Constructive Approximation

, Volume 35, Issue 3, pp 273–291

Approximation of Analytic Sets with Proper Projection by Algebraic Sets

Authors

    • Department of Mathematics and Computer ScienceJagiellonian University
Open AccessArticle

DOI: 10.1007/s00365-012-9156-0

Cite this article as:
Bilski, M. Constr Approx (2012) 35: 273. doi:10.1007/s00365-012-9156-0

Abstract

Let X be an analytic subset of U×Cn of pure dimension k such that the projection of X onto U is a proper mapping, where UCk is a Runge domain. We show that X can be approximated by algebraic sets. Next we present a constructive method for local approximation of analytic sets by algebraic ones.

Keywords

Analytic setAlgebraic setNash setApproximationHolomorphic chainMarkov’s inequalityNormalization

Mathematics Subject Classification (2010)

32C2541A1032E3014Q99

1 Introduction

The problem of polynomial approximation of holomorphic mappings has been thoroughly studied by several mathematicians (see a survey article [20] by N. Levenberg and the list of references therein).

In many cases, a holomorphic map f, for which approximations are looked for, is given implicitly; i.e., the graph of f (contained in some open set UCm) is defined by \(\operatorname{graph}(f)=\{F=0\}\), where F:UCq is another holomorphic map. This leads to a generalization of the above mentioned problem by asking whether analytic sets can be constructively approximated by algebraic sets. An important motivation for such a question comes from algebraic geometry, where computational methods have been rapidly developed in recent years (see the book [15] by G.-M. Greuel and G. Pfister and references therein). These methods could be transferred to analytic geometry if one could suitably approximate analytic sets by algebraic ones.

The aim of the present paper is to show that every purely k-dimensional analytic subset of U×Cn whose projection onto U is a proper mapping, where U is a Runge domain in Ck, can be approximated by purely k-dimensional algebraic sets (see Theorem 3.1). Here, by a Runge domain we mean a domain of holomorphy UCk such that every function \(f\in\mathcal {O}(U)\) can be uniformly approximated on every compact subset of U by polynomials in k complex variables (cf. [16], pp. 36, 52). The approximation is expressed in terms of the convergence of holomorphic chains; i.e., analytic sets are treated as holomorphic chains with components of multiplicity one (see Sect. 2.2). (In the considered context, the convergence of holomorphic chains could be equivalently replaced by the convergence of the currents of integration over analytic sets (see [12], pp. 141, 206–207).)

One of the direct consequences of Theorem 3.1 is the existence of local algebraic approximations for every purely dimensional analytic set. This is because, due to Noether normalization, for every point of such a set X there is a neighborhood U such that XU is with proper projection onto an open subset of some linear space of dimension dimX (see Corollary 3.7).

Proving Theorem 3.1, we considerably strengthen the results of [4, 5], where it is shown that purely k-dimensional analytic subsets of U×Cn with proper projection onto a Runge domain UCk can be approximated by complex Nash sets. The latter fact is the starting point for our considerations. More precisely, it allows us to reduce the proof of Theorem 3.1 to the case where the approximated object is a complex Nash set. The problem in the reduced case is solved by Proposition 3.2. This proposition states that algebraic approximation of such a set is possible under milder hypotheses than those of Theorem 3.1, and therefore it is of independent interest. (In particular, the assumption that the approximated Nash set is an analytic cover is not necessary here.)

In the last section, a constructive method for local approximation of analytic sets by algebraic ones is given. The method is based on three main tools. These are a theorem on constructive approximation of holomorphic maps whose domains are Markov’s sets by J.-P. Calvi and N. Levenberg [11] (see also [10]), (constructive) normalization of algebraic sets for which the reader is referred to a book [15] by G.-M. Greuel and G. Pfister, and constructive approximation of analytic sets by Nash ones as described in [5].

Let us finish the introduction by recalling that the number q of equations defining an analytic set X={xU:F1(x)=⋯=Fq(x)=0}, where U is an open subset of Cm, may be greater than the codimension of X in Cm. In particular, there exist analytic sets defined (even locally) only by such “overdetermined” systems of equations. (An example of an analytic set for which there does not exist a description such that the number of defining functions equals the codimension of the set is given in [3], pp. 58–59.) In such a case, algebraic sets of the form \(\{x\in U:\tilde{F}_{1}(x)=\cdots=\tilde{F}_{q}(x)=0\}\), where \(\tilde{F}_{i}\) is any polynomial approximating Fi, are not good approximations for X because their dimension is usually smaller than required. For this reason, the problem of algebraic approximation of analytic sets is not a straightforward generalization of the problem of polynomial approximation of holomorphic maps, and new methods are necessary.

The organization of this paper is as follows. In Sect. 2, preliminary material is presented. Section 3 contains the proofs of Theorem 3.1 and Proposition 3.2. In the last section, we give a constructive procedure for local approximation of analytic sets and illustrate it by an example.

2 Preliminaries

2.1 Nash Sets

Let Ω be an open subset of Cn, and let f be a holomorphic function on Ω. We say that f is a Nash function at x0Ω if there exist an open neighborhood U of x0 and a polynomial P:Cn×CC, P≠0, such that P(x,f(x))=0 for xU. A holomorphic function defined on Ω is said to be a Nash function if it is a Nash function at every point of Ω. A holomorphic mapping defined on Ω with values in CN is said to be a Nash mapping if each of its components is a Nash function.

A subset Y of an open set ΩCn is said to be a Nash subset of Ω if and only if for every y0Ω there exists a neighborhood U of y0 in Ω and there exist Nash functions f1,…,fs on U such that
$$Y\cap U=\bigl\{x\in U: f_1(x)=\cdots=f_s(x)=0\bigr\}.$$

The fact from [28] stated below explains the relation between Nash and algebraic sets.

Theorem 2.1

LetXbe an irreducible Nash subset of an open setΩCn. Then there exists an algebraic subsetYofCnsuch thatXis an analytic irreducible component ofYΩ. Conversely, every analytic irreducible component ofYΩis an irreducible Nash subset ofΩ.

2.2 Convergence of Closed Sets and Holomorphic Chains

Let U be an open subset in Cm. By a holomorphic chain in U, we mean the formal sum A=∑jJαjCj, where αj≠0 for jJ are integers and {Cj}jJ is a locally finite family of pairwise distinct irreducible analytic subsets of U (see [29], cp. also [2, 12]). The set ⋃jJCj is called the support of A and is denoted by |A|, whereas the sets Cj are called the components of A with multiplicities αj. The chain A is called positive if αj>0 for all jJ. If all the components of A have the same dimension n, then A will be called an n-chain.

Below we introduce the convergence of holomorphic chains in U. To do this, we first need the notion of the local uniform convergence of closed sets. Let Y,Yν be closed subsets of U for νN. We say that {Yν} converges to Y locally uniformly if:
  1. (1l)

    For every aY there exists a sequence {aν} such that aνYν and aνa in the standard topology of Cm.

     
  2. (2l)

    For every compact subset K of U such that KY=∅, it holds that KYν=∅ for almost all ν.

     
Then we write YνY. For details concerning the topology of local uniform convergence see [30].
We say that a sequence {Zν} of positive n-chains converges to a positive n-chain Z if:
  1. (1c)

    |Zν|→|Z|.

     
  2. (2c)

    For each regular point a of |Z| and each submanifold T of U of dimension mn transversal to |Z| at a such that \(\overline{T}\) is compact and \(|Z|\cap\overline{T}=\{a\}\), we have deg(ZνT)=deg(ZT) for almost all ν.

     
Then we write ZνZ. (By ZT we denote the intersection product of Z and T (cf. [29]). Observe that the chains ZνT and ZT for sufficiently large ν have finite supports and the degrees are well defined. Recall that for a chain \(A=\sum_{j=1}^{d}\alpha_{j}\{a_{j}\}\), \(\deg(A)=\sum_{j=1}^{d}\alpha_{j}\)).

2.3 Normalization of Algebraic Sets

Let us recall that every affine algebraic set, regarded as an analytic set, has an algebraic normalization (see [21], p. 471). Therefore (in view of the basic properties of normal spaces, see [21], pp. 337, 343), the following theorem, which will be useful in the proof of the main result, holds true.

Theorem 2.2

Let\(\tilde{Y}\)be an algebraic subset ofCm. Then there are an integernand an algebraic subsetZofCm×Cnwith\(\pi(Z)=\tilde{Y}\), whereπ:Cm×CnCmis the natural projection, satisfying the following properties:
  1. (01)

    Z, regarded as an analytic set, is locally irreducible.

     
  2. (02)

    π|Z:ZCmis a proper map.

     
  3. (03)

    \(\pi|_{Z\cap(\pi^{-1}(\tilde{Y}\setminus \operatorname{Sing}(\tilde{Y})))}:Z\cap(\pi^{-1}(\tilde{Y}\setminus \operatorname{Sing}(\tilde{Y})))\rightarrow\tilde{Y}\)is an injective map.

     

2.4 Runge Domains

We say that P is a polynomial polyhedron in Cn if there exist polynomials in n complex variables q1,…,qs and real constants c1,…,cs such that
$$P=\bigl\{x\in\mathbf{C}^n:\bigl|q_1(x)\bigr|\leq c_1,\ldots,\bigl|q_s(x)\bigr|\leq c_s\bigr\}.$$

The following lemma is a straightforward consequence of Theorem 2.7.3 and Lemma 2.7.4 from [16].

Lemma 2.3

LetΩCnbe a Runge domain. Then for everyΩ0Ω, there exists a compact polynomial polyhedronPΩsuch that\(\varOmega_{0}\Subset \operatorname{Int}P\).

Theorem 2.7.3 from [16] immediately implies the following:

Claim 2.4

LetPbe any polynomial polyhedron inCn. Then\(\operatorname{Int}P\)is a Runge domain inCn.

The following fact from [16] (Theorem 2.7.7, p. 55) will also be useful to us.

Theorem 2.5

Letfbe a holomorphic function in a neighborhood of a polynomially convex compact setKCn. Thenfcan be uniformly approximated onKby polynomials inncomplex variables.

3 Approximation of Analytic Sets

The following theorem is the first main result of this paper.

Theorem 3.1

LetUbe a Runge domain inCk, and letXbe an analytic subset ofU×Cnof pure dimensionkwith proper projection ontoU. Then there is a sequence {Xν} of algebraic subsets ofCk×Cnof pure dimensionksuch that {Xν∩(U×Cn)} converges toXin the sense of holomorphic chains.

The proof of Theorem 3.1 is based on two results. First, every purely dimensional analytic set with proper and surjective projection onto a Runge domain can be approximated by Nash sets (a precise statement will be recalled later). Second, every complex Nash set with proper projection onto a Runge domain can be approximated by algebraic sets as stated in the following:

Proposition 3.2

LetYbe a Nash subset ofΩ×Cof pure dimensionk<m, with proper projection ontoΩ, whereΩis a Runge domain inCm−1. Then there is a sequence {Yν} of algebraic subsets ofCm−1×Cof pure dimensionksuch that {Yν∩(Ω×C)} converges toYin the sense of holomorphic chains.

Proof of Proposition 3.2

Let \(\hat{l}\) be a positive integer, and let \(\|\cdot\|_{\hat{l}}\) denote a norm in \(\mathbf{C}^{\hat {l}}\). Set \(B_{\hat{l}}(r)=\{x\in\mathbf{C}^{\hat{l}}:\|x\|_{\hat{l}}<r\}\). For any analytic subset X of an open subset of \(\mathbf{C}^{\hat{l}}\), let X(q) denote the union of all q-dimensional irreducible components of X.

To prove the proposition, it is clearly sufficient to show that for every open Ω0Ω and for every real number r>0, the following holds:
  1. (*)

    There exists a sequence {Yν} of purely k-dimensional algebraic subsets of Cm−1×C such that {Yν∩(Ω0×B1(r))} converges to Y∩(Ω0×B1(r)) in the sense of chains.

     

Fix an open relatively compact subset Ω0 of Ω and a real number r>0. Let \(\tilde{\pi}:\mathbf{C}^{m-1}\times\mathbf{C}\rightarrow\mathbf{C}^{m-1}\) denote the natural projection.

Claim 3.3

There exists a purelyk-dimensional algebraic subset\(\tilde{Y}\)ofCm−1×Csuch thatY∩(Ω0×B1(r)) is the union of some of the analytic irreducible components of\(\tilde{Y}\cap(\varOmega_{0}\times B_{1}(r))\). Moreover, the mapping\(\tilde{\pi}|_{\tilde{Y}}:\tilde{Y}\rightarrow\mathbf{C}^{m-1}\)may be assumed to be proper.

Proof of Claim 3.3

By Lemma 2.3, we can fix a compact polynomial polyhedron PΩ such that Ω0ΓΩ, where \(\varGamma=\operatorname{Int}P\). The complex manifold \(\operatorname{Reg}_{\mathbf{C}}(Y\cap(\varGamma\times\mathbf{C}))\) is a semi-algebraic subset of R2m, hence it has a finite number of connected components. Consequently, Y∩(Γ×C) has finitely many analytic irreducible components. Therefore, by Theorem 2.1 there exists a purely k-dimensional algebraic subset \(\tilde{Y}\) of Cm−1×C such that Y∩(Γ×C) is the union of some of the analytic irreducible components of \(\tilde{Y}\cap(\varGamma\times\mathbf{C})\). Then, clearly, Y∩(Ω0×B1(r)) is the union of some of the analytic irreducible components of \(\tilde{Y}\cap(\varOmega_{0}\times B_{1}(r))\).

If the mapping \(\tilde{\pi}|_{\tilde{Y}}:\tilde{Y}\rightarrow\mathbf{C}^{m-1}\) is proper, then the proof is completed. Otherwise, using the facts that \(Y\cap(\varGamma\times\mathbf{C})\subset\tilde{Y}\cap(\varGamma\times\mathbf{C})\) and Ω0Γ, we show that there are a C-linear isomorphism Φ:CmCm, a Runge domain Ω1 in Cm−1, and a real number s>0 such that the following hold:
  1. (a)

    The projection of \(\varPhi(\tilde{Y})\subset\mathbf{C}^{m-1}\times\mathbf{C}\) onto Cm−1 is a proper mapping.

     
  2. (b)

    Φ(Ω0×B1(r))⊂Ω1×B1(s).

     
  3. (c)

    Φ(Y)∩(Ω1×B1(s)) is a Nash subset of Ω1×C whose projection onto Ω1 is a proper mapping.

     
  4. (d)

    Φ(Y)∩(Ω1×B1(s)) is the union of some of the irreducible components of \(\varPhi(\tilde{Y})\cap(\varOmega_{1}\times B_{1}(s))\).

     

If there exists a sequence {Zν} of purely k-dimensional algebraic subsets of Cm−1×C such that {Zν∩(Ω1×B1(s))} converges to Φ(Y)∩(Ω1×B1(s)) in the sense of chains, then Y∩(Ω0×B1(r)) is approximated, in view of (b), by {Φ−1(Zν)∩(Ω0×B1(r))}. Moreover, (c) implies that Ω1 and Φ(Y)∩(Ω1×B1(s)) taken in place of Ω and Y, respectively, satisfy the hypotheses of Proposition 3.2. Since, in view of (a) and (d), \(\varPhi(\tilde{Y})\) is a purely k-dimensional algebraic subset of Cm−1×C with proper projection onto Cm−1, containing Φ(Y)∩(Ω1×B1(s)), the proof of the claim is completed provided there are Φ, Ω1, and s satisfying (a), (b), (c), and (d).

Take Ω1 to be any Runge domain in Cm−1 with Ω0Ω1Γ. (The existence follows by Lemma 2.3 and Claim 2.4.) Now, since \(\dim(\tilde{Y})=k<m\), by the Sadullaev theorem (see [21], p. 389), the set \(\mathcal{S}_{\tilde{Y}}\) of one-dimensional linear subspaces l of Cm such that the projection of \(\tilde{Y}\) along l onto the orthogonal complement l of l in Cm is proper, is open and dense in the Grassmannian G1(Cm). Consequently, for every ε>0, there is \(l\in\mathcal{S}_{\tilde{Y}}\) so close to {0}m−1×C that there is a C-linear isomorphism Φε:CmCm transforming l,l onto {0}m−1×C, Cm−1×{0}, respectively, such that
$$\|\varPhi_{\varepsilon}-\operatorname{Id}_{\mathbf{C}^m}\|<\varepsilon,$$
where \(\operatorname{Id}_{\mathbf{C}^{m}}\) is the identity on Cm.
Clearly, Φ=Φε satisfies (a) (for every ε>0). Now, by the facts that Y is a Nash subset of Ω×C such that \(\tilde{\pi}|_{Y}:Y\rightarrow\varOmega\) is a proper map and Ω1Ω, there is a real number s>r such that
$$\bigl(\overline{\varOmega_1}\times\partial B_1(s)\bigr)\cap Y=\emptyset.$$
This implies that Φ=Φε satisfies (c) if ε is sufficiently small. Next, the facts Ω0Ω1 and s>r imply that Φ=Φε satisfies (b) for small ε. Finally, by Ω1Γ, we get \(\varPhi_{\varepsilon}^{-1}(\varOmega_{1}\times B_{1}(s))\subset\varGamma\times\mathbf{C}\) if ε is small enough. Therefore,
$$\varPhi_{\varepsilon}(Y)\cap\bigl(\varOmega_1\times B_1(s)\bigr)\subset\varPhi_{\varepsilon}(\tilde{Y})\cap\bigl(\varOmega_1\times B_1(s)\bigr),$$
which easily implies that (d) is satisfied with Φ=Φε. Thus the proof is completed. □

Proof of Proposition 3.2 (continuation)

By Theorem 2.2, there are an integer n and a locally irreducible (regarded as an analytic space), purely k-dimensional algebraic subset Z of Cm×Cn such that the restriction π|Z of the natural projection π:Cm×CnCm is a proper mapping, \(\pi(Z)=\tilde{Y}\) and
$$\pi|_{Z\cap(\pi^{-1}(\tilde{Y}\setminus \operatorname{Sing}(\tilde{Y})))}:Z\cap\bigl(\pi^{-1}\bigl(\tilde{Y}\setminus \operatorname{Sing}(\tilde{Y})\bigr)\bigr)\rightarrow\tilde{Y}$$
is an injective mapping.
We may assume that \((\tilde{Y}\setminus Y)\cap(\varOmega\times\mathbf{C})\neq\emptyset\), because otherwise
$$Y\cap\bigl(\varOmega_0\times B_1(r)\bigr)=\tilde{Y}\cap \bigl(\varOmega_0\times B_1(r)\bigr),$$
and one can take \(Y_{\nu}=\tilde{Y}\) for every νN.
Now, the Nash subsets E and F of Ω×C×Cn defined by
$$E=\bigl(Z\cap\pi^{-1}(Y)\bigr)_{(k)} \quad \mbox{and}\quad F=\bigl(Z\cap\pi^{-1}(\overline{\tilde{Y}\setminus Y})\cap\bigl(\varOmega \times\mathbf{C}\times\mathbf{C}^n\bigr)\bigr)_{(k)},$$
where the closure is taken in Ω×C, satisfy EF=∅. Indeed, if there exists some aEF, then Z (regarded as an analytic space) is not irreducible at a because
$$Z\cap\bigl(\varOmega\times\mathbf{C}\times\mathbf{C}^n\bigr)=E\cup F,$$
and dim(EF)<k. Consequently, the sets
$$\tilde{E}=E\cap\bigl(P\times\mathbf{C}\times\mathbf{C}^n\bigr) \quad \mbox{and}\quad \tilde{F}=F\cap\bigl(P\times\mathbf{C}\times\mathbf{C}^n\bigr)$$
also satisfy \(\tilde{E}\cap\tilde{F}=\emptyset\), where PΩ is a fixed compact polynomial polyhedron such that Ω0P. (The existence of P follows by Lemma 2.3.)

By Claim 3.3, we may assume that the mapping \(\tilde{\pi}|_{\tilde{Y}}\) is proper. Then the mapping \(\hat{\pi}|_{Z}:Z\rightarrow\mathbf{C}^{m-1}\) is proper as well, where \(\hat{\pi}=\tilde{\pi}\circ\pi\). This implies that both \(\tilde{E}\) and \(\tilde{F}\) are compact. Moreover, the mapping \((\hat{\pi},p)|_{Z}:Z\rightarrow\mathbf{C}^{m}\) is proper for every polynomial p:Cm×CnC.

Now the idea of the proof is to find a sequence {pν} of polynomials defined on Cm×Cn with the following properties:
  1. (0)

    \(\{p_{\nu}|_{\tilde{E}}\}\) converges uniformly to the mapping (x1,…,xm,y1,…,yn)↦xm.

     
  2. (1)

    \(\inf_{b\in\tilde{F}}|p_{\nu}(b)|>r\) for almost all ν.

     
Then we show that the sequence \(\{(\hat{\pi},p_{\nu})(Z)\}\) of purely k-dimensional algebraic subsets of Cm is as required in the condition (*): \(\{(\hat{\pi},p_{\nu})(Z)\cap(\varOmega_{0}\times B_{1}(r))\}\) converges to Y∩(Ω0×B1(r)) in the sense of chains.

Claim 3.4

There exists a sequence {pν} of polynomials inm+ncomplex variables, satisfying (0) and (1).

Proof of Claim 3.4

First, by the fact that \(\tilde{E}\cap\tilde{F}=\emptyset\), there is an open subset U of Cm×Cn such that U=U1U2, where U1,U2 are open subsets of Cm×Cn, U1U2=∅, and \(\tilde{E}\subset U_{1},\tilde{F}\subset U_{2}\).

Second, abbreviate (x,y)=(x1,…,xm,y1,…,yn) and note that the function f:UC defined by f(x,y)=xm on U1 and f(x,y)=r+1 on U2 is holomorphic.

Third, observe that, since Z is an algebraic subset of Cm×Cn with proper projection onto Cm, and \(\pi(Z)=\tilde{Y}\) is an algebraic subset of Cm−1×C with proper projection onto Cm−1, and P is a compact polynomial polyhedron in Cm−1, the union \(\tilde{E}\cup\tilde{F}\) is a compact polynomial polyhedron (and hence a polynomially convex compact set) in Cm×Cn. Indeed, there are closed bounded polydiscs P′⊂C,P″⊂Cn such that
$$\tilde{E}\cup\tilde{F}=Z\cap\bigl(P\times P'\times P''\bigr),$$
and the right-hand side of the latter equation is clearly described by a finite number of inequalities of the form |Q(x,y)|≤c, where Q is a complex polynomial and c is a real constant (possibly equal to zero).

Lastly, since f is a holomorphic function in a neighborhood of a polynomially convex compact set \(\tilde{E}\cup\tilde{F}\), it is sufficient to apply Theorem 2.5 to obtain a sequence {pν} of complex polynomials in m+n variables converging uniformly to f on \(\tilde{E}\cup\tilde{F}\). Clearly, every such sequence satisfies (0) and (1). □

Proof of Proposition 3.2 (end)

Let {pν} be a sequence of polynomials satisfying the assertion of Claim 3.4. We check that the sequence {Yν} defined by
$$Y_{\nu}=(\hat{\pi},p_{\nu}) (Z),\quad \mbox{for\ }\nu\in \mathbf{N},$$
satisfies the condition (*), which is sufficient to complete the proof of Proposition 3.2.

Every Yν is a purely k-dimensional algebraic subset of Cm because the mapping \((\hat{\pi},p_{\nu}):\mathbf{C}^{m}\times\mathbf{C}^{n}\rightarrow\mathbf{C}^{m}\) is polynomial, its restriction \((\hat{\pi},p_{\nu})|_{Z}\) is proper, and Z is a purely k-dimensional algebraic subset of Cm×Cn. Hence it remains to check that {Yν∩(Ω0×B1(r))} converges to Y∩(Ω0×B1(r)) in the sense of chains.

To see that {Yν∩(Ω0×B1(r))} converges to Y∩(Ω0×B1(r)) locally uniformly (cf. (1l) and (2l), Sect. 2.2), it is sufficient to observe that
$$Y_{\nu}\cap\bigl(\varOmega_0\times B_1(r)\bigr)=(\hat{\pi},p_{\nu}) (\tilde{E})\cap \bigl(\varOmega_0\times B_1(r)\bigr)$$
for ν large enough, whereas
$$Y\cap\bigl(\varOmega_0\times B_1(r)\bigr)=\pi(\tilde{E})\cap\bigl(\varOmega_0\times B_1(r)\bigr).$$
The first equality follows directly by (0), (1), and the definitions of \(\tilde{E}\) and \(\tilde{F}\). The latter one is an obvious consequence of the definition of \(\tilde{E}\). Moreover, (0) implies that \((\hat{\pi},p_{\nu})|_{\tilde{E}}\) converges to \(\pi|_{\tilde{E}}\) uniformly, which in turn implies that both (1l) and (2l) are satisfied.
To finish the proof, let us verify that {Yν∩(Ω0×B1(r))} and Y∩(Ω0×B1(r)) satisfy (2c) (of Sect. 2.2). Suppose that it is not true. Then there exist a k-dimensional C-linear subspace l of Cm−1×{0} and open balls C1,C2 in l,l respectively, where l denotes the orthogonal complement of l in Cm, such that \(\overline{C_{1}+C_{2}}\subset\varOmega_{0}\times B_{1}(r)\) and the following hold:
  1. (a)

    \(Y\cap(\overline{C_{1}}+\partial C_{2})=\emptyset\) and \((\overline{\tilde{Y}\setminus Y})\cap\overline{(C_{1}+C_{2})}=\emptyset\).

     
  2. (b)

    Every fiber of the projection of Y∩(C1+C2) onto C1 is 1-element.

     
  3. (c)

    The generic fiber of the projection of Yν∩(C1+C2) onto C1 is at least 2-element for infinitely many ν.

     

The existence of lCm (and C1, C2) as above is a direct consequence of the assumption that (2c) does not hold. Since the projection of \(\tilde{Y}\subset\mathbf{C}^{m-1}\times\mathbf{C}\) onto Cm−1 is a proper mapping, the subspace l can be chosen in such a way that it is contained in Cm−1×{0}.

The conditions (a), (b), and the facts that \(\pi|_{Z\cap(\pi^{-1}(\tilde{Y}\setminus \operatorname{Sing}(\tilde{Y})))}\) is injective and π|Z is proper imply that
$$Z\cap\bigl(({C_1}+{C_2})\times\mathbf{C}^n\bigr)=\operatorname{graph}(G),$$
where \(G\in\mathcal{O}({C_{1}},C_{2}\times\mathbf{C}^{n})\). Consequently, by (0) and the inclusion lCm−1×{0}, for ν large enough,
$$T_{\nu}:=(\hat{\pi},p_{\nu}) \bigl(\operatorname{graph}(G)\bigr)$$
is a k-dimensional analytic subset of C1+C2 whose projection onto C1 has 1-element fibers. Now, we show that, for almost all ν,
$$Y_{\nu}\cap(C_1+C_2)\subset T_{\nu},$$
which contradicts (c). The latter inclusion holds because, as observed previously, for large ν we have
$$Y_{\nu}\cap(C_1+ C_2)= (\hat{\pi},p_{\nu}) (\tilde{E})\cap(C_1+C_2).$$
Moreover, by (0), (a), and the fact that lCm−1×{0}, the following holds:
$$(\hat{\pi},p_{\nu}) (\tilde{E})\cap(C_1+C_2)=(\hat{\pi},p_{\nu}) \bigl(\tilde {E}\cap\bigl(({C_1}+{C_2})\times\mathbf{C}^n\bigr)\bigr),$$
which implies the inclusion as
$$\tilde{E}\cap\bigl(({C_1}+{C_2})\times \mathbf{C}^n\bigr)\subset \operatorname{graph}(G).$$

Thus {Yν∩(Ω0×B1(r))} and Y∩(Ω0×B1(r)) satisfy (2c), and the proof of Proposition 3.2 is completed. □

Proof of Theorem 3.1

Let us first recall that for analytic covers, there exist Nash approximations:

Theorem 3.5

LetUbe a connected Runge domain inCk, and letXbe an analytic subset ofU×Cnof pure dimensionkwith proper projection ontoU. Then for every open relatively compact subsetVofUthere is a sequence of Nash subsets ofV×Cnof pure dimensionkwith proper projection ontoV, converging toX∩(V×Cn) in the sense of holomorphic chains.

Papers [4, 5] contain detailed proofs of Theorem 3.5. Here let us just mention that this theorem is related to the problem of approximation of holomorphic maps between complex (algebraic) spaces, for which the reader is referred to [1, 9, 13, 14, 18, 19, 2527].

Let us return to the proof of Theorem 3.1. Fix a Runge domain U in Ck and an analytic subset X of U×Cn of pure dimension k with proper projection onto U. Clearly, in order to prove Theorem 3.1, it is sufficient to check the following:

Claim 3.6

For every openVU, there exists a sequence {Xν} of purelyk-dimensional algebraic subsets ofCk×Cnsuch that {Xν∩(V×Cn)} converges toX∩(V×Cn) in the sense of chains.

Let us prove the claim. Fix an open VU. Since, without loss of generality, V can be replaced by a larger relatively compact Runge subdomain of U (cf. Sect. 2.4), we may assume that V is a Runge domain. By Theorem 3.5, there is a sequence {Tν} of purely k-dimensional Nash subsets of V×Cn, with proper projection onto V, converging to X∩(V×Cn) in the sense of chains. (Formally in Theorem 3.5, U is assumed to be connected, but this assumption can be easily omitted by treating every connected component of U separately.)

For every ν, by Proposition 3.2 applied with Y=Tν, Ω=V×Cn−1, and m=n+k, there is a sequence {Yν,μ} of algebraic subsets of Ck×Cn of pure dimension k such that {Yν,μ∩(V×Cn)} converges to Tν in the sense of chains. Clearly, there is a function α:NN such that {Yν,α(ν)∩(V×Cn)} converges to X∩(V×Cn). Thus the proofs of Claim 3.6 and Theorem 3.1 are completed. □

An immediate consequence of Theorem 3.1 is the following:

Corollary 3.7

LetXbe a purelyk-dimensional analytic subset of some openΩCm. Then for everyaX, there are an open neighborhoodUofainΩand a sequence {Xν} of purelyk-dimensional algebraic subsets ofCmsuch that {XνU} converges toXUin the sense of holomorphic chains.

Proof

Every point of a purely k-dimensional analytic subset X of some open ΩCm has a neighborhood UΩ such that, after a linear change of the coordinates, the projection of XU onto an open ball in Ck×{0}mk is a proper mapping. Then, having applied Theorem 3.1, we obtain the corollary. □

4 Constructive Approximation of Analytic Sets

Let X be a purely k-dimensional analytic subset of an open set ΩCm such that 0∈X. Our aim is to construct a sequence {Xν} of purely k-dimensional algebraic subsets of Cm such that {XνΩ0} converges to XΩ0 in the sense of chains for some open neighborhood Ω0 of 0∈Cm and, moreover, every irreducible component E of XΩ0 is the limit of a sequence {Eν}, where Eν is an analytic irreducible component of XνΩ0. The construction is based on the proof of Theorem 3.1 and for that reason we need constructive versions of Theorem 2.5 and Theorem 2.2 involved in the proof (details below).

Let us first prepare the setup. Applying a linear change of the coordinates and shrinking Ω, if necessary, we assume that Ω=U×B, where U,B are open balls in Ck, Cn respectively, m=k+n, and the projection of X onto U is a proper map. Then, for some integer r, there exist polynomials \(p_{1},\ldots,p_{r}\in\mathcal{O}(U)[z_{1},\ldots,z_{n}]\), such that
$$X=\bigl\{(x,z_1,\ldots,z_n)\in U\times \mathbf{C}^n: p_1(x,z_1,\ldots ,z_n)=\cdots=p_r(x,z_1,\ldots,z_n)=0\bigr\}.$$
(For example, p1,…,pr may be the canonical defining functions of X, so one may also assume that Ω=U×Cn.) The functions p1,…,pr constitute the input for our method.

Before presenting the details, let us briefly sketch the idea of the construction of Xν. The first stage is to construct a purely k-dimensional algebraic subset \(\tilde{X}_{\nu}\) of Cm with the following properties. For some open connected neighborhood U0U of 0 (independent of ν), every irreducible component of X∩(U0×Cn) is approximated by some analytic irreducible component of \(\tilde{X}_{\nu}\cap (U_{0}\times\mathbf{C}^{n})\). Moreover, \(\tilde{X}_{\nu}\subset\mathbf{C}^{k}\times\mathbf{C}^{n}\approx\mathbf{C}^{m}\) is with proper projection onto Ck. Steps 1–4 of the procedure (written below) are responsible for this first stage. (Note that \(\tilde{X}_{\nu}\cap(U_{0}\times\mathbf{C}^{n})\) may contain irreducible components not corresponding to any irreducible components of X∩(U0×Cn), so in further steps \(\tilde{X}_{\nu}\) will have to be modified so that these additional components can be removed.)

The fact that \(\tilde{X}_{\nu}\), defined in Step 4, is with proper projection onto Ck is a consequence of \(\tilde{q}_{i}\) being unitary in zi for i=1,…,n. Let us explain that \(\tilde{X}_{\nu}\) has the other required properties. For i∈{1,…,n}, define
$$q_{i,\nu}(x,z_i):=z^{n_i}_i+z^{n_i-1}_ia_{i,1,\nu}(x)+\cdots + a_{i,n_i,\nu}(x),$$
where \(a_{i,1,\nu},\ldots,a_{i,n_{i},\nu}:C_{0}\rightarrow\mathbf{C}\) are Nash functions described in Step 3, and let qi(x,zi) be the polynomial defined in Step 1. Next we write
https://static-content.springer.com/image/art%3A10.1007%2Fs00365-012-9156-0/MediaObjects/365_2012_9156_Equw_HTML.gif
and take any open U0C0 containing zero. By Steps 1–3, we can invoke Theorem 3.1 of [5] (with X=W,Xν=Wν), which implies that every analytic irreducible component of W∩(U0×Cn) is approximated by some analytic irreducible component of Wν∩(U0×Cn) (if ai,j,ν is close to ai,j for every i∈{1,…,n},j∈{1,…,ni}). Now, X∩(U0×Cn)⊂W∩(U0×Cn), and \(W_{\nu}\cap(U_{0}\times\mathbf{C}^{n})\subset\tilde{X}_{\nu}\cap(U_{0}\times \mathbf{C}^{n})\), and all these sets are purely k-dimensional, which imply that \(\tilde{X}_{\nu}\) has all the required properties.

Given \(\tilde{X}_{\nu}\), we follow the proof of Proposition 3.2 with Ω=U0×Cn−1 and Y taken to be the union of those analytic irreducible components of \(\tilde{X}_{\nu}\cap(U_{0}\times\mathbf{C}^{n})\) for which Y approximates X∩(U0×Cn). More precisely, the situation is now simpler than in Proposition 3.2 because \(\tilde{X}_{\nu}\subset\mathbf{C}^{k}\times\mathbf{C}^{n}\) is with proper projection onto Ck. Consequently, no global coordinate changes are necessary. We just proceed by computing a purely k-dimensional algebraic subset Zν of Ck×Cn×Cd, for some dN, such that Zν is a locally irreducible analytic set, \(\pi|_{Z_{\nu}}\) is a proper map, and \(\pi(Z_{\nu})=\tilde{X}_{\nu}\), where π:Ck×Cn×CdCk×Cn is the natural projection (Step 5).

By the proof of Proposition 3.2, we know that Zν∩(U0×Cn×Cd)=EνFν, where Eν,Fν are purely k-dimensional Nash sets such that π(Eν) approximates X∩(U0×Cn) and Eν,Fν are separated from each other. In the last step, we replace π by a polynomial map πν such that Xν=πν(Zν) is an algebraic set approximating X as described in the first paragraph of this section (where Ω0=U0×Cn). To this end, πν is chosen in such a way that πν(Eν) is close to π(Eν), whereas πν(ZνEν)∩(U0×Bn(ν))=∅, where Bn(ν) is the ball in Cn centered at zero with radius ν.

Let us present the details. For hints on how the approximation can be carried out in practice, see the paragraphs following the procedure below. Let X,p1,…,pr be as in the second paragraph of this section.

Construction of Xν

  1. 1.
    For every i∈{1,…,n}, compute the unitary polynomial
    $$q_i=z_i^{n_i}+z_i^{n_i-1}a_{i,1}+\cdots +a_{i,n_i}\in\mathcal{O}(U)[z_i],$$
    with nonzero discriminant such that πi(X)={(x,zi)∈U×Ci:qi(x,zi)=0}, where \(\pi_{i}:U\times\mathbf{C}^{n}_{z_{1},\ldots,z_{n}}\rightarrow U\times\mathbf{C}_{z_{i}}\) is the natural projection.
     
  2. 2.
    For every i∈{1,…,n}, compute the discriminant \(\varDelta_{i}\in\mathbf{C}[a_{i,1},\ldots,a_{i,n_{i}}]\) of qi. Next find functions \(F_{1},\ldots,F_{\hat{s}}\) holomorphic in some neighborhood of 0∈Ck such that for every i∈{1,…,n},
    $$\varDelta_i\bigl(a_{i,1}(x),\ldots,a_{i,n_i}(x)\bigr)=F_1^{\beta_{i,1}}(x)\cdot\cdots \cdot F_{\hat{s}}^{\beta_{i,\hat{s}}}(x),$$
    where \(\beta_{i,1},\ldots,\beta_{i,\hat{s}}\) is a sequence of nonnegative integers; moreover, for every \(l\in\{1,\ldots,\hat{s}\}\), {Fl=0} has no multiple irreducible components, and for every \(j,l\in\{1,\ldots,\hat{s}\}\) with jl, {Fj=0} and {Fl=0} have no common irreducible components.
     
  3. 3.
    For every i∈{1,…,n}, j∈{1,…,ni}, introduce a new variable \(\tilde{z}_{i,j}\) and construct a nonzero polynomial \(P_{i,j,\nu}\in(\mathbf{C}[x])[\tilde{z}_{i,j}]\) of degree in \(\tilde{z}_{i,j}\) independent of ν, unitary in \(\tilde{z}_{i,j}\) such that there are Nash functions ai,j,ν, \(F_{1,\nu},\ldots, F_{\hat{s},\nu}\) approximating uniformly \(a_{i,j}, F_{1},\ldots, F_{\hat{s}}\) on an open neighborhood C0 of zero (independent of ν), and
    https://static-content.springer.com/image/art%3A10.1007%2Fs00365-012-9156-0/MediaObjects/365_2012_9156_Equz_HTML.gif
    for every xC0. Such a construction is described in detail in [5] (p. 327).
     
  4. 4.
    For every i∈{1,…,n}, let \(\tilde{q}_{i}\) be the polynomial obtained by replacing every coefficient ai,j of qi (except for the coefficient 1 standing at \(z_{i}^{n_{i}}\)) by the variable \(\tilde{z}_{i,j}\) (appearing in Pi,j,ν). Now let \(\tilde{X}_{\nu}\) be the image of the projection of the algebraic set
    $$V_{\nu}=\{\tilde{q}_i=0, P_{i,j,\nu}=0, \mbox{ for }i=1,\ldots,n \mbox{ and } j=1,\ldots, n_i\}$$
    onto \(\mathbf{C}^{k}_{x}\times\mathbf{C}^{n}_{z_{1}\ldots z_{n}}\). Hence the equations defining \(\tilde{X}_{\nu}\) can be obtained by eliminating \(\tilde{z}_{i,j}\)’s from the equations defining Vν.
     
  5. 5.

    For some dN, compute a purely k-dimensional algebraic subset Zν of Ck×Cn×Cd locally irreducible as an analytic set such that \(\pi|_{Z_{\nu}}\) is a proper map and \(\pi(Z_{\nu})=\tilde{X}_{\nu}\), where π:Ck×Cn×CdCk×Cn is the natural projection. Let U0C0 be an open polydisc containing 0∈Ck, and let \(\overline{U_{0}}\) be its closure in C0. Then \(Z_{\nu}\cap(\overline{U_{0}}\times\mathbf{C}^{n}\times\mathbf{C}^{d})=\tilde {E_{\nu}}\cup \tilde{F_{\nu}}\), where \(\tilde{E_{\nu}}, \tilde{F_{\nu}}\) are compact sets such that \(\tilde{E_{\nu}}\cap\tilde{F_{\nu}}=\emptyset\). Moreover, \(E_{\nu}={\tilde{E}_{\nu}}\cap(U_{0}\times\mathbf{C}^{n}\times\mathbf{C}^{d})\) is a purely k-dimensional Nash set such that π(Eν) approximates X∩(U0×Cn).

     
  6. 6.
    Compute a polynomial map Qν:Ck×Cn×CdCn approximating uniformly the natural projection \(\check{\pi}:\mathbf{C}^{k}\times\mathbf{C}^{n}\times\mathbf{C}^{d}\rightarrow \mathbf{C}^{n}\) on \(\tilde{E_{\nu}}\) such that
    $$\inf_{x\in\tilde{F_{\nu}}}\bigl\|Q_{\nu}(x)\bigr\|_{\mathbf{C}^n}>\nu.$$
    Next compute \(X_{\nu}=(\acute{\pi}, Q_{\nu})(Z_{\nu})\), where \(\acute{\pi}:\mathbf{C}^{k}\times\mathbf{C}^{n}\times\mathbf{C}^{d}\rightarrow \mathbf{C}^{k}\) is the natural projection.  □
     

Before discussing how this construction could be carried out in practice, let us note that not every analytic function can constitute (a part of) input data. Only objects which can be encoded as finite sequences of symbols can be considered. A large class of sets for which approximation could be useful are algebraic (or Nash) sets described by polynomials of very high degrees. Then the task would be to find approximations of such sets described by polynomials of lower degrees. Also in many cases in which analytic sets are described by (compositions) of elementary analytic functions whose properties we know, the construction could be carried out quite fast.

In general, one could consider the model in which for every function f depending on the variables x1,…,xk, there is a finite procedure \(\operatorname{Expand}_{f}\) which for every tuple (n1,…,nk)∈Nk returns the coefficient of the Taylor expansion of f around zero, standing at \(x_{1}^{n_{1}}\cdot\cdots\cdot x_{k}^{n_{k}}\). The input data corresponding to the function f consist of the procedure \(\operatorname{Expand}_{f}\), the size of the polydisc neighborhood Uf of zero on which the Taylor expansion of f is convergent. Furthermore, |f| is assumed to be bounded on Uf, and we know the bound Mf.

Observe that having input data for two functions f,g, we can recover the corresponding data for f+g, fg, and \(\frac{1}{f}\) (the latter if f(0)≠0). If we could test whether f is identically zero, then we could carry out all the first three steps of the construction in which transcendental objects appear. (Step 2 requires some further explanations given below. For the moment, let us only note that, clearly, under these assumptions we would also have procedures for expansions of functions appearing in the Weierstrass preparation theorem).

Unfortunately, given only \(\operatorname{Expand}_{f}\), Uf, Mf, it is not possible to test in a finite number of steps whether f=0. But for every ε>0, it is possible to check whether \(\sup_{U_{f}}|f|<\varepsilon\); hence, we can pick small ε>0 at the beginning and every time \(\sup_{U_{f}}|f|<\varepsilon\) assume that f=0. Of course, if f≠0, then the output may not be correct. If we have some extra information about X which allows us to exclude incorrect outputs Xν, then we can repeat the procedure with smaller ε. In such a general model, however, the existence of “badly conditioned” problems seems to be inevitable.

Carrying out the construction in Steps 2, 5, and 6 requires further explanation. First, to compute \(F_{1},\ldots,F_{\hat{s}}\) in Step 2, we may assume that in some neighborhood of zero for every i∈{1,…,n}, \(\varDelta_{i}(a_{i,1}(x),\ldots ,a_{i,n_{i}}(x))=W_{i}(x)H_{i}(x)\), where Hi is a holomorphic function, Hi(0)≠0 and Wi is the unitary polynomial in xk with holomorphic coefficients depending on x′=(x1,…,xk−1) vanishing at 0∈Ck−1. (Otherwise we apply a generic linear change of the variables in a neighborhood of 0∈Ck and the Weierstrass preparation theorem.) We may write F1=H1,…,Fn=Hn. Now we are left to find holomorphic functions \(F_{n+1},\ldots,F_{\hat{s}}\), whose zero sets have the properties specified in Step 2, such that for every i∈{1,…,n},
$$W_i=F_{n+1}^{\beta_{i,n+1}}\cdot\cdots\cdot F_{\hat{s}}^{\beta_{i,\hat{s}}},$$
where \(\beta_{i,n+1},\ldots,\beta_{i,\hat{s}}\) is a sequence of nonnegative integers. Since every Wi is a unitary polynomial in xk (with holomorphic coefficients), we can obtain \(F_{n+1},\ldots,F_{\hat{s}}\) by applying the division algorithm for polynomials to the Wi’s and their (higher order) partial derivatives with respect to xk.

In Step 5, Zν can be effectively obtained, for example, by computing the normalization of \(\tilde{X}_{\nu}\) (see [15] for the algorithm of normalization; the fact that the algebraic set Zν normalizing \(\tilde{X}_{\nu}\) is a locally irreducible analytic set follows by the standard properties of normal spaces, see [21], pp. 337, 343, 471).

The fact that Step 6 is constructive is a direct consequence of the existence of an effective procedure for computing the polynomial map Qν. Let K1,K2 be any closed bounded neighborhoods of \(\tilde{E_{\nu}}, \tilde{F_{\nu}}\), respectively, such that K1K2=∅. Observe that any polynomial map Qν sufficiently close (on K1K2) to the holomorphic map f:K1K2Cn given by \(f|_{K_{1}}=\check{\pi}|_{K_{1}}\) and \(f|_{K_{2}}=c\), where cCn, \(\|c\|_{\mathbf{C}^{n}}>\nu\), is good for our purpose. Hence what we need is to construct K1K2 and approximate f.

The union K1K2 will be constructed in such a way that it satisfies Markov’s inequality. Then holomorphic functions defined on K1K2 can be constructively approximated by polynomials (see [11]). More precisely, an approximating polynomial \(\tilde{g}\) (of a given degree) for a holomorphic function \(g\in\mathcal{O}(K)\), where K is a Markov’s set, can be taken to be the one minimizing the sum \(\sum_{c\in\tilde{K}}|g(c)-\tilde{g}(c)|^{2}\), where \(\tilde{K}\) is a suitably chosen finite subset of K. (The cardinality of \(\tilde{K}\) depends on the exponent in Markov’s inequality.) For details, the reader is referred to [11]. This method is related to the concept of using the Lagrange interpolating polynomials to approximate holomorphic functions (see [68, 24]).

As for constructing K1K2, one can take a special polynomial polyhedron, i.e., a set of the form
$$\bigl\{u\in\mathbf{C}^{k}\times\mathbf{C}^n\times \mathbf{C}^d: \bigl|g_1(u)\bigr|\leq1,\ldots,\bigl|g_{k+n+d}(u)\bigr|\leq1\bigr\},$$
where g1,…,gk+n+d are complex polynomials in k+n+d variables such that \(\{u\in\mathbf{C}^{k}\times\mathbf{C}^{n}\times\mathbf{C}^{d}:\hat {g}_{1}(u)=\cdots=\hat{g}_{k+n+d}(u)=0\}=\{0\}^{k+n+d}\), where \(\hat{g}_{1},\ldots,\hat{g}_{k+n+d}\) are homogeneous polynomials with \(\deg(g_{i})=\deg(\hat{g}_{i})\) and \(\deg(g_{i}-\hat{g}_{i})<\deg(g_{i})\) for i=1,…,k+n+d. Using the techniques of [22] (pp. 369–374), one can construct a special polynomial polyhedron \(\mathcal{P}\) such that \(\tilde{E_{\nu}}\cup\tilde{F_{\nu}}\subset \mathcal{P}\) and \(\tilde{E_{\nu}}\cup\tilde{F_{\nu}}\) is approximated (in the sense of the Hausdorff distance) by the union of some connected components of \(\mathcal{P}\). When this approximation is close enough, then \(\mathcal{P}\) decomposes as \(\mathcal{P}=K_{1}\cup K_{2}\) where K1,K2 are compact sets such that \(\tilde{E_{\nu}}\subset K_{1}\), \(\tilde{F_{\nu }}\subset K_{2}\), and K1K2=∅. Let us recall that every special polynomial polyhedron satisfies Markov’s inequality. Moreover, for such a set, the formula for the Siciak extremal function is known (see [17], p. 37), which allows one to compute the Markov’s exponent (see [20], p. 129). (For other examples of Markov’s sets, see [23].)

Finally, let us consider the following:

Example

Define
$$Y=\bigl\{(x,y,z,w)\in\mathbf{C}^4:z^5-2f(x,y)e^{(\frac{x+y^2}{10})^{30}}=w^5+g(x,y)w-2f(x,y)=0\bigr\},$$
where
https://static-content.springer.com/image/art%3A10.1007%2Fs00365-012-9156-0/MediaObjects/365_2012_9156_Equaf_HTML.gif
For any r>0, put Kr={xC:|x|<r}. One can easily check that \(\tilde{Y}=Y\cap(K_{1}\times K_{1}\times K_{1.4}\times K_{1.4})\) is a purely 2-dimensional analytic subset of K1×K1×K1.4×K1.4 with proper projection onto K1×K1, whose generic fiber over K1×K1 has 25 points. As we shall see, \(\tilde{Y}\) is reducible.
Let X be the irreducible component of \(\tilde{Y}\) such that
https://static-content.springer.com/image/art%3A10.1007%2Fs00365-012-9156-0/MediaObjects/365_2012_9156_Equag_HTML.gif
where for any BCn and a=(a1,…,an)∈Cn, \(\operatorname{dist}(B,a)=\inf\{\|a-b\|:b\in B\}\), and ∥a∥=maxi=1,…,n|ai|. We shall see subsequently that X has 5 points in the generic fiber over K1×K1.
Our aim is to construct an algebraic subset X1 of C4 approximating X. More precisely, we shall construct p1,p2,p3C[x,y,z,w] such that
$$X_{1}=\bigl\{(x,y,z,w)\in\mathbf {C}^4:p_1(x,y,z,w)=p_2(x,y,z,w)=p_3(x,y,z,w)=0\bigr\},$$
and \(\operatorname{dist}({X_{1}}\cap T,{X}\cap T)<5\cdot10^{-4}\), where T=K1×K1×K1.14×K1.4 and \(\operatorname{dist}(M,N)=\max\{\sup_{m\in M}\operatorname{dist}(N,m),\sup_{n\in N}\operatorname{dist}(M,n)\}\). Moreover, every branch of XT will be approximated by precisely one branch of X1T (as required in the definition of the convergence of holomorphic chains).
Note that we are not given equations defining X and it does not seem to be easy to obtain these equations from the definition of X. However, in this example one does not need them to show (see below) that
https://static-content.springer.com/image/art%3A10.1007%2Fs00365-012-9156-0/MediaObjects/365_2012_9156_Equai_HTML.gif
where πw,πz denote the natural projections of \(\mathbf {C}^{4}_{xyzw}\) onto \(\mathbf{C}^{3}_{xyw}\), \(\mathbf{C}^{3}_{xyz}\), respectively, and to show that polynomials qw,qz computed in the first step of the construction are qw(x,y,w)=w5+g(x,y)w−2f(x,y), \(q_{z}(x,y,z)=z^{5}-2f(x,y)e^{(\frac {x+y^{2}}{10})^{30}}\).
In the second step, we compute and decompose the discriminants Δw,Δz of qw,qz to obtain
https://static-content.springer.com/image/art%3A10.1007%2Fs00365-012-9156-0/MediaObjects/365_2012_9156_Equaj_HTML.gif
where h(x,y)=x+2.2⋅10−3y+10−30x1500cos(y), \(a(x,y)=2f(x,y)e^{(\frac{x+y^{2}}{10})^{30}}\), and \((g_{z}^{-1}(0)\cup g_{w}^{-1}(0))\cap(K_{1}\times K_{1})=\emptyset\).
In the third step, we approximate h,gz,gw,a,f,g by Nash functions h1,gz,1, gw,1,a1,f1,g1 in such a way that the equations
https://static-content.springer.com/image/art%3A10.1007%2Fs00365-012-9156-0/MediaObjects/365_2012_9156_Equak_HTML.gif
are satisfied. Then \(\tilde{Y}_{1}=\{(x,y,w,z)\in K_{1}\times K_{1}\times K_{1.4}\times K_{1.4}:z^{5}-a_{1}(x,y)=w^{5}+g_{1}(x,y)w-2f_{1}(x,y)=0\}\) has the following property. Every irreducible component of \(\tilde{Y}\) is approximated by some irreducible component of \(\tilde{Y}_{1}\) (and we are able to extract from \(\tilde{Y}_{1}\) the component which approximates X). If the equations were not satisfied, then \(\tilde{Y}_{1}\) might turn out to be irreducible.

Take h1(x,y)=x+2.2⋅10−3y, f1(x,y)=(h1(x,y))2, g1(x,y)=(h1(x,y))3 and a1(x,y)=2f1(x,y). Observe that f1,g1,a1 are so close to f,g,a respectively that \(\operatorname{dist}(\tilde{Y}_{1},\tilde {Y})<5\cdot10^{-6}\). Moreover, there exist gz,1, gw,1 satisfying the equations, and from the proof of Theorem 3.1 of [5], it follows that \(\tilde{Y}_{1}\) has the required property.

Since in our example f1,g1,a1 are polynomials, it is not necessary to introduce Pi,j,1 in the third step, and in the fourth step, we can take
$$\tilde{X}_{1}=\bigl\{(x,y,w,z)\in\mathbf{C}^4:z^5-2f_{1}(x,y)=w^5+g_{1}(x,y)w-2f_{1}(x,y)=0\bigr\}.$$

Now we have to remove from T all the analytic irreducible components of \(\tilde{X}_{1}\cap T\) except the one which approximates XT. To do this, we shall construct an algebraic subset Z1 of some C4×Cd such that the projection \(\pi|_{Z_{1}}:Z_{1}\rightarrow\tilde{X}_{1}\) is a proper map (bijective when restricted to the pre-image of the regular part of \(\tilde{X}_{1}\)), \(\pi (Z_{1})=\tilde{X}_{1}\), and the analytic irreducible components of Z1∩(T×Cd) are separated from each other.

Before constructing Z1, let us describe some properties of \(\tilde {X}_{1}\) which will be useful to us. First, introduce a new variable u, and consider a complex curve C in C3 given by
$$C=\bigl\{(u,z,w)\in\mathbf{C}^3: z^5-2u^2=w^5+u^3w-2u^2=0\bigr\}.$$
Fix u0K1, u0≠0. Then for every z0C such that \(z_{0}^{5}-2u_{0}^{2}=0\) there is w0C such that \(w_{0}^{5}+u_{0}^{3}w_{0}-2u_{0}^{2}=0\) and \(|z_{0}-w_{0}|\leq\frac{1}{6}\sqrt[5]{2|u_{0}|^{2}}\). To see this, it is sufficient to observe that for a fixed z0C such that \(z_{0}^{5}-2u_{0}^{2}=0\) and for every ϕ∈[0,2π],
$$\biggl|\biggl(z_0+\frac{1}{6}e^{i\phi}\sqrt[5]{2|u_0|^2}\biggr)^5-2u^2_0\biggr|> \biggl|\biggl(z_0+\frac{1}{6}e^{i\phi}\sqrt[5]{2|u_0|^2}\biggr)u_0^3\biggr|.$$
Now the claim follows immediately by the Rouche theorem.
The previous paragraph implies that the map
https://static-content.springer.com/image/art%3A10.1007%2Fs00365-012-9156-0/MediaObjects/365_2012_9156_Equao_HTML.gif
which assigns to every (u,z) the unique point (u,w) such that \(|z-w|\leq\frac{1}{6}\sqrt[5]{2|u|^{2}}\) is a biholomorphism, which in turn implies that
$$C'=\overline{\bigl\{(u,z,w)\in K_1\times \mathbf{C}^2:u\neq0, F(u,z)=(u,w)\bigr\}},$$
is an irreducible analytic subset of K1×C2. Consequently, X′=J−1(C′)∩T, where J(x,y,z,w)=(x+2.2⋅10−3y,z,w), is the irreducible component of \(\tilde{X}_{1}\cap T\), and, clearly, this is the one approximating XT. (Now it is also clear that X has 5 points in the generic fiber over K1×K1 and that \(\pi_{z}(X)=\pi_{z}(\tilde{Y})\), \(\pi_{w}(X)=\pi_{w}(\tilde{Y})\).)

Note that if \(z_{1}^{5}-2u_{0}^{2}=z_{2}^{5}-2u_{0}^{2}=0\) and z1z2, then \(|z_{1}-z_{2}|\geq2\sin(\frac{\pi}{5})\sqrt[5]{2|u_{0}|^{2}}\). Consequently, in view of the previous paragraph, for every (x,y,z,w)∈T, z≠0, the following hold. If (x,y,z,w)∈X′, then \(|\frac{(z-w)^{5}}{z^{5}}|\leq\frac{1}{6^{5}}\) hence \(|2\frac {(z-w)^{5}}{z^{5}}|\leq3\cdot 10^{-4}\), whereas if \((x,y,z,w)\in(\tilde{X}_{1}\cap T)\setminus X'\), then \(|\frac{(z-w)^{5}}{z^{5}}|\geq(2\sin(\frac{\pi}{5})-\frac{1}{6})^{5}\) hence \(|2\frac{(z-w)^{5}}{z^{5}}|\geq2\). Moreover, \(|\frac{(z-w)^{5}}{z^{5}}|\) is bounded from above on \(\tilde{X}_{1}\cap T\).

Now consider
$$\tilde{Z}_{1}=\bigl\{(x,y,z,w,t)\in\mathbf{C}^5:(x,y,z,w)\in\tilde{X}_{1}, tz^5=2(z-w)^5\bigr \},$$
and observe that \(Z_{1}=\overline{\tilde{Z}_{1}\setminus N}\), where N={(x,y,z,w,t)∈C5:z=0}, has all the required properties.

The last step is to find a polynomial map P:C5C4 such that P(Z1) is an algebraic set approximating X′ on T with \(\operatorname{dist}(P(Z_{1})\cap T,X')<4.9\cdot10^{-4}\). It is clear that here one can take P(x,y,z,w,t)=(xt,y,z,w).

Let us calculate polynomials describing P(Z1). First observe that
$$P(\tilde{Z}_{1})=\bigl\{(x,y,z,w)\in\mathbf {C}^4:p_1(x,y,z,w)=p_2(x,y,z,w)=0\bigr\},$$
where
https://static-content.springer.com/image/art%3A10.1007%2Fs00365-012-9156-0/MediaObjects/365_2012_9156_Equas_HTML.gif
and that \(P(Z_{1})=\overline{P(\tilde{Z}_{1})\setminus M}\) and \(M\subset P(\tilde{Z}_{1})\), where M={(x,y,z,w)∈C4:z=w=0}. To remove M from \(P(\tilde{Z}_{1})\), set u=x+2.2⋅10−3y, and define
https://static-content.springer.com/image/art%3A10.1007%2Fs00365-012-9156-0/MediaObjects/365_2012_9156_Equat_HTML.gif
Let
$$\tilde{V}=\bigl\{(u,z,w)\in\mathbf{C}^3:\tilde{p}_1(u,z,w)=\tilde {p}_2(u,z,w)=0\bigr\},$$
and let \(V=\overline{\tilde{V}\setminus\{z=w=0\}}\). Clearly, \(\hat{\pi}|_{\tilde{V}}:\tilde{V}\rightarrow\mathbf{C}^{2}_{u,z}\) is a proper map, where \(\hat{\pi}:\mathbf{C}^{3}_{u,z,w}\rightarrow\mathbf{C}^{2}_{u,z}\) is the natural projection. Moreover, \((u,0,w)\in\tilde{V}\) implies w=0 for every u. Therefore if \(\hat{\pi}(V)=\{(u,z)\in\mathbf{C}^{2}:\tilde{p}_{3}(u,z)=0\}\) for some \(\tilde{p}_{3}\in\mathbf{C}[u,z]\), then \(V=\{(u,z,w)\in\mathbf{C}^{3}:\tilde{p}_{1}(u,z,w)=\tilde{p}_{2}(u,z,w)=\tilde {p}_{3}(u,z)\}\).
Let us compute \(\tilde{p}_{3}\). Let rC[u,z] be the resultant of \(\tilde{p}_{1},\tilde{p}_{2}\in(\mathbf{C}[u,z])[w]\), and let nN,sC[u,z], be such that r(u,z)=zns(u,z) and s(u,0) is a nonzero polynomial. Since
$$\hat{\pi}(V)=\overline{\hat{\pi}(V)\setminus\{z=0\}}=\overline{\hat{\pi }(\tilde{V})\setminus\{z=0\}}=\bigl\{(u,z)\in\mathbf{C}^2:s(u,z)=0\bigr \},$$
we can take \(\tilde{p}_{3}=s\). Now it is clear that
$$P(Z_{1})=\bigl\{(x,y,z,w)\in\mathbf {C}^4:p_1(x,y,z,w)=p_2(x,y,z,w)=p_3(x,y,z)=0\bigr\},$$
where \(p_{3}(x,y,z)=\tilde{p}_{3}(x+2.2\cdot10^{-3}y,z)\).

Let us finish this example by the remark that s can be effectively computed by a computer algebra system. Moreover, for any fixed yK1,zK1.14∖{0}, one can solve numerically the system p1(x,y,z,w)=p2(x,y,z,w)=0 and the system z5−2f1(x,y)=w5+g1(x,y)w−2f1(x,y)=0 to observe that every solution of the first one which stays in T satisfies the inequality \(|z-w|\leq\frac{1}{6}\sqrt[5]{2|x+2.2\cdot10^{-3}y|^{2}}\) and approximates the corresponding solution of the latter one with the required precision.

Acknowledgements

I am grateful to Professor Marek Jarnicki for introducing me to the subject of polynomial approximation of holomorphic functions. I thank Professor Mirosław Baran and Dr. Leokadia Białas-Cież for information on Markov’s inequality. I also thank Dr. Marcin Dumnicki and Dr. Rafał Czyż for helpful discussions.

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© The Author(s) 2012