Skip to content
BY 4.0 license Open Access Published by De Gruyter July 29, 2022

Hartogs-type theorems in real algebraic geometry, I

Marcin Bilski , Jacek Bochnak and Wojciech Kucharz EMAIL logo

Abstract

Let f : X be a function defined on a connected nonsingular real algebraic set X in n . We prove that regularity of f can be detected by controlling the restrictions of f to either algebraic curves or algebraic surfaces in X. If dim X 2 and k is a positive integer, then f is a regular function whenever the restriction f | C is a regular function for every algebraic curve C in X that is a 𝒞 k submanifold homeomorphic to the unit circle and is either nonsingular or has precisely one singularity. Moreover, in the latter case, the singularity of C is equivalent to the plane curve singularity defined by the equation x p = y q for some primes p < q . If dim X 3 , then f is a regular function whenever the restriction f | S is a regular function for every nonsingular algebraic surface S in X that is homeomorphic to the unit 2-sphere. We also have suitable versions of these results for X not necessarily connected.

1 Introduction and main results

The purpose of the present paper is to prove that regularity of real-valued functions defined on a nonsingular real algebraic set X in n can be detected on mildly singular algebraic curves or nonsingular algebraic surfaces in X. All theorems announced in this section are proved in Section 3.

We refer to either [4] or [26] for the general theory of real algebraic sets, real regular functions, and related topics. Unless explicitly stated otherwise, we consider n and all its subsets endowed with the Euclidean topology induced by the standard norm on n .

Let Z be an algebraic set in n . The notions of nonsingular point and singular point of Z will be always understood in the standard algebraic sense. Thus Z is nonsingular at a point a Z if there exist k polynomial functions on n , vanishing identically on Z, such that their set of common zeros coincides with Z in a neighborhood of a in n and their Jacobian matrix at a has rank k, see [4, 26].

The algebraic complexification of Z is the smallest complex algebraic subset Z of n that contains Z ( n is viewed as a subset of n ). Note that Z is nonsingular at a if and only if Z is nonsingular at a (in the algebraic sense), which according to [28, Proposition 4] is equivalent to Z being a complex analytic manifold in an open neighborhood of a in n . Assuming that Z is nonsingular at a, for any real analytic map-germ φ : ( Z , a ) ( m , 0 ) , the complexification of φ is the uniquely determined complex analytic map-germ φ : ( Z , a ) ( m , 0 ) whose restriction to Z is equal to φ (in the case under consideration, this is equivalent to the usual definition involving the analytic complexification of the germ of Z at a).

Let X be an irreducible nonsingular real algebraic set in n of dimension m 2 , and C a real algebraic curve in X. Let ( p , q ) be a pair of primes, 2 < p < q , and

D p , q := { ( x , y , z 1 , , z m - 2 ) m : x p - y q = 0 , z 1 = = z m - 2 = 0 } .

We say that C is of type ( p , q ) at a C if there exists a real analytic diffeomorphism-germ σ : ( X , a ) ( m , 0 ) whose complexification σ maps the germ of C at a onto the germ of D p , q at 0 m . We say that C is a ( p , q ) -curve if it has a unique singular point a and it is of type ( p , q ) at a. If k is a nonnegative integer with p k < q , then D p , q is a 𝒞 k submanifold of m , hence any ( p , q ) -curve in X is a 𝒞 k submanifold.

1.1 Main results (simplified versions)

The main two results of this paper are stated in a simplified form as Theorems 1.1 and 1.3 below.

Theorem 1.1.

Let k be a positive integer and let f : X R be a function defined on a connected nonsingular algebraic set X in R n , with dim X 2 . Then the following conditions are equivalent:

  1. f is regular on X.

  2. For every algebraic curve C in X that is homeomorphic to the unit circle and is either nonsingular or a ( p , q ) - curve for some odd primes p , q with p k < q (so in particular is a 𝒞 k submanifold), the restriction f | C is a regular function.

A real-valued function on n , for n 2 , need not be regular (or even continuous) despite the fact that all its restrictions to nonsingular algebraic curves in n are regular.

Example 1.2.

Let g : n , for n 2 , be the function defined by

g ( x 1 , , x n ) = { x 1 8 + x 2 ( x 1 2 - x 2 3 ) 2 x 1 10 + ( x 1 2 - x 2 3 ) 2 + x 3 2 + + x n 2 for  ( x 1 , , x n ) ( 0 , , 0 ) , 0 for  ( x 1 , , x n ) = ( 0 , , 0 ) .

The restriction of g to every nonsingular algebraic curve in n is a regular function, and the restriction of g to every one-dimensional real analytic submanifold of n is a real analytic function, see [17, Example 2.3] for the case n = 2 . However, g is not regular on n because it is not even locally bounded on the curve defined by x 1 2 - x 2 3 = 0 , x 3 = 0 , , x n = 0 . Hence in Theorem 1.1, as well as in Theorems 1.6 and 1.7 below, we have to allow algebraic curves with singularities of some type.

In the special case X = n , a result sharper than Theorem 1.1 is contained in Theorem 1.6, where the algebraic curves used for testing regularity of functions are given explicitly. A more general result for X irreducible but not necessarily connected is given in Theorem 1.7. Theorems 1.6 and 1.7 are optimal in the sense made precise in Remark 1.10. They can be viewed as a rather surprising continuation of the research project undertaken in [17], whose principal aim has been a characterization of continuous real rational functions by their restrictions to algebraic curves or arcs of such curves. In particular, by [17, Theorem 1.7], if the restriction of a function f to every algebraic curve contained in its domain is continuous and rational, then f is continuous and rational.

Theorem 1.3.

Let f : X R be a function defined on a connected nonsingular algebraic set X in R n , with dim X 3 . Then the following conditions are equivalent:

  1. f is regular on X.

  2. For every nonsingular algebraic surface S in X that is homeomorphic to the unit 2 - sphere, the restriction f | S is a regular function.

This result also has a sharper version, Theorem 1.8 (for X = n ), and a generalization, Theorem 1.9 (for X not necessarily connected). Each of Theorems 1.3 and 1.9 is a significant improvement upon [17, Theorem 6.2]. The latter states that a function f is regular if and only if for every real algebraic surface S contained in its domain, the restriction of f to the set of nonsingular points of S is a regular function. Theorems 1.3, 1.8 and 1.9 are algebraic analogs of the results obtained in [6] for the real analytic category. However, the transition to the algebraic setting is not obvious at all and requires new methods. The results of the present paper can be interpreted as real algebraic variants of the classical Hartogs theorem on separately holomorphic functions of several complex variables [10].

Given nonnegative integers 0 r k and 𝒞 k manifolds M , N , we consider the space 𝒞 r ( M , N ) of 𝒞 r maps M N endowed with the weak 𝒞 r topology defined in [15, p. 34], which will be referred to as the 𝒞 r topology for short.

It seems that a comment is in order for condition (b) in Theorems 1.1 and 1.3 (relevant also for condition (c) in Theorem 1.7 and condition (b) in Theorem 1.8). Let S be an algebraic set in n or, more generally, a connected component of such a set. Assume that S is a 𝒞 k submanifold homeomorphic to the unit d-sphere 𝕊 d , where k 1 , and d = 1 or d = 2 . Then there exists a 𝒞 k diffeomorphism φ : S 𝕊 d such that φ is semialgebraic and the restriction of φ to the subset of nonsingular points in S is real analytic. Indeed, according to the classification theorem for manifolds of dimension 1 or 2, there exists a 𝒞 1 diffeomorphism ψ : S 𝕊 d . By the Weierstrass approximation theorem, ψ regarded as a map into d + 1 { 0 } can be approximated in the 𝒞 1 topology by a polynomial map σ : S d + 1 { 0 } d + 1 . If σ is sufficiently close to ψ, then the composite map φ := ϱ σ : S 𝕊 d of σ with the radial projection ϱ : d + 1 { 0 } 𝕊 d has the required properties.

Besides [6, 23], the key ingredient in our proof of Theorem 1.9 is Theorem 1.7 engaging ( p , q ) -curves introduced above. Let us briefly discuss the property of these curves, called faithfulness, that plays a decisive role in the proof of Theorem 1.9. The concept in question refers in fact to real algebraic sets of any dimension. Given a real algebraic set Z in n and a point b Z , we regard the germ Z b of Z at b as a real analytic set-germ. The analytic complexification of Z b is the smallest complex analytic set-germ at b n that contains Z b , see [27, pp. 91–92]. We say that Z is faithful at b if the analytic complexification of Z b is equal to the germ at b of the algebraic complexification Z of Z; we say that Z is faithful if it is faithful at each of its points. Here “faithful” replaces “quasi-regular” used in [29]. If Z is additionally a topological manifold with all connected components of the same dimension, then Z is faithful at b if and only if the complex analytic germ of Z at b is irreducible.

By [28, Proposition 4], Z is faithful at b if and only if, in the ring of real analytic function-germs ( n , b ) , the ideal of function-germs vanishing on Z b is generated by the germs at b of polynomial functions n vanishing on Z. Thus, Z is nonsingular at b if and only if Z is faithful at b and is a real analytic manifold in an open neighborhood of b in n , which is equivalent to Z being faithful at b and being a 𝒞 manifold in an open neighborhood of b in n (by [25, Chapter VI, Proposition 3.11]). Obviously if Z is faithful, it is coherent as a real analytic set. The following example illustrates the issues at hand quite well and, in addition, shows that Z can be coherent but not faithful.

Example 1.4.

The real polynomial P ( x , y ) = x 4 - 2 x 2 y - y 3 is irreducible and its gradient vanishes only at ( 0 , 0 ) . It follows that the real algebraic curve

E := { ( x , y ) 2 : P ( x , y ) = 0 }

is irreducible and has the only singular point at ( 0 , 0 ) . On the other hand, E is a real analytic submanifold of 2 . It suffices to justify this claim in a neighborhood of ( 0 , 0 ) in 2 , which is straightforward because for y > - 1 the curve E is given by the real analytic equation φ ( x , y ) := x 2 - y ( 1 + 1 + y ) = 0 , and φ y ( 0 , 0 ) 0 . Consequently, E is not faithful at ( 0 , 0 ) . Note that E is coherent as a real analytic set (as is any real analytic curve).

This example shows that in the real case (unlike in the complex one) an algebraic set can be both singular and an analytic manifold.

Let C be a real algebraic curve in an irreducible nonsingular real algebraic subset X of n with dim X = m 2 . Let ( p , q ) be a pair of primes, 2 < p < q . Then C is a ( p , q ) -curve if and only if it has a unique singular point a, it is faithful at a, and there exists a real analytic diffeomorphism-germ σ : ( X , a ) ( m , 0 ) such that σ ( C a ) is equal to the germ of D p , q at 0 m .

1.2 Main results (full generality)

Given a real-valued function α on some set Ω, we denote by 𝒵 ( α ) the zero set of α, that is, 𝒵 ( α ) = { x Ω : α ( x ) = 0 } .

It will be convenient to consider regular functions in a more general context than usual. Let A be an arbitrary subset of n . A function f : A is said to be regular at a point a A if there exist two polynomial functions φ , ψ : n such that ψ ( a ) 0 and f ( x ) = φ ( x ) ψ ( x ) for all x A 𝒵 ( ψ ) ; as expected, f is said to be regular on A if it is regular at every point in A. Actually, assuming that f is regular on A, one can find two polynomial functions Φ , Ψ : n with

A n 𝒵 ( Ψ )    and    f ( x ) = Φ ( x ) Ψ ( x ) for all  x A ,

see the proof of [4, Proposition 3.2.3].

A function g : A defined on a subset A of n is said to be real analytic if for every point a A there exist an open neighborhood U n of a and a real analytic function G : U (in the usual sense) such that G and g agree on A U . Obviously, every regular function on A is real analytic.

For integers n and d, with 1 d n - 1 , an algebraic set Σ in n is called a Euclidean d-sphere if it can be expressed as

Σ = { x n : x - x 0 = r } Q ,

where Q is an affine ( d + 1 ) -plane in n , x 0 Q , and r > 0 . A Euclidean 1-sphere is also called a Euclidean circle.

Next we define certain collections of real algebraic curves and surfaces in order to formulate our results in an optimal way. Let Π denote the set of all prime numbers. For any positive integer k, put

Λ k := { ( p , q ) Π × Π : 2 < p  and  p k < q } .

Notation 1.5.

Let X be an irreducible nonsingular algebraic set in n , for n 2 , and let U be a nonempty open subset of X.

(1.5.1) Collection S d ( U ) . For an integer d, with 1 d dim X - 1 , we denote by 𝒮 d ( U ) the collection of all d-dimensional irreducible nonsingular algebraic sets S in X, contained in U, satisfying one of the following two conditions:

  1. If U is connected, then S is homeomorphic to the unit d-sphere.

  2. If U is disconnected, then S has at most two connected components, each homeomorphic to the unit d-sphere.

Only the cases d = 1 and d = 2 will be relevant.

(1.5.2) Collection F k ( U ) . Given a positive integer k, we denote by k ( U ) the collection of all ( p , q ) -curves C in X for ( p , q ) Λ k such that C is contained in U and is homeomorphic to the unit circle. Recall that such curves are 𝒞 k submanifolds of U.

(1.5.3). For any primes 2 < p < q , define

H p , q ( x , y ) := x p - y q ,
E p , q := 𝒵 ( H p , q ) = { ( x , y ) 2 : H p , q ( x , y ) = 0 }

and

F p , q ( x , y ) := x p - y q + x 2 q + y 2 q ,
C p , q := 𝒵 ( F p , q ) = { ( x , y ) 2 : F p , q ( x , y ) = 0 } .

For ( a , b ) 2 , the translates

( a , b ) + E p , q and ( a , b ) + C p , q

are algebraic curves in 2 defined by the equations

H p , q ( x - a , y - b ) = 0 and F p , q ( x - a , y - b ) = 0 ,

respectively.

(1.5.4) Collections H k ( R n ) and G k ( R n ) . For any affine 2-plane Q in n , we choose once and for all an affine-linear isomorphism φ Q : Q 2 . If n = 2 , then Q = 2 and φ 2 is chosen to be the identity map. Given a positive integer k, we denote by k ( n ) (resp. 𝒢 k ( n ) ) the collection of all algebraic curves C in n for which there exist an affine 2-plane Q in n containing C and a pair of primes ( p , q ) Λ k such that φ Q ( C ) is a translate of E p , q (resp. C p , q ). In particular, k ( 2 ) (resp. 𝒢 k ( 2 ) ) is the collection of all translates of the curves E p , q (resp. C p , q ) for ( p , q ) Λ k . The curves in k ( 2 ) and in 𝒢 k ( 2 ) are therefore given by explicit and simple polynomial equations. By Lemmas 2.2 and 2.6, 𝒢 k ( n ) k ( n ) .

Here is a characterization of regular functions on n , for n 2 .

Theorem 1.6.

Let k , n be two integers, k 1 , n 2 . Then, for a function f : R n R , the following conditions are equivalent:

  1. f is regular on n .

  2. The restriction of f to every irreducible algebraic curve in n is a regular function.

  3. The restriction of f to every algebraic curve, which is either a Euclidean circle in n or is in the collection 𝒢 k ( n ) , is a regular function.

  4. The restriction of f to every Euclidean circle in n is a regular function, and the restriction of f to every algebraic curve in the collection 𝒢 k ( n ) is a real analytic function.

  5. The restriction of f to every algebraic curve, which is either an affine line parallel to one of the coordinate axes of n or is in the collection k ( n ) , is a regular function.

  6. The restriction of f to every affine line parallel to one of the coordinate axes of n is a regular function, and the restriction of f to every algebraic curve in the collection k ( n ) is a real analytic function.

Let us note that each curve E p , q in (1.5.3) is rational. Indeed, the map E p , q , t ( t q , t p ) is regular with rational inverse ( x , y ) x l y k , where k , l are (possibly negative) integers satisfying k p + l q = 1 . Consequently, each curve in the collection k ( n ) is rational. Thus, according to Theorem 1.6, regularity of real-valued functions on n can be detected on rational curves. This is not the case if n is replaced by an arbitrary nonsingular real algebraic set. For example, the nonsingular real cubic curve

C := { ( x , y ) 2 : y 2 = x 3 - 1 }

is not rational, so the Cartesian product X := C n does not contain any rational curve. Observe that C is diffeomorphic to , hence X is diffeomorphic to n .

In the general setting we have the following result.

Theorem 1.7.

Let k be a positive integer, X an irreducible nonsingular algebraic set in R n , with dim X 2 , and U a nonempty open subset of X. Then, for a function f : U R , the following conditions are equivalent:

  1. f is regular on U.

  2. The restriction of f to every irreducible algebraic curve in X , contained in U , is a regular function.

  3. The restriction of f to every algebraic curve, which is either in the collection 𝒮 1 ( U ) or in the collection k ( U ) , is a regular function.

  4. The restriction of f to every algebraic curve in the collection 𝒮 1 ( U ) is a regular function, and the restriction of f to every algebraic curve in the collection k ( U ) is a real analytic function.

By the definition of the collections 𝒮 1 ( U ) and k ( U ) , Theorem 1.7 is both more general and stronger than Theorem 1.1. Taking U = X would not simplify the proof of Theorem 1.7 in a significant way. In Theorems 1.6 and 1.7, the implications (a)   (b)   (c)   (d) and (a)   (e)   (f) are obvious; we prove the implications (d)   (a) and (f)   (a) in Section 3. Let us note that deducing from (d) or (f) continuity of f, which is seemingly a much simpler task, is not at all obvious.

Regular functions on n , with n 3 , can also be characterized as follows.

Theorem 1.8.

For a function f : R n R , with n 3 , the following conditions are equivalent:

  1. f is regular on n .

  2. The restriction of f to every Euclidean 2 - sphere in n is a regular function.

As an application of Theorem 1.7, we will obtain the following.

Theorem 1.9.

Let X be an irreducible nonsingular algebraic set in R n , with dim X 3 , and let U be a nonempty open subset of X. Then, for a function f : U R , the following conditions are equivalent:

  1. f is regular on U.

  2. The restriction of f to every algebraic surface in the collection 𝒮 2 ( U ) is a regular function.

Next, we give some comments on the assumptions in our theorems.

Remark 1.10.

Our results are optimal in the following sense.

(1.10.1). The real algebraic curves in Theorem 1.1 and in Theorems 1.6 and 1.7 (except part (b)) are faithful, have at most one singular point, and are 𝒞 k manifolds, where k is any given positive integer. Additionally, the singularities that occur have a simple analytic description. As we have already noted, an algebraic set in n is nonsingular if and only if it is faithful and is a 𝒞 manifold. So, according to Example 1.2, if algebraic curves used for testing regularity of functions are to be faithful, then they cannot simultaneously be 𝒞 manifolds. Faithfulness is a desirable distinguishing condition for those real algebraic sets that have particularly good local algebraic properties typical for complex algebraic sets. These properties are indispensable for our proofs of Theorems 1.3 and 1.9.

(1.10.2). Suppose that the set U in Theorems 1.7 and 1.9 is disconnected. Let U 0 be a connected component of U, and let f : U be the function defined by f = 0 on U 0 and f = 1 on U U 0 . Obviously, f is not a regular function on U, but the restriction of f to every connected algebraic set in X, contained in U, is a constant (hence regular) function. Therefore it is essential that the algebraic sets in the collections 𝒮 d ( U ) , with d = 1 and d = 2 , are not necessarily connected.

Remark 1.11.

In the second part of the present paper, currently in preparation, we prove that it is sufficient to restrict attention to algebraic curves which are analytic manifolds (that admit singularities and therefore cannot be faithful). More precisely, among the results we obtain the following: A function f : X defined on a connected nonsingular real algebraic set X in n (with dim X 2 ) is regular if and only if for every algebraic curve C in X, which has at most one singular point and is a real analytic submanifold homeomorphic to the unit circle, the restriction f | C is a regular function.

Our results fit into the research program in real algebraic geometry focusing on continuous rational functions, regulous functions and piecewise-regular functions [2, 3, 5, 12, 13, 17, 18, 19, 22] (see also the recent surveys [20, 21] and the references therein).

The paper is organized as follows. In Section 2 we prove, by a rather intricate argument, a criterion for analyticity of some real meromorphic functions on nonsingular real algebraic surfaces. This enables us to apply in a novel way the tools developed in [6, 9, 17, 23, 8], leading to the proofs of Theorems 1.6, 1.7, 1.8 and 1.9 (hence also Theorems 1.1 and 1.3) in Section 3.

2 Real meromorphic functions on algebraic surfaces

The following result will play the key role in Section 3.

Proposition 2.1.

Let k be a positive integer, X an irreducible nonsingular algebraic set in R n , with dim X = 2 , and f : U R a function defined on an open subset U of X. Let a be a point in U and let g , h : U R be two real analytic functions such that

𝒵 ( h ) = { a }    𝑎𝑛𝑑    f ( b ) = g ( b ) h ( b ) for all  b U { a } .

Assume that for every algebraic curve C in the collection F k ( U ) , with singular point at a, the restriction f | C is a real analytic function. Then f is real analytic on U.

Recall that the collection k ( U ) is defined in (1.5.2). The proof of Proposition 2.1 requires some preparation and will be preceded by several auxiliary results. Suggestions conveyed to us by S. Donaldson allowed us to simplify the original proof. The case X = 2 will be analyzed first.

2.1 Real meromorphic functions of two variables

Let 𝕂 denote either the field of real numbers or the field of complex numbers. Let 𝒪 n 𝕂 denote the ring of all 𝕂 -analytic function-germs at the origin in 𝕂 n . If no confusion is possible, we will make no distinction between function-germs and their representatives. The ring 𝒪 n 𝕂 is a local ring whose maximal ideal is denoted by 𝔪 𝕂 , n . As usual, given a positive integer i and an ideal 𝔪 , we denote by 𝔪 i the ith power of 𝔪 .

In the next four lemmas we discuss some properties of the polynomial functions F p , q and H p , q defined in (1.5.3).

Lemma 2.2.

Let 2 < p < q be prime numbers. Then for every v m K , 2 2 q there exists a K -analytic diffeomorphism-germ G : ( K 2 , 0 ) ( K 2 , 0 ) such that F p , q , H p , q regarded as elements of m K , 2 satisfy

F p , q + v = H p , q G .

Proof.

Clearly, F p , q + v = x p ( 1 + f ( x , y ) ) + y q ( - 1 + g ( x , y ) ) , for some 𝕂 -analytic function-germs f , g vanishing at the origin. Hence, it is sufficient to take

G ( x , y ) = ( x φ ( x , y ) , - y ψ ( x , y ) ) ,

where φ , ψ are 𝕂 -analytic function germs satisfying φ p = 1 + f  and  ψ q = - 1 + g . ∎

Lemma 2.3.

Let 2 < p < q be prime numbers. Let B be an open disc in R 2 centered at the origin, ψ 1 , , ψ m real-valued analytic functions on B that vanish at the origin, and let v r = ( ψ 1 + + ψ m ) r for positive integers r. Then there are an open neighborhood V B of the origin, a positive integer r 0 , and real analytic maps H r : V B for r r 0 such that

  1. the sequence { H r } r r 0 converges to the inclusion map V B in the 𝒞 1 topology, and

  2. for each r r 0 the map H r : V H r ( V ) is an analytic diffeomorphism for which

    H r ( 0 , 0 ) = ( 0 , 0 )    𝑎𝑛𝑑    F p , q + v r = F p , q H r on  V .

Proof.

Since we have ψ i ( 0 , 0 ) = 0 for i = 1 , , m , there are real analytic functions λ i , μ i : B such that

ψ i ( x , y ) = x λ i ( x , y ) + y μ i ( x , y ) for all  ( x , y ) B .

Therefore taking r 1 = p + q , we get

v r 1 ( x , y ) = x p u r 1 ( x , y ) + y q w r 1 ( x , y ) ,

where u r 1 , w r 1 : B are real analytic functions. For r > r 1 define real analytic functions u r , w r : B by the equations u r = u r 1 v 1 r - r 1 and w r = w r 1 v 1 r - r 1 , and note that

v r ( x , y ) = v r 1 ( x , y ) v 1 r - r 1 ( x , y ) = x p u r ( x , y ) + y q w r ( x , y ) for all  ( x , y ) B .

Since v 1 ( 0 , 0 ) = 0 , there is an open disc B 0 in 2 centered at the origin such that its closure is contained in B and the sequences { u r | B 0 } , { w r | B 0 } converge to 0 in the 𝒞 1 topology.

Shrinking B 0 and choosing a sufficiently large integer r 0 > r 1 , we see that the formulas

G 0 ( x , y ) = ( x 1 + x 2 q - p p , - y - 1 + y q q )

and

G r ( x , y ) = ( x 1 + x 2 q - p + u r ( x , u ) p , - y - 1 + y q + w r ( x , y ) q )

for r r 0 define real analytic maps from B 0 into 2 , and G 0 : B 0 G 0 ( B 0 ) is a real analytic diffeomorphism. By construction, the sequence { G r } r r 0 converges to G 0 in the 𝒞 1 topology. By [15, Lemma 1.3, p. 36]), increasing r 0 and shrinking B 0 , we may assume that the map G r : B 0 G r ( B 0 ) is a real analytic diffeomorphism for r r 0 . Now, let V 2 be an open neighborhood of the origin with closure contained in B 0 . Increasing r 0 once again, we get G r ( V ) G 0 ( B 0 ) for r r 0 . Since on V, we have

F p , q = H p , q G 0 and F p , q + v r = H p , q G r ,

for r r 0 , the real analytic maps H r = G 0 - 1 ( G r | V ) , for r r 0 , satisfy the required conditions. ∎

Lemma 2.4.

Let 2 < p < q be prime numbers. Then for every v m K , 2 2 q the K -analytic function-germ F p , q + v is irreducible.

Proof.

The 𝕂 -analytic function-germ of H p , q at the origin is irreducible. Hence the assertion is an immediate consequence of Lemma 2.2. ∎

For any germ A of a subset of 2 at the origin, let I ( A ) denote the ideal of all function-germs in 𝒪 2 vanishing on A. The following lemma formulated in the real setting clearly has its complex analogue.

Lemma 2.5.

Let 2 < p < q be prime numbers. Then for every v m R , 2 2 q the real analytic set-germ Z ( F p , q + v ) is irreducible of dimension 1, and

I ( 𝒵 ( F p , q + v ) ) = ( F p , q + v ) 𝒪 2 .

Proof.

The real analytic set-germ 𝒵 ( H p , q ) is irreducible of dimension 1, and

I ( 𝒵 ( H p , q ) ) = ( H p , q ) 𝒪 2 .

Hence, the assertion follows immediately by Lemma 2.2. ∎

In the following lemma we establish some useful properties of the real algebraic curves C p , q defined in (1.5.3).

Lemma 2.6.

Let k be a positive integer and p , q primes with 2 < p and p k < q . Then the following hold:

  1. C p , q = graph ( ϕ + ) graph ( ϕ - ) , where ϕ + , ϕ - : [ α , β ] are given by

    ϕ + ( x ) = 1 2 + 1 2 1 - 4 ( x p + x 2 q ) q

    and

    ϕ - ( x ) = 1 2 - 1 2 1 - 4 ( x p + x 2 q ) q ,

    and α < β are (the only) real roots of the polynomial 1 - 4 ( x p + x 2 q ) . In particular, C p , q is contained in the open disc in 2 centered at ( 0 , 0 ) with radius independent of p , q .

  2. The origin is the only singular point of the real algebraic curve C p , q .

  3. C p , q is a 𝒞 k manifold homeomorphic to the unit circle.

  4. The real algebraic curve C p , q is faithful.

Proof.

i Put F ~ ( x , t ) := x p - t + x 2 q + t 2 . Solving the equation F ~ = 0 with respect to t, we obtain t + , t - : [ α , β ] given by

t + ( x ) = 1 2 + 1 2 1 - 4 ( x p + x 2 q )

and

t - ( x ) = 1 2 - 1 2 1 - 4 ( x p + x 2 q ) ,

where α < β are real roots of 1 - 4 ( x p + x 2 q ) . Clearly we have 𝒵 ( F ~ ) = graph ( t + ) graph ( t - ) . Since F p , q ( x , y ) = F ~ ( x , y q ) , the first part of i follows. Note that - 1 < α < β < 1 and sup [ α , β ] | ϕ - | sup [ α , β ] | ϕ + | < 2 , for every p , q , which implies the second part of i.

ii Elementary calculation shows that ( 0 , 0 ) is the unique critical point of F p , q lying on C p , q . Moreover, the real polynomial F p , q is irreducible in [ x , y ] as its homogeneous form of the highest degree is irreducible over .

iii By ii, C p , q { ( 0 , 0 ) } is an analytic manifold. Lemma 2.2 shows that the germ of C p , q at ( 0 , 0 ) is analytically equivalent to the germ of 𝒵 ( H p , q ) at ( 0 , 0 ) . Since p k < q , the curve 𝒵 ( H p , q ) is a 𝒞 k manifold. Hence, C p , q is also a 𝒞 k manifold.

By i, C p , q is homeomorphic to the unit circle.

iv The algebraic complexification C p , q of C p , q is given by the complex solutions of the equation F p , q = 0 . Hence, the complex analytic set-germ of C p , q at the origin is irreducible by Lemma 2.4. Therefore C p , q is faithful in view of ii and iii. ∎

The following general observation will also be needed.

Lemma 2.7.

Let Ω R 2 be an open neighborhood of ( 0 , 0 ) R 2 , and g , h : Ω R real analytic functions with Z ( h ) = { ( 0 , 0 ) } . Assume that the real analytic function

f : Ω { ( 0 , 0 ) } , ( x , y ) g ( x , y ) h ( x , y )

does not have a real analytic extension to Ω. Then there is a positive integer m 0 such that for every real analytic function F : Ω R with F m R , 2 m 0 and ( F ) O 2 R = I ( Z ( F ) ) the restriction f | Z ( F ) { ( 0 , 0 ) } does not have a real analytic extension to Z ( F ) .

Proof.

By nonextendability of f, the germ of g at the origin is a nonzero element of 𝒪 2 . After shrinking Ω and dividing g and h by their greatest common divisor we may assume that the germs of g , h are relatively prime in 𝒪 2 and, again by nonextendability of f, we still have h ( 0 , 0 ) = 0 .

Let U be a polydisc about the origin in 2 on which the complexification of h is well defined. Set V := { ( x , y ) U : h ( x , y ) = 0 } and note that the germ of V at the origin is of complex dimension 1 as h ( 0 , 0 ) = 0 . Since g , h are relatively prime in 𝒪 2 , there exists an irreducible complex analytic curve-germ ( T , ( 0 , 0 ) ) contained in ( V , ( 0 , 0 ) ) such that g does not vanish identically on ( T , ( 0 , 0 ) ) . Indeed, otherwise g would vanish identically on ( V , ( 0 , 0 ) ) . Hence, by the Nullstellensatz, there would be an integer t and a v 𝒪 2 with g t = v h . Since g , h 𝒪 2 , we would have v 𝒪 2 , contradicting the fact that g , h are relatively prime in 𝒪 2 .

By the Puiseux theorem, we have a holomorphic normalization ϕ : ( , 0 ) ( T , ( 0 , 0 ) ) . Then g ϕ : ( , 0 ) ( , 0 ) is a nonzero holomorphic function. Define m 0 to be any integer strictly greater than the order of zero of g ϕ at 0.

Let F : Ω be a real analytic function satisfying the hypotheses of the lemma. Suppose that f | 𝒵 ( F ) { ( 0 , 0 ) } has a real analytic extension to 𝒵 ( F ) . Then there is a u 𝒪 2 such that g - u h I ( 𝒵 ( F ) ) . By assumption there is a w 𝒪 2 such that g - u h = w F . Passing to the complexifications of g , F and w in a small neighborhood of the origin, we obtain g ϕ = ( w ϕ ) ( F ϕ ) in a neighborhood of 0. This is a contradiction because the order of zero at 0 of the right-hand side of the latter equation is greater than that of the left-hand side. ∎

2.2 Generalizing to algebraic surfaces

Let us recall that a smooth map is transverse to a point c precisely when c is its regular value (in the sense of differential topology). In the sequel we will make use of the fact that if π : m × k m is the canonical projection and X m × k is a nonsingular algebraic set of pure dimension m such that π | X is a proper map with finite fibers, then the following holds. For every b m to which π | X is transverse, there is an open neighborhood Ω in m such that X ( Ω × k ) is a finite analytic covering over Ω.

The following lemma is a modification of the Noether normalization lemma.

Lemma 2.8.

Let X be an irreducible nonsingular real algebraic set in R n , with dim X = m 1 , and let a be a point in X. Then there exists a linear map π : R n R m for which the following hold:

  1. The restriction π | X : X m is a proper map with finite (some possibly empty) fibers.

  2. The map π | X is transverse to π ( a ) .

Proof.

Let Y X - a be the translate of X. Let L ( n , m ) be the space of all linear maps from n to m . By a suitable version of the Noether normalization theorem, see [11, Theorem 13.3] and its proof, there is a nonempty Zariski open subset Ω of L ( n , m ) such that every β Ω is a surjective map whose restriction β | Y is proper with finite fibers. Moreover, there is γ Ω such that the derivative of φ γ | Y : Y m at the origin 0 Y is an isomorphism. After a linear coordinate change, we may assume that γ is the canonical projection n = m × n - m m .

For any constant ε > 0 , we set

M ε { ( t i j ) m × n : | t i j | < ε  for  1 i m ,  1 j n }

and consider the map Φ : n × M ε m defined by

Φ ( x , t ) = ( x 1 + j = 1 n t 1 j x j , , x m + j = 1 n t m j x j ) ,

where x = ( x 1 , , x n ) n and t = ( t i j ) M ε . If ε is sufficiently small, then Φ | Y × M ε is a submersion since for every point x Y { 0 } the restriction of Φ to { x } × M ε is a submersion, and φ is a submersion at the origin 0 Y . Hence, according to the standard consequence of Sard’s theorem [1, p. 48, Theorem 19.1], the map Φ t : Y m , Φ t ( x ) = Φ ( x , t ) , is transverse to the origin 0 m for some t M ε . Finally, define π L ( n , m ) by π ( x ) = Φ ( x , t ) and note that π | Y = Φ t . If ε is small, then π belongs to Ω and has the required properties. ∎

For the sake of clarity we include the following observation.

Lemma 2.9.

Let a , b be two points in R m and let ε > 0 be a constant. For the Euclidean norm on R m , if a 2 > ε and b 2 < ε 9 , then a - b 2 > 4 ε 9 .

Proof.

The conclusion follows immediately from the inequality

a - b a - b .

Finally, we are ready to complete the main task of this section.

Proof of Proposition 2.1.

Our aim is to prove analyticity of f in some neighborhood of a in U.

Preparation. Since the problem is local, we are allowed to shrink U if convenient. One may assume that k 2 and the point a U is the origin in n . We will reduce the problem to the case where X ( Ω × n - 2 ) is a finite analytic covering over an open subset Ω of 2 with U being one of the sheets, satisfying additional properties useful in the sequel.

By Lemma 2.8, there is a linear map π : n 2 such that the restriction π | X : X 2 is a proper map with finite fibers and is transverse to ( 0 , 0 ) 2 . After a linear coordinate change in n we may assume that π : n = 2 × n - 2 2 is the canonical projection. Now we choose an open neighborhood Ω 2 of ( 0 , 0 ) such that

( π | X ) - 1 ( Ω ) = U 1 U s ,

where the U l are pairwise disjoint open subsets of X, each π | U l : U l Ω is a real analytic diffeomorphism, and a U 1 U . Let ( x , y , z ) , where z = ( z 1 , , z n - 2 ) , be the variables in n . The inverse map ( π | U l ) - 1 : Ω U l is of the form ( x , y ) ( x , y , φ l ( x , y ) ) , where

φ l = ( φ 1 l , , φ n - 2 l ) : Ω n - 2

is a real analytic map. Shrinking U and Ω we may assume that U 1 = U and, for some open subset U ~ of n and polynomial functions P 1 , , P n - 2 on n , we have

(2.1) det ( ( P 1 , , P n - 2 ) ( z 1 , , z n - 2 ) ( b ) ) 0 for all  b U ,
U = X U ~ = 𝒵 ( P 1 ) 𝒵 ( P n - 2 ) U ~ .

By construction,

τ : Ω U , ( x , y ) ( x , y , φ 1 ( x , y ) )

is the inverse of the map π | U : U Ω . Note that τ ( 0 , 0 ) = a , where a is the origin in n .

By Lemma 2.6i there is an open disc B 2 centered at ( 0 , 0 ) such that C p , q B for all primes 2 < p < q . Rescaling the coordinates in 2 we may assume that Ω contains the closure B ¯ ; for any subset A of m , by A ¯ we denote the closure of A in the Euclidean topology. From now on, U , Ω and B are kept fixed. This completes the preparation stage.

Claim 2.10.

The restriction f | U { a } has a real analytic extension to U.

Proof of Claim 2.10.

Suppose that f | U { a } does not have a real analytic extension. We will construct an algebraic curve C k ( U ) (defined in (1.5.2)) with a singular point at a such that the restriction f | C { a } does not have a real analytic extension to C. The existence of such a C contradicts the hypothesis of Proposition 2.1.

First note that f τ | Ω { ( 0 , 0 ) } does not have a real analytic extension to Ω and

( f τ ) ( x , y ) = ( g τ ) ( x , y ) ( h τ ) ( x , y ) for all  ( x , y ) Ω { ( 0 , 0 ) } .

Let m 0 be a positive integer provided by Lemma 2.7 applied to g τ and h τ . Fix a pair ( p , q ) Λ k such that m 0 < p (for Λ k introduced in the paragraph preceding Notation 1.5). Now we consider two cases depending on the number s of sheets of the finite covering

π | X ( Ω × n - 2 ) : X ( Ω × n - 2 ) Ω .

The case s = 1 is easy. Indeed, by Lemma 2.5 (with v = 0 ) and by Lemma 2.7 the restriction f τ | C p , q { ( 0 , 0 ) } does not have a real analytic extension to C p , q . Next, by Lemma 2.6, we conclude that the algebraic curve C = ( π | X ) - 1 ( C p , q ) is homeomorphic to the unit circle and a is the unique singular point of C. Moreover, by Lemma 2.2, the germ C a is -analytically equivalent to the germ of E p , q at ( 0 , 0 ) . Furthermore, the complex analytic set-germ of the algebraic complexification of C at a is irreducible because, by Lemma 2.4, the complex analytic set-germ of the algebraic complexification of C p , q at ( 0 , 0 ) is irreducible, hence C is faithful at a. Thus C is an element of the collection k ( U ) , with singular point at a, such that the restriction f | C { a } does not have a real analytic extension to C.

Henceforth assume that s 2 . First we construct the curve C and then we discuss its properties allowing us to complete the proof of Claim 2.10. Choose a constant ε > 0 satisfying

(2.2) inf 2 l s inf ( x , y ) B ¯ i = 1 n - 2 ( φ i 1 ( x , y ) - φ i l ( x , y ) ) 2 > ε .

(The sum above does not vanish at any point of Ω, because the U l are pairwise disjoint subsets of X. Hence, its infimum over the compact subset B ¯ of Ω is strictly positive.)

By the Weierstrass approximation theorem, for i = 1 , , n - 2 , there exist polynomial functions Φ i : 2 with Φ i ( 0 , 0 ) = φ i 1 ( 0 , 0 ) = 0 and

(2.3) sup ( x , y ) B ¯ i = 1 n - 2 ( φ i 1 ( x , y ) - Φ i ( x , y ) ) 2 < ε 9 .

Given a positive integer r, define two functions K r , W r on n by

K r ( x , y , z ) = ( i = 1 n - 2 3 ε ( z i - Φ i ( x , y ) ) 2 ) r ,
W r ( x , y , z ) = F p , q ( x , y ) + K r ( x , y , z ) .

We will prove that one can take

C = 𝒵 ( W r | X )

for r sufficiently large. The proof will be preceded by a discussion of some properties of the polynomial function W r .

Subclaim 2.10.1.

For r large enough, the gradient of W r τ does not vanish at any point of Z ( W r τ | B ) { ( 0 , 0 ) } . Moreover, there exists a real analytic diffeomorphism-germ σ : ( R 2 , ( 0 , 0 ) ) ( R 2 , ( 0 , 0 ) ) such that σ ( Z ( W r τ ) ( 0 , 0 ) ) = Z ( H p , q ) ( 0 , 0 ) .

Proof of Subclaim 2.10.1.

By Lemma 2.3 with v r = K r τ and by Lemma 2.2 with v = 0 there is an open neighborhood V B of the origin such that for r large enough, there is a real analytic diffeomorphism σ r : V σ r ( V ) , σ r ( 0 , 0 ) = ( 0 , 0 ) , with W r τ = H p , q σ r on V. Then σ r ( 𝒵 ( W r τ ) ( 0 , 0 ) ) = 𝒵 ( H p , q ) ( 0 , 0 ) . Moreover, the gradient of W r τ does not vanish at any point of ( 𝒵 ( W r τ ) V ) { ( 0 , 0 ) } because the gradient of H p , q vanishes only at ( 0 , 0 ) . It remains to check that the gradient of W r τ does not vanish at any point of 𝒵 ( W r τ | B ) V .

By (2.3), for ( x , y ) B ¯ , we obtain

(2.4) K r ( τ ( x , y ) ) = ( i = 1 n - 2 3 ε ( φ i 1 ( x , y ) - Φ i ( x , y ) ) 2 ) r < ( 1 3 ) r .

Since W r τ - F p , q = K r τ on Ω, it readily follows from (2.4) that the sequence { W r τ | B ¯ } converges to F p , q | B ¯ in the 𝒞 1 topology. Now recall that the gradient of F p , q does not vanish at any point of 𝒵 ( F p , q ) V = C p , q V . Consequently, for large r, the gradient of W r τ does not vanish at any point of 𝒵 ( W r τ | B ) V . ∎

Subclaim 2.10.2.

For r large enough, Z ( W r τ | B ) is homeomorphic to C p , q .

Proof of Subclaim 2.10.2.

First let us recall that for any one-dimensional 𝒞 2 manifold M 2 , by a tubular neighborhood of M we mean a pair ( N , η ) , where N is an open neighborhood of M and η : N M is a 𝒞 1 map such that for every x M , η - 1 ( x ) is a segment normal to M at x. It is clear that for every x 0 M there is an open neighborhood E of x 0 in M admitting a tubular neighborhood ( N E , η E ) in 2 of constant radius (that is, there is a c > 0 such that η E - 1 ( x ) is a segment centered at x of length c for every x E ).

For a tubular neighborhood ( N , η ) of C p , q with N B we will prove that the restriction η | 𝒵 ( W r τ ) N : 𝒵 ( W r τ ) N C p , q is a homeomorphism if r is large enough (so that in particular 𝒵 ( W r τ ) N = 𝒵 ( W r τ | B ) ). Since the function F p , q defining C p , q is not a submersion (having the origin as a critical point), we first carry out the construction locally in a neighborhood of every point of C p , q and then show that it can be globalized.

Recall that the gradient of F p , q does not vanish at any element b C p , q { ( 0 , 0 ) } . Hence for every b C p , q { ( 0 , 0 ) } there exist an ε b > 0 and a tubular neighborhood ( N b , η b ) of N b C p , q b , with N b B , of constant radius such that

  1. for every x N b C p , q we have

    inf | ( F p , q α x ) | > ε b ,

    where α x : ( - 1 , 1 ) η b - 1 ( x ) is an affine linear map from the interval ( - 1 , 1 ) onto the segment η b - 1 ( x ) .

By (2.4), the sequence { W r τ } converges to F p , q in the 𝒞 1 topology on every N b ¯ . By Lemma 2.3 with v r = K r τ , there are an open neighborhood V B of the origin and a sequence { H r : V B } of real analytic maps converging to the inclusion map V B in the 𝒞 1 topology such that H r : V H r ( V ) is an analytic diffeomorphism and F p , q H r = W r τ on V for r large enough. We may assume, shrinking V if necessary, that there is a 𝒞 1 function f : V whose gradient does not vanish at any point of V such that 𝒵 ( f ) = C p , q V . Therefore, it follows that there exist an ε ( 0 , 0 ) > 0 and a tubular neighborhood ( N ( 0 , 0 ) , η ( 0 , 0 ) ) of N ( 0 , 0 ) C p , q ( 0 , 0 ) , with N ( 0 , 0 ) ¯ V , of constant radius such that

  1. for every x N ( 0 , 0 ) C p , q we have

    inf | ( f α x ) | > ε ( 0 , 0 ) ,

    where α x : ( - 1 , 1 ) η ( 0 , 0 ) - 1 ( x ) is an affine linear map from the interval ( - 1 , 1 ) onto the segment η ( 0 , 0 ) - 1 ( x ) .

As C p , q is compact, there are b 1 , , b s C p , q { ( 0 , 0 ) } such that C p , q j = 0 s N b j , where b 0 = ( 0 , 0 ) . Define f r , j on N b j to be f H r if j = 0 and W r τ if j = 1 , , s . Set ε := min j = 0 , , s ε b j . Since f r , 0 converges to f | N b 0 and f r , j converges to F p , q | N b j , for j = 1 , , s , we conclude that for r large enough, by ( * ) and ( * * ),

  1. for every j = 0 , , s and x N b j C p , q we have inf | ( f r , j α x ) | > ε 2 .

Consequently, for r large enough, for every j = 0 , , s and x N b j C p , q the function f r , j has precisely one zero on η b j - 1 ( x ) , hence W r τ has precisely one zero on η b j - 1 ( x ) (in view of 𝒵 ( f r , j ) = 𝒵 ( W r τ ) N b j , for j = 0 , , s ).

Let ( N , η ) be a tubular neighborhood of C p , q with N j = 0 s N b j . Then for r large enough, by (2.4) and 𝒵 ( F p , q ) = C p , q B , we have 𝒵 ( W r τ ) ( B ¯ N ) = , and by the previous paragraph, W r τ has precisely one zero on η - 1 ( x ) for every x C p , q . In particular, 𝒵 ( W r τ ) N = 𝒵 ( W r τ | B ) and the map η | 𝒵 ( W r τ ) N : 𝒵 ( W r τ ) N C p , q is bijective. Clearly, this map is continuous, hence it is a homeomorphism in view of compactness of its domain. ∎

Subclaim 2.10.3.

For r large enough, the real algebraic curve Z ( W r | X ) is contained in U and is homeomorphic to the unit circle.

Proof of Subclaim 2.10.3.

By Subclaim 2.10.2 and Lemma 2.6iii, for r large enough, the analytic curve 𝒵 ( W r τ | B ) is homeomorphic to the unit circle. Since τ : Ω U is a real analytic diffeomorphism, it is sufficient to show that

(2.5) 𝒵 ( W r | X ) τ ( B )

for large r.

Let us recall that C p , q = 𝒵 ( F p , q ) B . Since F p , q > 0 on the unbounded component of 2 𝒵 ( F p , q ) , we have