of Proposition 2.10.

Denote by $g_z$ the complex differential of g with respect to $z=(z_1,\ldots , z_n)$ . Let U be such that the closure $\overline {U} \subset B_r$ and $|g|>c > 0$ , $|g_z| < C$ on U for some $c,C.$ We argue as follows in the spirit of Thom’s parametric transversality theorem.

The function $|g|^2= \bar {g}g$ is Morse on U whenever $\bar {g} g_z$ is transverse to $0$ as a mapping $U \to \mathbb {C}^n.$ Indeed, in this case by the proof of Proposition 2.8, at every critical point of $|g|^2,$ its Hessian will be the real part of a non-degenerate complex-valued symmetric bilinear form $2\bar {g} g_{zz},$ and as such, it will be non-degenerate.

Denote by $A_i$ the standard basis in the space ${\mathrm {Mat}}(m \times n,\mathbb {C})$ of $m \times n$ complex matrices. Choose $\alpha>0$ small enough, and consider an $\alpha $ -net $b_j$ in a compact in $\mathbb {C}^m$ containing $A_i \left (U \right )$ for all i. Put

$$ \begin{align*}v_{ij}(z) = A_iz-b_j\;.\end{align*} $$

We look at the perturbation

$$ \begin{align*}G(z,\epsilon) = g(z) + \sum \epsilon_{ij} v_{ij}(z)\;.\end{align*} $$

The derivative of $\bar {G}G_z$ over $\epsilon _{ij}$ is

(2.2) $$ \begin{align} A_i\bar{g} + \overline{(A_iz-b_j)}g_z\;. \end{align} $$

Fix any point $z \in U$ , and choose j so that the second summand in (2.2) is smaller than $C\alpha $ . At the same time, at least one of the components of $\bar {g}(z)$ has absolute value $> c'$ , where $c'$ depends only on c and n. Thus, the collection $\{A_i \bar {g}\}$ contains elements $c_1e_1, \dots , c_me_m$ , where $e_1, \dots , e_m$ is the standard basis of $\mathbb {C}^m$ , and $|c_i|> c'$ . As invertible matrices form an open set in ${\mathrm {Mat}}(m \times m, \mathbb {C}),$ after the perturbation $\overline {(A_iz-b_j)}g_z$ of norm at most $C\alpha $ , these vectors still span $\mathbb {C}^m$ , provided $\alpha $ is small enough. It follows that $\bar {G}G_z$ is transverse to $0$ . Thus, by Thom’s parametric transversality theorem, for almost all $\epsilon $ , we have that $|G_\epsilon |^2$ is Morse. Choosing $g' = G_{\epsilon }$ for $\epsilon $ sufficiently small then finishes the proof.

6.1. Persistence modules and barcodes

Recall that for a Morse function $f:M \to \mathbb {R}$ on a compact manifold and a coefficient field ${\mathbb {K}}$ , its barcode in degree ${q} \in \mathbb {Z}$ is a finite multi-set ${\mathcal {B}}_{q}(f;{\mathbb {K}})$ of intervals with multiplicities $(I_j,m_j),$ where $m_j \in \mathbb {N}$ and $I_j $ is finite, that is of the form $[a_j,b_j)$ or infinite, that is of the form $[c_j,\infty ).$ The number of infinite bars is equal to the Betti number $b_{q}(M;{\mathbb {K}}) = \dim H_{q}(M;{\mathbb {K}}).$

This barcode is obtained algebraically from the persistence module $V_{q}(f;{\mathbb {K}})$ consisting of vector spaces $V_{q}(f;{\mathbb {K}})^t = H_{q}(\{f \leq t\}; {\mathbb {K}})$ parametrized by $t\in \mathbb {R}$ and connecting maps $\pi ^{s,t}: V_{q}(f;{\mathbb {K}})^s \to V_{q}(f;{\mathbb {K}})^t$ induced by the inclusions $\{f \leq s \} \hookrightarrow \{f \leq t \}$ for $s\leq t.$ These maps satisfy the structure relations of a persistence module: $\pi ^{s,s} =\mathrm {id}_{V_{q}(f;{\mathbb {K}})^s}$ for all s and $\pi ^{s_2,s_3} \circ \pi ^{s_1,s_2} = \pi ^{s_1,s_3}$ for all $s_1 \leq s_2 \leq s_3.$ The total barcode of f is set to be

$$\begin{align*}{\mathcal{B}}(f;{\mathbb{K}}) = \sqcup_{{q} \in \mathbb{Z}} {\mathcal{B}}_{q}(f;{\mathbb{K}}),\end{align*}$$

where $\sqcup $ stands for the sum operation on multisets. This is the barcode of the persistence module

$$\begin{align*}V(f;{\mathbb{K}}) = \oplus_{{q} \in \mathbb{Z}} V_{q}(f;{\mathbb{K}}).\end{align*}$$

On a compact manifold M with boundary $\partial M$ and a Morse function $f: M \to \mathbb {R}$ in the sense of manifolds with boundary, we may define the persistence module and barcode of f as above.

One simple property of the barcode of this persistence module is that the number of bars in the barcode coincides with the number of their starting points. Another property is that the number of bars containing a given interval $[a,b]$ is $\dim {\mathrm {Im}}(\pi ^{a,b}).$

Recall that the length of a finite bar $[a,b)$ is $b-a$ and the length of an infinite bar $[c,\infty )$ is $+\infty .$ We require the following notion: for $\delta \geq 0,$ let ${\mathcal {N}}_{\delta }(f;{\mathbb {K}})$ denote the number of bars of length $>\delta $ in the barcode ${\mathcal {B}}(f;{\mathbb {K}}).$ Similarly, ${\mathcal {N}}_{{q},\delta }(f;{\mathbb {K}})$ is the number of bars of length $>\delta $ in the barcode ${\mathcal {B}}_{q}(f;{\mathbb {K}})$ and ${\mathcal {N}}_{\delta }(f;{\mathbb {K}}) = \sum _{q} {\mathcal {N}}_{{q},\delta }(f;{\mathbb {K}}).$ The definitions of ${\mathcal {N}}_{{q},\delta }(f;{\mathbb {K}})$ and ${\mathcal {N}}_{\delta }(f;{\mathbb {K}})$ extend to any continuous function f on a compact manifold with boundary, as explained in detail in [Reference Buhovsky, Payette, Polterovich, Polterovich, Shelukhin and Stojisavljević9, Section 2.2].

We refer to [Reference Polterovich, Rosen, Samvelyan and Zhang30] for a systematic introduction to persistence modules with a view toward applications in topology and analysis. The only result which we require here is the following direct consequence of the algebraic isometry theorem [Reference Bauer and Lesnick5] (see [Reference Polterovich, Rosen, Samvelyan and Zhang30, Theorem 2.2.8, Equation (6.4)]).

6.2. Bounds on barcodes of analytic functions

For a continuous map $f: \mathbb {C}^n \to \mathbb {C}^m,$ and $r>0,$ set

$$\begin{align*}{\mathcal{N}}_{\delta}(f,r) = {\mathcal{N}}_{\delta}( |f | \big |_{B_r}),\;\;\; {\mathcal{N}}_{q,\delta}(f,r) = {\mathcal{N}}_{q, \delta}(|f| \big |_{B_r}).\end{align*}$$

Consider the invariants $\zeta _d(f,r,\delta )$ from Section 1.5.1. Note that

$$\begin{align*}\zeta_d(f,r, \delta) \leq {\mathcal{N}}_{\delta}(f,r),\;\; \zeta(f,r,\delta) \leq {\mathcal{N}}_{0,\delta}(f,r),\end{align*}$$

as shown in Lemma 6.4 and Remark 6.5 in [Reference Buhovsky, Payette, Polterovich, Polterovich, Shelukhin and Stojisavljević9]. Using the terminology and methods of persistence, we can prove the following result, which therefore generalizes Theorems 1.2 and 2.6.

Theorem 6.2. For any analytic map $f: \mathbb {C}^n \to \mathbb {C}^m$ , $m\leq n$ and any $a>1$ , $r>0$ , and $\delta \in (0, \frac {\mu (f, ar)}{2})$ , we have

(6.1) $$ \begin{align} {\mathcal{N}}_{\delta}(f) \leq C \left(\log\left(\frac{\mu(f, ar)}{\delta}\right)\right)^{2n}, \end{align} $$

(6.2) $$ \begin{align} {\mathcal{N}}_{0, \delta}(f) \leq C \left(\log\left(\frac{\mu(f, ar)}{\delta}\right)\right)^{2n-1}, \end{align} $$

where the constant C depends only on a and n.

Since, given Theorem 6.1, this result reduces essentially to our proofs above and does not influence the main results of the paper, we only briefly sketch its proof.