A GEOMETRY OF REAL POLYNOMIAL MAPPINGS^{∗)}

by Zbigniew Jelonek

IMUJ PREPRINT 1999/12 KRAKÓW

Abstract In this paper we study the set of points at which a real polynomial

mapping is not proper.

1 Introduction

Let f : X → Y be a continuous map of locally compact spaces. We say that the mapping f is not proper at a point y ∈ Y, if there is no a neighborhood U of a point y such that the set f^{−1}(cl(U)) is compact.

The set S_{f} of points at which the map f is not proper indicates how the map f differs from a proper map. In particular f is proper if and only if this set is empty, moreover if f(X) is open then S_{f} contains the border of the set f(X). The set S_{f} is a minimal set S with a property that the mapping f : X \ f^{−1}(S) → Y \ S is proper.

Key words: real polynomial mapping, quasi-finite mapping, the set of points over which a polynomial mapping is not proper.

*) This paper is partially supported by the grant of KBN number 2 PO3A 037 14.

Further, if f : ℝ^{n} → ℝ^{n} is a generically finite polynomial mapping, then the mapping f has a constant topological degree over every connected component of the set ℝ^{n} \ S_{f}.

In our previous paper [6] we described the set S_{f} in the case of complex polynomial mappings f : ℂ^{n} → ℂ^{m} and in the paper [7] we described this set in the case of real polynomial mappings f : ℝ^{2} → ℝ^{2}.

The aim of this paper is to do the same in the general case of real polynomial generically finite mappings f : ℝ^{n} → ℝ^{m}. Our main result is the following

Theorem. Let f : ℝ^{n} → ℝ^{m} be a polynomial generically-finite mapping. Then the set S_{f} is closed, semi-algebraic and for every non-empty connected component S ⊂ S_{f} we have 1 ≤ dim S ≤ n − 1.

Moreover, the set S_{f} is ℝ-uniruled. It means, that for every point a ∈ S_{f} there is a non-constant polynomial mapping ϕ : ℝ → S_{f} such that ϕ(0) = a. Further, there is a real polynomial P ∈ ℝ[x_{1}, ..., x_{m}] of degree at most (degf)^{n−1} − 1, such that S_{f} ⊂ P^{−1}(0).

Corollary. If f : ℝ^{n} → ℝ^{m} is a polynomial generically-finite mapping, then every non-empty connected component of S_{f} is unbounded.

In this paper we use methods from our recent papers [4] and [5].

2 Terminology.

Let X ⊂ ℝ^{n}(ℙ^{n}(ℝ)) be a semi-algebraic set. The Zariski closure of X will be denoted by cl_{Z}(X). By ℂX we denote a complexification of X, i.e., ℂX is a Zariski closure of the set X in ℂ^{n} (ℙ^{n}(ℂ)).

More generally, if X ⊂ ℙ^{n}(ℝ) × ℙ^{n}(ℝ) is an algebraic set, then by its complexification ℂX we mean a Zariski closure of the set X in ℙ^{n}(ℂ) × ℙ^{n}(ℂ) etc.

Let X ⊂ ℝ^{n} be an algebraic set. A mapping f : X → ℝ is called polynomial if it is a restriction of a polynomial F : ℝ^{n} → ℝ, F ∈ ℝ[x_{1}, ..., x_{n}]. More generally, a mapping f : X → ℝ^{m} is called polynomial if f = (f_{1}, ..., f_{m}) where mappings f_{i} are polynomial mappings. More generally, if X ⊂ ℙ^{n}(ℝ), Y ⊂ ℙ^{m}(ℝ) are semialgebraic sets and f : X → Y is a regular mapping then f is called polynomial if it has an extension to a regular mapping ℂf : ℂX → ℂY.

For an algebraic set X ⊂ ℝ^{n} we consider the coordinate ring of X: ℝ[X] := ℝ/I(X), where I(X) = {f ∈ ℝ[x_{1}, ..., x_{n}] : f(X) = {0}}.

A polynomial mapping f : X → ℝ^{m} is called generically finite if for generic x ∈ X the fiber f^{−1}(f(x)) is finite. It is easy to see that f is generically-finite if and only if the mapping f : ℂX → ℂ^{m} is genericallyfinite.

If g = ∑_{|α|≤n} a_{α}x^{α} ∈ ℂ[x_{1}, ..., x_{n}], then \overline{g} := ∑_{|α|≤n} \overline{a_{α}}x^{α} ∈ ℂ[x_{1}, ..., x_{n}], where the overband denotes the complex conjugate.

If f = (f_{1}, ..., f_{m}) : ℝ^{n} → ℝ^{m} is a polynomial mapping, then by deg f we mean the number max_{j=1}^{m} deg f_{j}.

3 Preliminaries.

In the beginning we recall some basic facts about polynomial mappings of ℂ^{2}. We start with the theorem about surfaces which contain ℂ^{2} as an open subset ([6]):

Theorem 3.1 Let X be a complete normal surface. Assume, that X contains a plane ℂ^{2} as an open, dense subset. Let W_{1}, W_{2} ⊂ X \ ℂ^{2} be two connected, complete curves without common components. Then the intersection W_{1} ∩ W_{2} is either the empty set or it is a point.

Definition 3.2 Let Γ be an affine curve such that there is a surjective polynomial mapping ϕ : ℂ → Γ. Then Γ is called to be an affine parametric line and the mapping ϕ is called to be a parametrization of Γ.

In the sequel we need also the following (see [6]):

Proposition 3.3 Let X be an irreducible affine variety of dimension ≥ 1. The following conditions are equivalent:

1) for every point x ∈ X there is a parametric affine line in X going through x;

2) there exists a Zariski-open, non-empty subset U of X, such that for every point x ∈ U there is a parametric affine line in X going through x;

3) there exists a subset U of X of the second Baire’s category, such that for every point x ∈ U there is a parametric affine line in X going through x;

4) there exists an affine variety W with dim W = dim X − 1 and a dominant polynomial mapping ϕ : W × ℂ → X.

Definition 3.4 An affine irreducible variety X is called ℂ-uniruled if it is of dimension ≥ 1, and satisfies one of equivalent conditions 1) − 4) listed in Proposition 3.7. More generally, an affine variety X is called ℂ-uniruled if all its irreducible components are ℂ-uniruled.

The next result was proved in our papers [5], [6]:

Theorem 3.5 Let f = (f_{1}, ..., f_{n}) : ℂ^{n} ⟶ ℂ^{n} be a dominant polynomial map. Then the set S_{f} of points at which f is either empty, or it is a ℂuniruled hypersurface. Moreover, its degree is not greater than

\frac{∏_{i=1}^{n} degf_{i} − μ(f)}{min_{i=1,...,n}deg f_{i}},

where μ(f) denotes the maximal number of points in (finite) fibers of f.

4 Mappings of the real plane.

In this section we study generically-finite polynomial mappings f : ℝ^{2} ⟶ ℝ^{m}. We start with the following:

Definition 4.1 Let Γ ⊂ ℝ^{n} be a semi-algebraic closed curve, such that there exists a polynomial surjective mapping ϕ : ℝ → Γ. Then Γ will be called a parametric semi-line and the mapping ϕ will be called a parametrization of Γ. If a curve Γ is algebraic it will be called a parametric line.

We come to our first result:

Theorem 4.2 Let f = (f_{1}, ..., f_{m}) : ℝ^{2} ⟶ ℝ^{m} be a generically finite polynomial mapping. Then the set S_{f} of points at which f is not proper is a union of finite number (possibly 0) of parametric semi-lines.

Proof. We can treat the mapping f as a rational mapping f : ℙ^{2}(ℝ) → ℙ^{n}(ℝ). Now resolve points of indeterminacy of f by sequence of blowing up’s. In this way we obtain a smooth surface Z and a regular mapping F : Z → ℙ^{n}(ℝ), such that res_{ℝ}^{2}F = f.

We can also consider the surface ℂZ and the mapping ℂF, which are canonical complexifications of Z and F (i.e. to obtain Z, F we consider complex extension of real blow up’s). Hence the mapping ℂF is regular in a neighborhood of Z. Without loss of generality we can assume that the mapping ℂF is regular on the whole of ℂZ (later we resolve only complex points of indeterminacy).

The set R := Z \ ℝ^{2} is connected and it is a union ⋃ R_{i} of circles R_{i} ≅ ℙ^{1}(ℝ). Moreover, the complexification ℂR_{i} := S_{i} is isomorphic to ℙ^{1}(ℂ) and it contains R_{i} as ℙ^{1}(ℝ), i.e., there is a (complex) biregular mapping Ψ_{i} : ℙ^{1}(ℂ) → S_{i} such that it induces a (real) biregular mapping ψ_{i} : ℙ^{1}(ℝ) → R_{i}. It follows from elementary properties of blowing up.

Let us denote by L_{∞} the hyperplane at infinity in ℝ^{m} and take Q := F^{−1}(L_{∞}). Let R_{j} be a component of R which is not in Q. We show that R_{j} has at most one common point with Q. Indeed, the curve Q′ := ℂF^{−1}(ℂL_{∞}) is connected as the complement of semi-affine surface ℂF^{−1}(ℂ^{m}) (see [6]). Hence our assertion follows from Theorem 3.1 (applied to the surface ℂZ).

Now let us note that the set S_{f} is exactly the set F (R \ Q). Let H be a connected (non-empty) component of the set S_{f}. Hence there are indices j_{1}, ..., j_{k} such that H = ⋃_{i=1}^{k} F (R_{j_{i}} \ Q). On the other hand we know that for every j the curve R_{j} ≅ ℙ^{1}(ℝ) has at most one common point with Q. Consequently, the restriction of the mapping F to R_{j} \ Q gives a regular mapping ϕ_{j} : ℝ → H ⊂ ℝ^{m}. Here we identify ℝ with R_{j} \Q or (if R_{j} ∩Q = ∅) with R_{j} minus one point. Without restriction of generality we can assume that all mappings ϕ_{j} are non-constant. Indeed, since the set R is connected, it means that the set F (R) ⊂ S_{f} ∪ L_{∞} is connected, too, hence H cannot be an isolated point. Consequently, we can excluded all constant mappings ϕ_{j}. Now, we show that the mapping ϕ_{j} must be polynomial.

Since the mapping ϕ_{j} is regular, we have ϕ_{j} = (P_{1}/Q_{1}, ..., P_{m}/Q_{m}), where P_{i}, Q_{i} are relatively prime real polynomials from ℝ[t] with Q_{i} ≠ 0 everywhere. (see [1], 3.1.9).

Moreover, the mapping ϕ_{j} extends uniquely to the mapping Φ_{j} : S_{j} ≅ ℙ^{1}(ℂ) → ℙ^{m}(ℂ). If ∏_{i=1}^{m} Q_{i} ≠ const, then S_{j} would have two or more (conjugate) complex common points with Q′ = ℂF^{−1}(ℂL_{∞}) (namely zeros of some of Q_{i}). By Theorem 3.1 it is a contradiction.

Hence, the mapping ϕ_{j} is indeed a polynomial mapping. This also implies that R_{j} ∩ Q ≠ ∅ (i.e. R_{j} \ Q ≅ ℝ), and consequently H = ⋃_{i=1}^{k} ϕ_{j_{i}}(ℝ).

Remark 4.3 Let S ⊂ ℝ^{m}, where m ≥ 2, be a parametric semi-line. We prove in the section 7 that there exists a polynomial mapping f : ℝ^{2} → ℝ^{m} with finite fibers for which S is exactly the set of points at which f is not proper (i.e., S = S_{f}).

5 Auxiliary results.

For a sequel we need some auxiliary results.

Theorem 5.1 Let X ⊂ ℝ^{m} be a real affine variety and Y ⊂ X be a closed subvariety. Let f : Y → ℝ^{n} be a polynomial mapping. Assume that dim X ≤ n. Then there exists a polynomial mapping F : X → ℝ^{n} such that

1) res_{Y} F = f

2) the mapping res_{ℂX\ℂY} ℂF : ℂX \ ℂY → ℂ^{n} is quasi-finite. In particular if the mapping f is quasi-finite, then the mapping F is quasifinite, too.

Proof. Let I(Y) = (h_{1}, ..., h_{r}) denote the ideal of Y in ℝ[X]. Then polynomials (h_{1}, ..., h_{r}) are also generators of the ideal of ℂY in ℂ[ℂX].

Let g = (g_{1}, …, g_{n}) be a polynomial extension of f to X. Let x_{1}, …, x_{m} be a coordinate polynomials in ℝ^{m}.

Take H_{ij} := h_{i} · x_{j}; i = 1, ..., r, j = 1, ..., m; and consider the map H = (g_{1}, …, g_{n}, h_{1}, ..., h_{r}, H_{11}, ..., H_{1m}, ..., H_{r1}, …, H_{rm}) : X → ℝ^{n+r+mr}. By construction ℂH is injective outside ℂY, H(ℂY) ⊂ ℂ^{n} × {0} and dim H(ℂX) = dim ℂX ≤ n. Take Γ := cl(H(ℂX)), it is a closed subset of ℂ^{n+r+mr}. It is easy to see that there exists a proper linear projection π : Γ → ℂ^{n} × {0}, such that π(H(X)) ⊂ ℝ^{n} × {0}.

Indeed, we have the canonical inclusion ℂ^{n+r+mr} ⊂ ℙ^{n+r+mr}(ℂ) and the set W of points at infinity of ℂY has dimension ≤ n−1. Hence we can find a real linear subspace L of the hyperplane at infinity of dimension r+mr−1 which is disjoint from W and from cl(ℂ^{n} × {0}). Let π_{ℂL} : ℙ^{n+r+mr}(ℂ) → cl(ℂ^{n} × {0}) be the projection determined by ℂL. Since L is a real subspace, we have π_{ℂL} : ℙ^{n+r+mr}(ℝ) → cl(ℝ^{n} × {0}) The restriction of π_{ℂL} to Γ is a finite map of Γ into ℂ^{n} × {0} and it is the projection π we were looking for.

Now it is enough to take F = π ∘ H.

Also the following lemma will be useful:

Lemma 5.2 Let {a_{1}, ..., a_{n}} ⊂ ℂ^{2}. Then there exists a polynomial automorphism Q = (Q_{1}, Q_{2}) : ℂ^{2} → ℂ^{2} such that

1) Q_{1}, Q_{2} ∈ ℝ[x, y]

2) Q(a_{i}) ∈ {(x, y) ∈ ℂ^{2} : x = 0}.

Proof. Let a_{i} = (x_{i}, y_{i}). There is a real linear transformation T of ℂ^{2} such that that T (a_{i}) = (x′_{i}, y′_{i}) and x′_{i} ≠ x′_{j} for i ≠ j and x′_{i} ≠ \overline{x′_{j}} for a_{i} ≠ \overline{a_{j}}. Indeed, let M be a non-zero real number such that M ≠ x_{i} − x_{j}/(y_{i} − y_{j}) for every y_{i} ≠ y_{j} and M ≠ x_{i} − \overline{x_{j}}/(y_{i} − \overline{y_{j}}) for every y_{i} ≠ \overline{y_{j}}. It is easy to see that the linear transformation T (x, y) = (x + My, y) has desired properties.

Hence, we can assume that x_{i} ≠ x_{j} for every i ≠ j and if x_{i} = \overline{x_{j}} then y_{i} = \overline{y_{j}}. Now let ϕ be a polynomial of a minimal degree such that ϕ(x_{i}) = y_{i} and ϕ(\overline{x_{i}}) = \overline{y_{i}}. By our assumption this polynomial exists and it has real coefficients (the latter we get computing these coefficients by Cramer’s rules). Now it is enough to take Q(x, y) = (x, y − ϕ(x)).

Lemma 5.3 Let X ⊂ ℝ^{n} be a real algebraic set. Let F : ℂX → ℂ^{m} be a polynomial mapping. If F (X) ⊂ ℝ^{m} then the mapping f : X ∋ x → F (x) ∈ ℝ^{m} is a real polynomial mapping. In particular F has a real extension to ℂ^{n}.

Proof. Let F = (F_{1}, ..., F_{m}) : ℂ^{n} → ℂ^{m} and let \overline{F} = (\overline{F_{1}}, ..., \overline{F_{m}}) : ℂ^{n} → ℂ^{m}. By our assumption mappings F and \overline{F} coincide on X. If we take G = (F + \overline{F})/2 then G : ℝ^{n} → ℝ^{m} is a real polynomial mapping and res_{X}G = f. Since X is Zariski dense in ℂX we have that F = res_{ℂX}G, i.e., F has an extension to a real polynomial mapping G.

Proposition 5.4 Let Λ ⊂ ℝ^{m} be a real algebraic curve. Then there exists a real algebraic curve X ⊂ ℝ^{2} and a real polynomial mappings f : X → Λ such that the mapping ℂf : ℂX → ℂΛ is an embedding and the set Λ \ f(X) is finite.

Proof. We can assume that ℂΛ is smooth and affine. It is a standard result that ℂΛ can be embedded in ℂ^{2} in a birational way. More precisely there exists a proper regular and birational map Φ : ℂΛ → ℂ^{2}, such that Λ′ := Φ(ℂΛ) has only finite number of singular points. Moreover, we can assume that the mapping Φ is a composition of (real) projections, hence it is described by real polynomials.

By Lemma 5.2 we can assume that all singular points of Λ′ lie on the line {x = 0}. Moreover, we can assume that (0, 0) ∉ R := {x = 0} ∩ Λ′.

Let us consider the map q : ℂ^{2} ∋ (x, y) → (x, xy) ∈ ℂ^{2}. Take X′ = q^{−1}(Λ′) and X = X′ ∩ ℝ^{2}. Now Ψ = q ∘ (Φ)^{−1} : X′ → ℂ^{n} is a closed embedding of X′ in ℂΛ. Moreover, we have Ψ(X) ⊂ ℝ^{n}. Indeed, assume that x ∈ X and y = Ψ(x) ∉ ℝ^{n}. We have Φ(y) = Φ(\overline{y}) = q(x) and y ≠ \overline{y}, which contradicts the fact that Φ induces an injection Φ^{−1}(X′) → X′. Finally, we can take f = res_{X}Ψ : X → Λ.

Corollary 5.5 Let Λ ⊂ ℝ^{m} be a real algebraic curve. Then there exist a finite subset S ⊂ Λ and a real algebraic curve X ⊂ ℝ^{2}, such that the curve X is biregulary equivalent with the curve Λ \ S.

Definition 5.6 Let S ⊂ ℙ^{n}(ℝ) be a closed semi-algebraic curve. The curve S is called ℝ-rational if there is a regular surjective mapping f : ℙ^{1}(ℝ) → S, which can be extended to a regular mapping ℂf : ℙ^{1}(ℂ) → ℂS.

Theorem 5.7 Let X be obtained from ℙ^{2}(ℝ) by a sequence of blowing up’s with centers at infinity. Then for every x ∈ X and for every non-empty Zariski open subset U ⊂ ℝ^{2} there is a ℝ-rational curve Γ such that

1) x ∈ Γ

2) Γ contains a point from U.

Proof. We use induction on the number p of elementary blowing up’s. For X_{0} = ℙ^{2}(ℝ) our theorem is obvious, hence for p = 0 theorem is true. Moreover every point x ∈ X_{0} has the following important property:

∗) there is an affine neighborhood U_{x} of the point x, which is polynomially isomorphic to ℝ^{2}, i.e., there is an isomorphism ϕ_{x} : ℝ^{2} → U_{x} which extends to an isomorphism ℂϕ_{x} : ℂ^{2} → ℂU_{x} ⊂ ℂX_{0}.

Let us note that if the surface X_{p} ⊂ ℙ^{N}(ℝ) has the property ∗) then also the blown surface X_{p+1} ⊂ ℙ^{M}(ℝ) has the same property. It follows from the well-known properties of σ process. Now our theorem is obvious.

Corollary 5.8 Let X ⊂ ℝ^{n} be an algebraic set, which is polynomially isomorphic with ℝ^{k}, k ≥ 1. Let a ∈ cl(X) ⊂ ℙ^{n}(ℝ), (where cl(X) denotes the closure in the strong topology) and U ⊂ X be a non-empty Zariski open subset. Then there is a ℝ-rational curve Γ such that

1) a ∈ Γ ⊂ cl(X)

2) Γ contains a point from U.

Proof. We can assume that k > 1. Let a ∈ cl(X) ⊂ ℙ^{n}(ℝ). By a curve selecting lemma there is an algebraic curve G ⊂ X such that a ∈ cl(G) ⊂ ℙ^{n}(ℝ). Now by Proposition 5.4 there is a real algebraic curve X ⊂ ℝ^{2} and a real polynomial mappings f : X → Λ such that the mapping ℂf : ℂX → ℂΛ is an embedding and the set Λ\f(X) is finite. By Theorem 5.1 we can extend the mapping f to a polynomial quasi-finite mapping F : ℝ^{2} → X ⊂ ℙ^{n}(ℝ). Moreover, we can assume that F (ℝ^{2}) ∩ U ≠ ∅.

By a construction we have a ∈ cl(F (X)). We can resolve indeterminacy of the mapping F by a sequence of blowing up’s. Consequently we obtain a blown projective plane X_{p} and a polynomial mapping F′ : X_{p} → cl(X) such that a ∈ F′(X_{p}). Now the statement follows from Theorem 5.7.

To prove the next result we need also some facts about extensions of finite mappings.

Definition 5.9 Let X ⊂ ℝ^{n}, Y ⊂ ℝ^{m} be irreducible algebraic sets. Let f = (f_{1}, ..., f_{m}) : X → Y be a polynomial mapping and let f^{∗} : ℝ[Y] ∋ h → h ∘ f ∈ ℝ[X] be an induced mapping.

The mapping f is called finite if the ring ℝ[X] is a finite f^{∗}(ℝ[Y]) algebra (or equivalently the ring ℝ[X] is a finite ℝ[f_{1}, ..., f_{m}] algebra).

Remark 5.10 a) A finite mapping is proper and it has finite fibers.

b) If a mapping f is proper then the mapping ℂf : ℂX → ℂY is also finite.

c) A composition of finite mappings is a finite mapping.

d) A non-constant polynomial mapping ϕ : ℝ → ℝ^{m} is finite.

Theorem 5.11 Let X ⊂ ℝ^{m} be a real affine algebraic variety and let Γ be a closed algebraic subvariety of X. Let f : Γ → ℝ^{n} be a finite polynomial map and assume n ≥ dim X. Then there exists a finite polynomial map F : X → ℝ^{n}, such that res_{Γ}F = f.

Proof. We follow closely our method from [5]. Let I(Γ) denote the ideal of Γ in ℝ[X]. Let g = (g_{1}, …, g_{n}) be a polynomial extension of f to X. Let x_{1}, …, x_{m} be coordinate polynomial functions in ℝ^{m}.

It follows from our assumptions that there exist polynomial functions a_{j}^{i} ∈ ℝ[X_{1}, …, X_{n}] such that H_{i} = ∑_{j=0}^{n_{i}} a_{j}^{i}(g)x_{i}^{n_{i}−j} = 0 mod I(Γ), and a_{0}^{i}(g) = const ≠ 0 for i = 1, …, m.

Consider the map H = (g_{1}, …, g_{n}, H_{1}, …, H_{m}) : X → ℝ^{n+m}. By construction H is a finite polynomial mapping and H(Γ) ⊂ ℝ^{n} × {0} and dim H(X) =dim X ≤ n. Of course the set H(X) is a closed (in Euclidian topology) subset of ℝ^{n+m}. Let Y := cl(H(X)) be a Zariski closure of H(X) in ℝ^{n+m}. Then dim Y = dim X. It is easy to see that there exists a finite linear projection π : Y → ℝ^{n} × {0}.

Indeed, we have the canonical inclusion ℝ^{n+m} ⊂ ℙ^{n+m}(ℝ) and the set W of points at infinity of Y has dimension ≤ n − 1. Hence we can find a linear subspace L of the hyperplane at infinity of dimension m − 1 which is disjoint from W ∪ cl(ℝ^{n} × {0}). Let π_{L} : ℙ^{n+m}(ℝ) → cl(ℝ^{n} × {0}) be the projection determined by L. The restriction of π_{L} to Y is a finite mapping of Y into ℝ^{n} × {0} and it is the projection π we were looking for.

Now it is enough to take F = p ∘ H to obtain a finite extension of f to the whole of X.

6 General case.

In the beginning we compare the real set S_{f} with the complex one. We have the following simple but basic fact:

Proposition 6.1 Let f : ℝ^{n} ⟶ ℝ^{n} be a dominant polynomial map and let ℂf : ℂ^{n} ⟶ ℂ^{n} be its complexification. Then the set S_{f} of points at which f is not proper is contained in the set S_{ℂf} of points at which the mapping ℂf is not proper.

Proof. The (easy) purely topological proof is left to the reader.

In the sequel the following definition will be useful:

Definition 6.2 Let S ⊂ ℝ^{n} be a semialgebraic set. The set S is called ℝuniruled if for every point a ∈ S there is a parametric semi-line in S through the point a. Moreover, the set S is called generically ℝ-uniruled if there is an open and dense subset U ⊂ S, such that for every point a ∈ U there is a parametric semi-line in S through this point.

Remark 6.3 Of course, every ℝ-uniruled set is generically ℝ-uniruled. Moreover, if a set S is generically ℝ-uniruled, then the set ℂS is ℂ-uniruled. It follows immediately from Proposition 3.3.

Lemma 6.4 Let S ⊂ ℙ^{n}(ℝ) be a ℝ-rational curve, such that S′ := S ∩ℝ^{n} ≠ ∅ and S is not completely contained in ℝ^{n}. Then there is an algebraic curve X ⊂ ℝ^{2} and a finite and surjective polynomial mapping G : X → S′.

Proof. Let f : ℙ^{1}(ℝ) → S, be a real surjective regular mapping, which can be extended to a regular mapping ℂf : ℙ^{1}(ℂ) → ℂS. Let b ∈ ℙ^{1}(ℝ) be a point such that f(b) ∉ ℝ^{n}. Hence in fact we have a rational surjective mapping f : ℝ → S′, which can be extended to a rational mapping F := ℂf : ℂ → ℂS′.

Let P := {a_{1}, ..., a_{n}} be a set of poles of the mapping ℂf : ℂ → ℂS′. Let us note that if a_{i} ∈ P then also \overline{a_{i}} ∈ P. Consequently the rational mapping ϕ : ℂ ∋ t → (t, 1/ ∏_{j=1}^{n}(t − a_{j}) ∈ ℂ^{2} has real coefficients. Let X′ = ϕ(ℂ \ P). Then X′ is a closed curve in ℂ^{2} and we have an isomorphism ψ : X′ → ℂ \ P. If we take G := F ∘ψ then G has real coefficients and it gives a finite mapping G : X′ → ℂS′. Now we can pass to the real part of X′ and lemma follows.

Now we are ready to prove our main theorem:

Theorem 6.5 Let f = (f_{1}, ..., f_{m}) : ℝ^{n} → ℝ^{m} be a generically finite polynomial mapping. Then the set S_{f} is closed, semi-algebraic and for every non-empty connected component S ⊂ S_{f} we have 1 ≤ dim S ≤ n − 1.

Moreover, the set S_{f} is ℝ-uniruled. Further, there is a real polynomial P ∈ ℝ[x_{1}, ..., x_{m}] of degree at most (degf)^{n−1} − 1, such that S_{f} ⊂ P^{−1}(0).

Proof. Let \overline{X} be a closure of the set X := graph(f) ⊂ ℝ^{n} × ℝ^{m} in ℙ^{n}(ℝ) × ℝ^{m}. Let us note that X is polynomially isomorphic to ℝ^{n}. The set \overline{X} is semi-algebraic, hence the set R := \overline{X} \ X is semialgebraic, too. Moreover, if π : ℙ^{n}(ℝ) × ℝ^{m} → ℝ^{m} is the canonical projection we have S_{f} = π(R), consequently the set S_{f} is closed and semi-algebraic.

Let b ∈ S_{f}. Then there is a point a ∈ R such that b = π(a). By Corollary 5.8 there is a ℝ-rational curve Λ ⊂ cl(X) ⊂ ℙ^{n}(ℝ)×ℙ^{m}(ℝ), which contains the point a. We can assume that Λ contains a point c ∈ X, which is smooth with respect to f. Let Λ′ := Λ ∩ X. By Lemma 6.4 there exist an algebraic curve Γ ⊂ ℝ^{2} and a finite and surjective polynomial mapping q : Γ → Λ′. By Theorem 5.11 we can extend q to a finite map Q : ℝ^{2} → ℝ^{n} ≅ X. Let us take F = ϕ ∘ Q, where ϕ : X → ℝ^{m} is a canonical projection. It is easy to see that S_{F} ⊂ S_{f} and b ∈ S_{F} . Now by Theorem 4.2 we get that there is a parametric semi-line in S_{f} through b.

Let π : ℝ^{m} → {0} × ℝ^{n} be a projection which is finite on cl_{Z}(f(X)) and take h = π ∘ f. Then h : ℝ^{n} → ℝ^{n} is a polynomial mapping and S_{h} = π(S_{f}). Let us consider the mapping ℂh : ℂ^{n} → ℂ^{n}. We know by Theorem 6.1 that S_{h} ⊂ S_{ℂh}. By Theorem 3.5 the set S_{ℂh} is described by a polynomial Q of degree less than (deg f)^{n−1}. Moreover, since the mapping h is real, we have that the polynomial Q is also real (it can be deduced from [4]). Now, the set S_{f} is contained in the zero set of polynomial P := Q ∘ π and theorem follows.

Corollary 6.6 Let f : ℝ^{n} → ℝ^{m} be a generically finite polynomial mapping. Then every non-empty connected component of S_{f} is ℝ-uniruled, in particular unbounded.

Corollary 6.7 Let f : ℝ^{n} → ℝ^{n} be a generically finite polynomial mapping. Let Q be a non-empty connected component of ℝ^{n} \S_{f}. Then H_{n−1}(Q, ℤ) = 0. In particular for n = 2 all these components are simply connected.

Corollary 6.8 Let f : ℝ^{n} → ℝ^{m} be a generically finite polynomial mapping. Assume that S is an irreducible algebraic component of a (real) Zariski closure of S_{f}. Then the algebraic variety ℂS is ℂ-uniruled. Further , if S ⊂ S_{f} and either S is smooth and connected or S is smooth and has pure dimension n − 1, then S is generically ℝ-uniruled.

Remark 6.9 If f : ℝ^{n} → ℝ^{m} is a generically finite polynomial mapping, then the real algebraic closure set S_{f} can be not generically ℝ-uniruled. In fact Gwoździewicz in [4] show that if f is the Pinchuk mapping, then the algebraic closure of the set S_{f} has a point as an isolated component.

7 Examples.

In this section we give some examples of the set S_{f}, as well as some applications of our results.

Theorem 7.1 Let S ⊂ ℝ^{m} be a semi-algebraic set and assume that there is a finite and surjective polynomial mapping ϕ : ℝ^{n−1} → S. Assume that m ≥ n. Then there exists a polynomial mapping f : ℝ^{n} → ℝ^{m} with finite fibers for which S is exactly the set of points at which f is not proper (i.e., S = S_{f}).

Proof. Indeed, let V = {(x_{1}, x_{2}, x_{3}, ..., x_{n+1}) ∈ ℝ^{n+1} : x_{3}^{2} − x_{1}x_{2} − 1 = 0}. Then V is an affine variety and there exists a (non-proper) embedding i : ℝ^{n} ∋ (u_{1}, v_{2}) → (u_{1}(2 + u_{1}u_{2}), u_{2}, 1 + u_{1}u_{2}, u_{3}, ..., u_{n}) ∈ V. Note that V \ i(ℝ^{n}) is the linear subspace l = {x_{3} = −1, x_{2} = 0} ≅ ℝ^{n−1}. If ϕ : l ∋ (x_{1}, 0, −1, x_{4}, ..., x_{n+1}) → ϕ(x) ∈ S is a finite polynomial parametrization of S, then by Theorem 5.11, the mapping ϕ can be extended to a finite polynomial mapping Φ : V → ℝ^{m} and it is enough to take f = Φ ∘ i.

Example 7.2 a) There is no polynomial mapping f : ℝ^{n} → ℝ^{n} such that f(ℝ^{n}) = {(x_{1}, ..., x_{n}) ∈ ℝ^{n} : x_{2} > 0, x_{3} > 0, ..., x_{n} > 0, x_{1}·...·x_{n} > 1}. Indeed, let U = f(ℝ^{n}) and S = bd(U) (a border of U). Then ℂS has dimension n − 1 and consequently it must be a component of the set S_{ℂf}. In particular it must be ℂ-uniruled. It means that the hypersurface {(x_{1}, ..., x_{n}) ∈ ℂ^{n} : x_{1}·...·x_{n} = 1} is ℂ-uniruled, and we get a contradiction.

b) There is no polynomial mapping f : ℝ^{n} → ℝ^{n} such that f(ℝ^{n}) = {(x_{1}, ..., x_{n}) ∈ ℝ^{n} : ∑_{i=1}^{n} x_{i}^{2} > 1}. Indeed, let U = f(ℝ^{n}) and S = bd(U) = {(x_{1}, ..., x_{n}) ∈ ℝ^{n} : ∑_{i=1}^{n} x_{i}^{2} = 1}. Then S is a real algebraic set of dimension n − 1 and consequently it must be an irreducible component of the set cl_{Z}(S_{f}). In particular it must be generically ℝ-uniruled, hence unbounded, a contradiction.

c) More generally, let U ⊂ ℝ^{n} be an open domain. Assume that bd(U) contains an algebraic set Γ of dimension n−1, such that either Γ is bounded, or Γ is smooth, connected and not generically ℝ-uniruled. Then there is no polynomial mapping f : ℝ^{n} → ℝ^{n} such that f(ℝ^{n}) = U.

Example 7.3 Let us consider a polynomial mapping f : ℝ^{n} → ℝ^{n}. If n = 2, then the set S_{f} (if non-empty) has codimension 1. But for n > 2 it is not longer true. Indeed, let

f_{k} : ℝ^{n} ∋ (x_{1}, ..., x_{n}) → (x_{1}, ..., x_{n−1}, (x_{1}^{2} + ... + x_{k}^{2})x_{n}),

where k = 1, ..., n − 1. Then S_{f_{k}} = {x ∈ ℝ^{n} : x_{1} = x_{2} = ... = x_{k} = 0}. In particular codim S_{f_{k}} = k, where k = 1, ..., n − 1.

8 Real Jacobian Conjecture

At the end of this paper is to worth to consider the case of a real polynomial mapping with the Jacobian which nowhere vanishes. We have the following:

Theorem 8.1 Let f : ℝ^{n} → ℝ^{n} be a real polynomial mapping with the Jacobian which nowhere vanishes. If codim S_{f} ≥ 3 then f is a bijection (and consequently S_{f} = ∅).

Proof. Let us consider sets X := ℝ^{n} \ S_{f} and Y := ℝ^{n} \ f^{−1}(S_{f}). Since codim S_{f} ≥ 3 we have that the set X is simply connected. Moreover, the mapping f : Y → X is proper and unramified, hence it is a topological covering. In particular it is a homeomorphism. Consequently, the general fiber of the mapping f is a one point. Since the mapping f is unramified, this implies that f is an injection. Now our statements follows by theorem of Białynicki-Rosenlicht - see [2].

On the other hand the example of Pinchuk shows that there are real polynomial mapping with the Jacobian which nowhere vanishes with codim S_{f} = 1. Hence the only interesting case is the case of codim S_{f} > 1 and we can state:

Real Jacobian Conjecture. Let f : ℝ^{n} → ℝ^{n} be a real polynomial mapping with the Jacobian which nowhere vanishes. If codim S_{f} > 1 then f is a bijection (and consequently S_{f} = ∅).

By Theorem 4.2 this conjecture is true in dimension two. The Real Jacobian Conjecture is closely connected with the following famous Jacobian Conjecture:

Jacobian Conjecture. Let f : ℂ^{n} → ℂ^{n} be a polynomial mapping with the Jacobian which nowhere vanishes. Then f is an isomorphism.

In fact we have:

Proposition 8.2 The Real Jacobian Conjecture in dimension 2n implies the Jacobian Conjecture in (complex) dimension n.

Proof. Indeed, let f : ℂ^{n} → ℂ^{n} be a polynomial mapping with a non-zero Jacobian. We can treat the mapping f as a real polynomial mapping f : ℝ^{2n} → ℝ^{2n}. Assume that f is not an isomorphism. In particular the mapping f can not be finite. Thus by [5], [6] we get that the set S_{f} has a complex codimension 1, hence it has a real codimension 2. Consequently, if the Real Jacobian Conjecture holds in dimension 2n, this gives a contradiction.

INSTYTUT MATEMATYKI

POLSKA AKADEMIA NAUK

ŚW. TOMASZA 30, 31-027 KRAKÓW

and

INSTYTUT MATEMATYKI

UNIWERSYTET JAGIELLOŃSKI

REYMONTA 4, 30-059 KRAKÓW

POLAND

E-MAIL: JELONEK@IM.UJ.EDU.PL

References

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[2] Białynicki-Birula, A, Rosenlicht, M, Injective morphisms of real algebraic varieties, Proc. Amer. Math. Soc., 13, (1962), 200-204.

[3] Dold, A, Lectures on Algebraic Topology, Springer, 1995.

[4] Gwoździewicz, J, Geometry of Pinchuk’s mapping, Bull. Pol. Acad. Sci. Math, to appear.

[5] Jelonek, Z, The set of points at which a polynomial map is not proper, Ann. Polon. Math., 58, (1993), 259-266.

[6] Jelonek, Z, Testing sets for properness of polynomial mappings, Math. Ann., (1999), to appear.

[7] Jelonek, Z, A geometry of real polynomial mappings of the plane, Bull. Pol. Acad. Sci. Math, to appear.