Let be a function defined on a connected nonsingular real algebraic set X in . 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 and k is a positive integer, then f is a regular function whenever the restriction is a regular function for every algebraic curve C in X that is a 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 for some primes . If , then f is a regular function whenever the restriction 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 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  or  for the general theory of real algebraic sets, real regular functions, and related topics. Unless explicitly stated otherwise, we consider and all its subsets endowed with the Euclidean topology induced by the standard norm on .
Let Z be an algebraic set in . 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 if there exist k polynomial functions on , vanishing identically on Z, such that their set of common zeros coincides with Z in a neighborhood of a in and their Jacobian matrix at a has rank k, see [4, 26].
The algebraic complexification of Z is the smallest complex algebraic subset of that contains Z ( is viewed as a subset of ). Note that Z is nonsingular at a if and only if is nonsingular at a (in the algebraic sense), which according to [28, Proposition 4] is equivalent to being a complex analytic manifold in an open neighborhood of a in . Assuming that Z is nonsingular at a, for any real analytic map-germ , the complexification of φ is the uniquely determined complex analytic map-germ 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 of dimension , and C a real algebraic curve in X. Let be a pair of primes, , and
We say that C is of type at if there exists a real analytic diffeomorphism-germ whose complexification maps the germ of at a onto the germ of at . We say that C is a -curve if it has a unique singular point a and it is of type at a. If k is a nonnegative integer with , then is a submanifold of , hence any -curve in X is a submanifold.
1.1 Main results (simplified versions)
Let k be a positive integer and let be a function defined on a connected nonsingular algebraic set X in , with . Then the following conditions are equivalent:
f is regular on X.
For every algebraic curve C in X that is homeomorphic to the unit circle and is either nonsingular or a - curve for some odd primes with (so in particular is a submanifold), the restriction is a regular function.
A real-valued function on , for , need not be regular (or even continuous) despite the fact that all its restrictions to nonsingular algebraic curves in are regular.
Let , for , be the function defined by
The restriction of g to every nonsingular algebraic curve in is a regular function, and the restriction of g to every one-dimensional real analytic submanifold of is a real analytic function, see [17, Example 2.3] for the case . However, g is not regular on because it is not even locally bounded on the curve defined by , . 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 , 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 , 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.
Let be a function defined on a connected nonsingular algebraic set X in , with . Then the following conditions are equivalent:
f is regular on X.
For every nonsingular algebraic surface in that is homeomorphic to the unit 2 - sphere, the restriction is a regular function.
This result also has a sharper version, Theorem 1.8 (for ), 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  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 .
Given nonnegative integers and manifolds , we consider the space of maps endowed with the weak topology defined in [15, p. 34], which will be referred to as the 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 or, more generally, a connected component of such a set. Assume that S is a submanifold homeomorphic to the unit d-sphere , where , and or . Then there exists a diffeomorphism 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 diffeomorphism . By the Weierstrass approximation theorem, ψ regarded as a map into can be approximated in the topology by a polynomial map . If σ is sufficiently close to ψ, then the composite map of σ with the radial projection has the required properties.
Besides [6, 23], the key ingredient in our proof of Theorem 1.9 is Theorem 1.7 engaging -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 and a point , we regard the germ of Z at b as a real analytic set-germ. The analytic complexification of is the smallest complex analytic set-germ at that contains , see [27, pp. 91–92]. We say that Z is faithful at b if the analytic complexification of is equal to the germ at b of the algebraic complexification of Z; we say that Z is faithful if it is faithful at each of its points. Here “faithful” replaces “quasi-regular” used in . 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 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 , the ideal of function-germs vanishing on is generated by the germs at b of polynomial functions 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 , which is equivalent to Z being faithful at b and being a manifold in an open neighborhood of b in (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.
The real polynomial is irreducible and its gradient vanishes only at . It follows that the real algebraic curve
is irreducible and has the only singular point at . On the other hand, E is a real analytic submanifold of . It suffices to justify this claim in a neighborhood of in , which is straightforward because for the curve E is given by the real analytic equation , and . Consequently, E is not faithful at . 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 with . Let be a pair of primes, . Then C is a -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 such that is equal to the germ of at .
1.2 Main results (full generality)
Given a real-valued function α on some set Ω, we denote by the zero set of α, that is, .
It will be convenient to consider regular functions in a more general context than usual. Let A be an arbitrary subset of . A function is said to be regular at a point if there exist two polynomial functions such that and for all ; 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 with
see the proof of [4, Proposition 3.2.3].
A function defined on a subset A of is said to be real analytic if for every point there exist an open neighborhood of a and a real analytic function (in the usual sense) such that G and g agree on . Obviously, every regular function on A is real analytic.
For integers n and d, with , an algebraic set Σ in is called a Euclidean d-sphere if it can be expressed as
where Q is an affine -plane in , , and . 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
Let X be an irreducible nonsingular algebraic set in , for , and let U be a nonempty open subset of X.
(1.5.1) Collection . For an integer d, with , we denote by the collection of all d-dimensional irreducible nonsingular algebraic sets S in X, contained in U, satisfying one of the following two conditions:
If U is connected, then S is homeomorphic to the unit d-sphere.
If U is disconnected, then S has at most two connected components, each homeomorphic to the unit d-sphere.
Only the cases and will be relevant.
(1.5.2) Collection . Given a positive integer k, we denote by the collection of all -curves C in X for such that C is contained in U and is homeomorphic to the unit circle. Recall that such curves are submanifolds of U.
(1.5.3). For any primes , define
For , the translates
are algebraic curves in defined by the equations
(1.5.4) Collections and . For any affine 2-plane Q in , we choose once and for all an affine-linear isomorphism . If , then and is chosen to be the identity map. Given a positive integer k, we denote by (resp. the collection of all algebraic curves C in for which there exist an affine 2-plane Q in containing C and a pair of primes such that is a translate of (resp. ). In particular, (resp. ) is the collection of all translates of the curves (resp. ) for . The curves in and in are therefore given by explicit and simple polynomial equations. By Lemmas 2.2 and 2.6, .
Here is a characterization of regular functions on , for .
Let be two integers, , . Then, for a function , the following conditions are equivalent:
f is regular on .
The restriction of f to every irreducible algebraic curve in is a regular function.
The restriction of f to every algebraic curve, which is either a Euclidean circle in or is in the collection , is a regular function.
The restriction of f to every Euclidean circle in is a regular function, and the restriction of f to every algebraic curve in the collection is a real analytic function.
The restriction of f to every algebraic curve, which is either an affine line parallel to one of the coordinate axes of or is in the collection , is a regular function.
The restriction of f to every affine line parallel to one of the coordinate axes of is a regular function, and the restriction of f to every algebraic curve in the collection is a real analytic function.
Let us note that each curve in (1.5.3) is rational. Indeed, the map , is regular with rational inverse , where are (possibly negative) integers satisfying . Consequently, each curve in the collection is rational. Thus, according to Theorem 1.6, regularity of real-valued functions on can be detected on rational curves. This is not the case if is replaced by an arbitrary nonsingular real algebraic set. For example, the nonsingular real cubic curve
is not rational, so the Cartesian product does not contain any rational curve. Observe that C is diffeomorphic to , hence X is diffeomorphic to .
In the general setting we have the following result.
Let k be a positive integer, X an irreducible nonsingular algebraic set in , with , and U a nonempty open subset of X. Then, for a function , the following conditions are equivalent:
f is regular on U.
The restriction of f to every irreducible algebraic curve in X , contained in U , is a regular function.
The restriction of f to every algebraic curve, which is either in the collection or in the collection , is a regular function.
The restriction of f to every algebraic curve in the collection is a regular function, and the restriction of f to every algebraic curve in the collection is a real analytic function.
By the definition of the collections and , Theorem 1.7 is both more general and stronger than Theorem 1.1. Taking 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 , with , can also be characterized as follows.
For a function , with , the following conditions are equivalent:
f is regular on .
The restriction of f to every Euclidean 2 - sphere in is a regular function.
As an application of Theorem 1.7, we will obtain the following.
Let X be an irreducible nonsingular algebraic set in , with , and let U be a nonempty open subset of X. Then, for a function , the following conditions are equivalent:
f is regular on U.
The restriction of f to every algebraic surface in the collection is a regular function.
Next, we give some comments on the assumptions in our theorems.
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 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 is nonsingular if and only if it is faithful and is a 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 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 be a connected component of U, and let be the function defined by on and on . 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 , with and , are not necessarily connected.
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 defined on a connected nonsingular real algebraic set X in (with ) 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 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.
Let k be a positive integer, X an irreducible nonsingular algebraic set in , with , and a function defined on an open subset U of X. Let a be a point in U and let be two real analytic functions such that
Assume that for every algebraic curve C in the collection , with singular point at a, the restriction is a real analytic function. Then f is real analytic on U.
Recall that the collection 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 will be analyzed first.
2.1 Real meromorphic functions of two variables
Let denote either the field of real numbers or the field of complex numbers. Let denote the ring of all -analytic function-germs at the origin in . If no confusion is possible, we will make no distinction between function-germs and their representatives. The ring is a local ring whose maximal ideal is denoted by . As usual, given a positive integer i and an ideal , we denote by the ith power of .
In the next four lemmas we discuss some properties of the polynomial functions and defined in (1.5.3).
Let be prime numbers. Then for every there exists a -analytic diffeomorphism-germ such that regarded as elements of satisfy
Clearly, , for some -analytic function-germs vanishing at the origin. Hence, it is sufficient to take
where are -analytic function germs satisfying . ∎
Let be prime numbers. Let B be an open disc in centered at the origin, real-valued analytic functions on B that vanish at the origin, and let for positive integers r. Then there are an open neighborhood of the origin, a positive integer , and real analytic maps for such that
the sequence converges to the inclusion map in the topology, and
for each the map is an analytic diffeomorphism for which
Since we have for , there are real analytic functions such that
Therefore taking , we get
where are real analytic functions. For define real analytic functions by the equations and , and note that
Since , there is an open disc in centered at the origin such that its closure is contained in B and the sequences , converge to 0 in the topology.
Shrinking and choosing a sufficiently large integer , we see that the formulas
for define real analytic maps from into , and is a real analytic diffeomorphism. By construction, the sequence converges to in the topology. By [15, Lemma 1.3, p. 36]), increasing and shrinking , we may assume that the map is a real analytic diffeomorphism for . Now, let be an open neighborhood of the origin with closure contained in . Increasing once again, we get for . Since on V, we have
for , the real analytic maps , for , satisfy the required conditions. ∎
Let be prime numbers. Then for every the -analytic function-germ is irreducible.
The -analytic function-germ of at the origin is irreducible. Hence the assertion is an immediate consequence of Lemma 2.2. ∎
For any germ A of a subset of at the origin, let denote the ideal of all function-germs in vanishing on A. The following lemma formulated in the real setting clearly has its complex analogue.
Let be prime numbers. Then for every the real analytic set-germ is irreducible of dimension 1, and
The real analytic set-germ is irreducible of dimension 1, and
Hence, the assertion follows immediately by Lemma 2.2. ∎
In the following lemma we establish some useful properties of the real algebraic curves defined in (1.5.3).
Let k be a positive integer and primes with and . Then the following hold:
, where are given by
and are (the only) real roots of the polynomial . In particular, is contained in the open disc in centered at with radius independent of .
The origin is the only singular point of the real algebraic curve .
is a manifold homeomorphic to the unit circle.
The real algebraic curve is faithful.
i Put . Solving the equation with respect to t, we obtain given by
ii Elementary calculation shows that is the unique critical point of lying on . Moreover, the real polynomial is irreducible in as its homogeneous form of the highest degree is irreducible over .
By i, is homeomorphic to the unit circle.
iv The algebraic complexification of is given by the complex solutions of the equation . Hence, the complex analytic set-germ of at the origin is irreducible by Lemma 2.4. Therefore is faithful in view of ii and iii. ∎
The following general observation will also be needed.
Let be an open neighborhood of , and real analytic functions with . Assume that the real analytic function
does not have a real analytic extension to Ω. Then there is a positive integer such that for every real analytic function with and the restriction does not have a real analytic extension to .
By nonextendability of f, the germ of g at the origin is a nonzero element of . After shrinking Ω and dividing g and h by their greatest common divisor we may assume that the germs of are relatively prime in and, again by nonextendability of f, we still have .
Let U be a polydisc about the origin in on which the complexification of h is well defined. Set and note that the germ of V at the origin is of complex dimension 1 as . Since are relatively prime in , there exists an irreducible complex analytic curve-germ contained in such that g does not vanish identically on . Indeed, otherwise g would vanish identically on . Hence, by the Nullstellensatz, there would be an integer t and a with . Since , we would have , contradicting the fact that are relatively prime in .
By the Puiseux theorem, we have a holomorphic normalization . Then is a nonzero holomorphic function. Define to be any integer strictly greater than the order of zero of at 0.
Let be a real analytic function satisfying the hypotheses of the lemma. Suppose that has a real analytic extension to . Then there is a such that . By assumption there is a such that . Passing to the complexifications of and w in a small neighborhood of the origin, we obtain 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 is the canonical projection and is a nonsingular algebraic set of pure dimension m such that is a proper map with finite fibers, then the following holds. For every to which is transverse, there is an open neighborhood Ω in such that is a finite analytic covering over Ω.
The following lemma is a modification of the Noether normalization lemma.
Let X be an irreducible nonsingular real algebraic set in , with , and let a be a point in X. Then there exists a linear map for which the following hold:
The restriction is a proper map with finite (some possibly empty) fibers.
The map is transverse to .
Let be the translate of X. Let be the space of all linear maps from to . 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 such that every is a surjective map whose restriction is proper with finite fibers. Moreover, there is such that the derivative of at the origin is an isomorphism. After a linear coordinate change, we may assume that γ is the canonical projection .
For any constant , we set
and consider the map defined by
where and . If ε is sufficiently small, then is a submersion since for every point the restriction of Φ to is a submersion, and φ is a submersion at the origin . Hence, according to the standard consequence of Sard’s theorem [1, p. 48, Theorem 19.1], the map , , is transverse to the origin for some . Finally, define by and note that . If ε is small, then π belongs to Ω and has the required properties. ∎
For the sake of clarity we include the following observation.
Let be two points in and let be a constant. For the Euclidean norm on , if and , then .
The conclusion follows immediately from the inequality
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 and the point is the origin in . We will reduce the problem to the case where is a finite analytic covering over an open subset Ω of with U being one of the sheets, satisfying additional properties useful in the sequel.
By Lemma 2.8, there is a linear map such that the restriction is a proper map with finite fibers and is transverse to . After a linear coordinate change in we may assume that is the canonical projection. Now we choose an open neighborhood of such that
where the are pairwise disjoint open subsets of X, each is a real analytic diffeomorphism, and . Let , where , be the variables in . The inverse map is of the form , where
is a real analytic map. Shrinking U and Ω we may assume that and, for some open subset of and polynomial functions on , we have
is the inverse of the map . Note that , where a is the origin in .
By Lemma 2.6 i there is an open disc centered at such that for all primes . Rescaling the coordinates in we may assume that Ω contains the closure ; for any subset A of , by we denote the closure of A in the Euclidean topology. From now on, and B are kept fixed. This completes the preparation stage.
The restriction has a real analytic extension to U.
Proof of Claim 2.10.
Suppose that does not have a real analytic extension. We will construct an algebraic curve (defined in (1.5.2)) with a singular point at a such that the restriction 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 does not have a real analytic extension to Ω and
Let be a positive integer provided by Lemma 2.7 applied to and . Fix a pair such that (for introduced in the paragraph preceding Notation 1.5). Now we consider two cases depending on the number s of sheets of the finite covering
The case is easy. Indeed, by Lemma 2.5 (with ) and by Lemma 2.7 the restriction does not have a real analytic extension to . Next, by Lemma 2.6, we conclude that the algebraic curve is homeomorphic to the unit circle and a is the unique singular point of C. Moreover, by Lemma 2.2, the germ is -analytically equivalent to the germ of at . 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 at is irreducible, hence C is faithful at a. Thus C is an element of the collection , with singular point at a, such that the restriction does not have a real analytic extension to C.
Henceforth assume that . 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 satisfying
(The sum above does not vanish at any point of Ω, because the are pairwise disjoint subsets of X. Hence, its infimum over the compact subset of Ω is strictly positive.)
By the Weierstrass approximation theorem, for , there exist polynomial functions with and
Given a positive integer r, define two functions , on by
We will prove that one can take
for r sufficiently large. The proof will be preceded by a discussion of some properties of the polynomial function .
For r large enough, the gradient of does not vanish at any point of . Moreover, there exists a real analytic diffeomorphism-germ such that .
Proof of Subclaim 2.10.1.
By Lemma 2.3 with and by Lemma 2.2 with there is an open neighborhood of the origin such that for r large enough, there is a real analytic diffeomorphism , , with on V. Then . Moreover, the gradient of does not vanish at any point of because the gradient of vanishes only at . It remains to check that the gradient of does not vanish at any point of .
By (2.3), for , we obtain
Since on Ω, it readily follows from (2.4) that the sequence converges to in the topology. Now recall that the gradient of does not vanish at any point of . Consequently, for large r, the gradient of does not vanish at any point of . ∎
For r large enough, is homeomorphic to .
Proof of Subclaim 2.10.2.
First let us recall that for any one-dimensional manifold , by a tubular neighborhood of M we mean a pair , where N is an open neighborhood of M and is a map such that for every , is a segment normal to M at x. It is clear that for every there is an open neighborhood E of in M admitting a tubular neighborhood in of constant radius (that is, there is a such that is a segment centered at x of length c for every ).
For a tubular neighborhood of with we will prove that the restriction is a homeomorphism if r is large enough (so that in particular ). Since the function defining 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 and then show that it can be globalized.
Recall that the gradient of does not vanish at any element . Hence for every there exist an and a tubular neighborhood of , with , of constant radius such that
for every we have
where is an affine linear map from the interval onto the segment .
By (2.4), the sequence converges to in the topology on every . By Lemma 2.3 with , there are an open neighborhood of the origin and a sequence of real analytic maps converging to the inclusion map in the topology such that is an analytic diffeomorphism and on V for r large enough. We may assume, shrinking V if necessary, that there is a function whose gradient does not vanish at any point of V such that . Therefore, it follows that there exist an and a tubular neighborhood of , with , of constant radius such that
for every we have
where is an affine linear map from the interval onto the segment .
As is compact, there are such that , where . Define on to be if and if . Set . Since converges to and converges to , for , we conclude that for r large enough, by ( ) and ( ),
for every and we have .
Consequently, for r large enough, for every and the function has precisely one zero on , hence has precisely one zero on (in view of , for ).
Let be a tubular neighborhood of with . Then for r large enough, by (2.4) and , we have , and by the previous paragraph, has precisely one zero on for every . In particular, and the map is bijective. Clearly, this map is continuous, hence it is a homeomorphism in view of compactness of its domain. ∎
For r large enough, the real algebraic curve is contained in U and is homeomorphic to the unit circle.