Algebraic surfaces determine analyticity of functions

Let f:X→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f :X \rightarrow \mathbb {R}$$\end{document} be a function defined on a nonsingular real algebraic set X of dimension at least 3. We prove that f is an analytic (resp. a Nash) function whenever the restriction f|S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f|_{S}$$\end{document} is an analytic (resp. a Nash) function for every nonsingular algebraic surface S⊂X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S \subset X$$\end{document} whose each connected component is homeomorphic to the unit 2-sphere. Furthermore, the surfaces S can be replaced by compact nonsingular algebraic curves in X, provided that dimX≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X \ge 2$$\end{document} and f is of class C∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}^{\infty }$$\end{document}.

where k ≥ 1 and the ϕ i : R m → R are polynomial functions, with ϕ 0 not identically 0 on any connected component of M . For this reason, Nash functions are called algebraic functions in the older literature, see for example [5].
Let X ⊂ R m be an irreducible nonsingular algebraic set of dimension n ≥ 0. We define d(X) to be the supremum of the number of points in the intersection X ∩ L, where L runs through the family of all affine (m − n)-planes in R m that are transverse to X. Clearly, d(X) is a positive integer, see [11,Theorem 11.5.3]. Given an integer k with 1 ≤ k ≤ n − 1, we denote by F k (X) the collection of all nonsingular algebraic subsets Z ⊂ X having at most d(X) connected components, each of which is homeomorphic to the unit k-sphere S k . In what follows, we only make use of F k (X) with k = 1 and k = 2.
Theorem 1. Let f : X → R be a function defined on an irreducible nonsingular algebraic set X ⊂ R m of dimension n ≥ 3. Assume that the restriction f | S is an analytic (resp. a Nash) function for every algebraic surface S ∈ F 2 (X). Then f is an analytic (resp. a Nash) function.
The proof of Theorem 1 requires some preparation. Along the way, we establish results which are of independent interest. We emphasize that the function f in Theorem 1 is not assumed to be continuous.
Let B n be the open unit ball in R n . For any integer k with 1 ≤ k ≤ n − 1, we denote by E k (B n ) the collection of all Euclidean k-spheres in B n passing through the origin, that is, all algebraic sets Σ k ⊂ R n of the form where c ∈ R n , 0 < c < 1 2 , and V ⊂ R n is a vector subspace of dimension k + 1. In our results only E k (B n ) with k = 1 and k = 2 are relevant.
Theorem 2. Let f : B n → R be a function defined on the open unit ball B n ⊂ R n . Assume that n ≥ 3 and the restriction f | Σ 2 is an analytic (resp. a Nash) function for every Euclidean 2-sphere Σ 2 ∈ E 2 (B n ). Then f is an analytic (resp. a Nash) function.
The analytic case in Theorem 2 is already settled in [3,Theorem 2]. It plays the key role in the proof of the Nash case.
The inversion is a biregular isomorphism. It maps B n \ {0} onto the complement R n \ B n of the closed unit ball B n and gives a one-to-one correspondence between the Euclidean k-spheres in E k (B n ) and the affine k-planes contained in R n \ B n .
Proof of Theorem 2. As indicated above, we may assume that f is an analytic function and the restriction f | Σ 2 is a Nash function for every Euclidean 2sphere Σ 2 ∈ E k (B n ). Hence the function is analytic and its restriction to any affine 2-plane contained in R n \ B n is a Nash function. Evidently, the restriction of g to any affine line contained in Vol. 118 (2022) Algebraic surfaces determine analyticity of functions 59 R n \ B n is a Nash function. It follows from [2, Theorem 2.4] that g is a Nash function on R n \ B n , and therefore so is f on B n .

Proposition 3.
Let X ⊂ R m be an irreducible nonsingular real algebraic set of dimension n ≥ 1, and let p be a point in X. Then there exists a linear map λ : R m → R n for which the following hold: (i) The restriction λ| X : X → R n is a proper map with finite fibers (some fibers may be empty).
Let L(m, n) be the space of all linear maps from R m to R n , which we identify with the space M (n, m) of all n-by-m matrices with real entries. By a suitable version of the Noether normalization theorem, see [6,Theorem 13.3

] and its proof, there is a nonempty Zariski open subset Ω
. It follows that β is surjective and the restriction β| Y is a proper map with finite fibers. Now we choose a linear map γ ∈ Ω such that the derivative of ϕ := γ| Y : Y → R n at the origin 0 ∈ Y is an isomorphism. After a coordinate change, we may assume that γ is the canonical projection For any constant ε > 0, we set If ε is sufficiently small, then Φ is a submersion since for each point x = 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 [7, p. 79, Theorem 2.7], the If ε is small, then λ belongs to Ω and has the required properties.
Proof of Theorem 1. Let p be a point in X and let λ : R m → R n be a linear map as in Proposition 3. We can choose a constant r > 0 such that where B(λ(p), r) ⊂ R n is the open ball centered at λ(p) with radius r, the U i are pairwise disjoint open subsets of X, λ| Ui : U i → B(λ(p), r) are Nash isomorphisms, and p ∈ U 1 . Clearly, l ≤ d(X). Define the map π : X → R n by Then π −1 (B n ) = U 1 ∪ · · · ∪ U l and the restriction π| Ui : U i → B n is a Nash isomorphism for i = 1, . . . , l.
Making use of local coordinate charts and applying Theorem 2, we immediately obtain the following.
Theorem 4. Let f : M → R be a function defined on an analytic (resp. a Nash) manifold M of dimension n ≥ 3. Assume that the restriction f | N is an analytic (resp. a Nash) function for every analytic (resp. Nash) submanifold N ⊂ M homeomorphic to S 2 . Then f is an analytic (resp. a Nash) function.

The analytic case in Theorem 4 is contained in [3, Theorem 1].
Replacing in Theorem 1 (resp. Theorem 2) F 2 (X) by F 1 (X) (resp. E 2 (B n ) by E 1 (B n )), one would get a false statement. Then the restriction of f is an analytic (resp. a Nash) function on each nonsingular analytic (resp. Nash) curve in R 3 , but f is not even continuous at (0, 0, 0). To establish the first part of the assertion, it suffices to prove that for any nonsingular analytic curve C ⊂ R 3 passing through (0, 0, 0), the restriction f | C is analytic at (0, 0, 0). Since C is nonsingular, it has near (0, 0, 0) a local analytic parametrization where x(0) = y(0) = z(0) = 0, and at least one of the analytic functions x(t), y(t), z(t) has a zero of order 1 at t = 0. It is not hard to check that the function f (x(t), y(t), z(t)) is analytic for t near 0 ∈ R. Thus f | C is analytic at (0, 0, 0), as required.
Clearly, the function f is not continuous at (0, 0, 0) since on the curve x 2 − y 3 = 0, z = 0, it is equal to 1 x 2 away from (0, 0, 0). However, for C ∞ functions, we have the following version of Theorem 1.
Theorem 6. Let f : X → R be a C ∞ function defined on an irreducible nonsingular algebraic set X ⊂ R m of dimension n ≥ 2. Assume that the restriction f | C is an analytic (resp. a Nash) function for every algebraic curve C ∈ F 1 (X). Then f is an analytic (resp. a Nash) function. Vol. 118 (2022) Algebraic surfaces determine analyticity of functions 61 Proof. We argue as in the proof of Theorem 1, substituting Theorem 7 below for Theorem 2.
Theorem 7. Let f : B n → R be a C ∞ function defined on the unit open ball B n ⊂ R n . Assume that n ≥ 2 and the restriction f | Σ 1 is an analytic (resp. a Nash) function for every Euclidean 1-sphere Σ 1 ∈ E 1 (B n ). Then f is an analytic (resp. a Nash) function.
Proof. To begin with, we consider the analytic case, assuming that the restriction f | Σ 1 is an analytic function for every Euclidean 1-sphere Σ 1 ∈ E 1 (B n ). First we prove analyticity of f on the punctured unit ball B n \ {0}. This is equivalent to proving analyticity of the function The problem is local, so fix a point b ∈ R n \ B n . Our goal is to show that g is analytic at b. Evidently, g is of class C ∞ and its restriction to any affine line contained in R n \ B n is analytic. Let L := the set of all affine lines contained in R n \ B n and passing through b.
The union C of all lines in L is a cone in R n and the set U := C \ {b} is open in R n . Consider the series Σ k P k of homogeneous polynomials in n variables . . . , α n ), |α| = α 1 + · · · + α n .
By construction, for each vector line Λ ⊂ R n with b + Λ ∈ L, the series Σ k P k (x) converges to g(b + x) for all x ∈ Λ in a neighborhood of 0 ∈ Λ. Hence, in view of Lemma 8 below, the series Σ k P k (x) converges to an analytic function γ : B → R defined on an open ball B ⊂ R n centered at the origin. Clearly, the function is analytic on the ball We claim that g = γ b in a neighborhood of b in R n . Indeed, fix an affine line L 0 ∈ L and let K be any affine line contained in R n \ B n that is parallel to L 0 ∈ L and satisfies K ∩ B b ∩ U = ∅. The functions g and γ b coincide on K ∩ B b ∩ U , hence the analytic functions g| K∩B b and γ b | K∩B b are equal. The union of the sets K ∩ B b , for all K as above, is a neighborhood of b in R n . It follows that g and γ b are equal in a neighborhood of b in R n , as claimed. So f is an analytic function on B n , except possibly at the origin.
To prove that f is analytic at the origin, we use the inversion is of class C ∞ , analytic except possibly at μ a (0), and its restriction is analytic on every affine line passing through μ a (0) and contained in U a . Arguing as above, we show that the Taylor series of h at μ a (0) converges to h in a neighborhood of μ a (0) in R n . Thus f is analytic at the origin, and therefore everywhere on B n . In the Nash case, we proceed as in the proof of Theorem 2.
We have used the following result, see [4,Lemma 3] for the proof.

Lemma 8. Let
∞ k P k be a series of real homogenous polynomials in n variables, deg P k = k. Assume that there exists a nonempty open subset Ω ⊂ S n−1 such that for every point a ∈ Ω, one can find a constant ρ a > 0 such that the series ∞ k P k (x) converges at x = ρ a a. Then there exist constants c > 0, r > 0 such that |P k (z)| ≤ c 2 k for z ∈ C n , z ≤ r, k ≥ 0. In particular, the function z → ∞ k P k (z) is holomorphic in the ball z < r, z ∈ C n .
Working on local coordinate charts, we derive from Theorem 7 the following.
Theorem 9. Let f : M → R be a C ∞ function defined on an analytic (resp. a Nash) manifold M of dimension n ≥ 2. Assume that the restriction f | C is an analytic (resp. a Nash) function for every analytic (resp. Nash) submanifold C ⊂ M homeomorphic to S 1 . Then f is an analytic (resp. a Nash) function.
One can compare Theorems 6 and 9 with the following example. is of class C ∞ and analytic with respect to each variable separately. However, f is not analytic as a function of two variables.