amathematicaltribute - ucmjosefer/otros/homenaje-montesinos.pdf · 2017. 1. 2. ·...

A MATHEMATICAL TRIBUTE

to ProfessorJose Marıa Montesinos Amilibia

DEPARTAMENTO DE GEOMETRIA Y TOPOLOGIAFACULTAD DE CIENCIAS MATEMATICAS – UCM

A MATHEMATICAL TRIBUTEto Professor Jose Marıa Montesinos Amilibia

c� Los autores de las comunicaciones

EditaDepartamento de Geometrıa y TopologıaFacultad de Ciencias Matematicas – UCMPlaza de Ciencias, 3Ciudad Universitaria28040 Madrid

CubiertaRaquel DıazTetraedro inscrito en un cubo(Doblese por las lıneas negras finas, cortese por las lıneas negras gruesas.)

ISBN: 978-84-608-1684-3Deposito Legal: M-2056-2016

ImprentaUlzama Digital

Impreso en Espana

Professor Jose Marıa Montesinos Amilibia

Oh my knots!

This volume contains the contributions presented by several colleagues as a tributeto the mathematical and human qualities of Jose Marıa Montesinos Amilibia on theoccasion of his seventieth birthday. The editors would like to express their thanksto the contributors and their very especial gratitude to Jose Marıa for his examplethrough many years of scientific and personal contact.

Marco CastrillonElena Martın-PeinadorJose M. Rodrıguez-SanjurjoJesus M. Ruiz

Contents

Lectures addressed on the 8 of September, 2015

Morphismes analytiques finis et revetements ramifies. . . . . . . . . . . . . . . . . . . . . . . . . 1Claude Weber

Algunas contribuciones matematicas del Profesor Jose MarıaMontesinos Amilibia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Maite Lozano

Jose Marıa Montesinos Amilibia. Biographical Sketches . . . . . . . . . . . . . . . . . . . . . . 33Sebastia Xambo

Contributions on the occasion of JMMA’s 70 anniversary

Proof of the rigidity of model nilpotent Lie algebras by means of theInternal Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

J. M. Ancochea Bermudez, R. Campoamor Stursberg

Una caracterizacion de las proyecciones de Lagrange . . . . . . . . . . . . . . . . . . . . . . . . . 75Jorge L. Andres, Jesus Otero

The dodecahedron: from intersections of quadrics to Borromean rings . . . . . . . . 85Enrique Artal Bartolo, Santiago Lopez de Medrano,Marıa Teresa Lozano

⇢-pairs in graphs representing surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105P. Bandieri

Fixed points, bounded orbits and attractors of planar flows . . . . . . . . . . . . . . . . . . 125Hector Barge, Jose M. R. Sanjurjo

The contact structure in the space of light rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133Alfredo Bautista, Alberto Ibort, Javier Lafuente

Universal groups and super regular tesselations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161G. Brumfiel, H. Hilden, M. T. Lozano,J. M. Montesinos-Amilibia, E. Ramirez, H. Short, D. Tejada,M. Toro

Groups of automorphisms of bordered 2-tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181Emilio Bujalance, Jose Javier Etayo, Ernesto Martınez

vii

Classifying PL 4-manifolds via crystallizations: results and open problems . . . 199Maria Rita Casali, Paola Cristofori, Carlo Gagliardi

On certain classes of closed 3-manifolds with di↵erent geometric structures . . . 227Alberto Cavicchioli, Fulvia Spaggiari

Algunas exploraciones matematicas del mundo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243Capi Corrales Rodriganez

A note on regular branched foldings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259Antonio F. Costa

Partially flat surfaces solving k-Hessian perturbed equations . . . . . . . . . . . . . . . . . 265Gregorio Dıaz, Jesus Ildefonso Dıaz

Hyperbolic surfaces of genus 3 with symmetry S4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 295Raquel Dıaz, Victor Gonzalez Aguilera

Modelo plano conforme del Plano Proyectivo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307Fernando Etayo, Ujue R. Trıas

Toroidal Dehn Surgeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317Mario Eudave-Munoz, Enrique Ramırez-Losada

The open quadrant problem: A topological proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337Jose F. Fernando, J. M. Gamboa, Carlos Ueno

A la busqueda de la espiritualidad perdida. Meditar itinerante acerca delnumero, el tacto, la duracion y el Arte so pretexto de las Matematicasy la matematizacion del mundo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

Jesus Fortea Perez

Persistencia uniforme de atractores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373Antonio Giraldo, Victor Fernandez Laguna,Jose M. Rodriguez Sanjurjo

2-dimensional stratifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395J. C. Gomez-Larranaga, F. Gonzalez-Acuna, Wolfgang Heil

Surface knot groups and 3-manifold groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407Francisco Gonzalez-Acuna, Arturo Ramırez

Some geometric properties of variable exponent Lebesgue spaces . . . . . . . . . . . . . 415Francisco L. Hernandez, Cesar Ruiz

Carousel wild knots are ambient homogeneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423Gabriela Hinojosa, Cynthia Verjovsky Marcotte,Alberto Verjovsky

Global bifurcation for Fredholm operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437Julian Lopez-Gomez

viii

Pascal’s triangle, Stirling numbers and the Euler characteristic . . . . . . . . . . . . . . . 453Ana Luzon, Manuel A. Moron, Felipe Prieto-Martınez

Probabilidades en espacios topologicos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463Juan Margalef-Roig, Salvador Miret-Artes,Enrique Outerelo-Domınguez

Compact Hausdor↵ group topologies for the additive group of real numbers . . 491Elena Martın-Peinador, Monserrat Bruguera Padro

Unraveling the Dogbone Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499Diego Mondejar Ruiz

SL(3,C)-character variety of torus knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515Vicente Munoz, Jonathan Sanchez

Spectral limits of semiclassical commuting self-adjoint operators . . . . . . . . . . . . . 527Alvaro Pelayo, S. Vu Ngo. c

Nontrivial twisted Alexander polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547Joan Porti

Matrices de rotaciones, simetrıas y roto-simetrıas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559Marıa Jesus de la Puente

La Biblioteca de El Escorial. Un culto a la matematica . . . . . . . . . . . . . . . . . . . . . . 565Marıa Concepcion Romo Santos

Geometria en el siglo XIV: los trabajos de Thomas Bradwardine . . . . . . . . . . . . . 571Juan Tarres Freixenet

ix

The open quadrant problem:A topological proof

Jose F. FERNANDO⇤, J.M. GAMBOA† and Carlos UENO‡

Departamento de

´

Algebra

Facultad de Ciencias Matematicas

Universidad Complutense de Madrid

28040 Madrid, Spain

[email protected] [email protected]

Dipartimento di Matematica

Universita degli studi di Pisa

56127 Pisa, Italy

[email protected]

Dedicated to Jose Marıa Montesinos on the occasion of his 70th birthday.

ABSTRACT

In this work we present a new polynomial map f := (f1, f2) : R2 ! R2 whoseimage is the open quadrant Q := {x > 0, y > 0} ⇢ R2. The proof of this factinvolves arguments of topological nature that avoid hard computer calculations.In addition each polynomial fi 2 R[x, y] has degree 16 and only 11 monomials,becoming the simplest known map solving the open quadrant problem.

2010 Mathematics Subject Classification: 14P10, 26C99, 52A10.Key words: Polynomial map, polynomial image, semialgebraic set, open quadrant,total degree, total number of monomials.

⇤Author supported by Spanish GAAR MTM2011-22435, Grupos UCM 910444 and the “NationalGroup for Algebraic and Geometric Structures, and their Applications” (GNASA - INdAM). His oneyear research stay in the Dipartimento di Matematica of the Universita di Pisa is partially supportedby MECD grant PRX14/00016.

†Author supported by Spanish GAAR MTM2011-22435 and Grupos UCM 910444.‡Author supported by ‘Scuola Galileo Galilei’ Research Grant at the Dipartimento di Matematica

of the Universita di Pisa and Spanish GAAR MTM2011-22435.

337

J. F. Fernando, J. M. Gamboa, C. Ueno The open quadrant problem

1. Introduction

Although it is usually said that the first work in Real Geometry is due to Harnack[13], who obtained an upper bound for the number of connected components of a non-singular real algebraic curve in terms of its genus, modern Real Algebraic Geometrywas born with Tarski’s article [15], where it is proved that the image of a semialgebraicset under a polynomial map is a semialgebraic set. We are interested in studying whatmight be called the ‘inverse problem’. In the 1990 Oberwolfach Reelle algebraischeGeometrie week [12] the second author proposed:

Problem 1.1 Characterize the (semialgebraic) subsets of Rm that are either polyno-mial or regular images of Rn.

A map f := (f1, . . . , fm) : Rn! Rm is a polynomial map if its components

fk 2 R[x] := R[x1, . . . , xn] are polynomials. Analogously, f is a regular map if itscomponents can be represented as quotients fk = gk

hkof two polynomials gk, hk 2 R[x]

such that hk never vanishes on Rn. A subset S ⇢ Rn is semialgebraic when it admits adescription by a finite boolean combination of polynomial equalities and inequalities.

Open semialgebraic sets deserve a special attention in connection with the realJacobian Conjecture [14]. In particular the second author stated in [12] the ‘openquadrant problem’:

Problem 1.2 Determine whether the open quadrant Q := {x > 0, y > 0} of R2 is apolynomial image of R2.

This problem stimulated the interest of many specialists in the field. However,only after twelve years a first solution was found in [4] and presented by the firstauthor in the 2002 Oberwolfach Reelle algebraische Geometrie week [2].

The open quadrant problem was the germ of a more systematic study of ‘Polyno-mial and regular images of Euclidean spaces’ developed by the authors during the lastdecade and which was the topic of the Ph.D. Thesis of the third author [16]. Sincethen we have worked on this issue with two main objectives:

• Finding obstructions to be an either polynomial or regular image.

• Proving (constructively) that large families of semialgebraic sets with piecewiselinear boundary (convex polyhedra, their interiors, complements and the in-teriors of their complements) are either polynomial or regular images of someEuclidean space. The positive answer to the open quadrant problem has beena recurrent starting point for this approach.

In [4, 5] we presented the first steps to approach Problem 1.1. A complete solutionto Problem 1.1 for the one-dimensional case appears in [3], whereas in [6, 8, 9, 17, 18]we approached constructive results concerning the representation as either polynomialor regular images of the semialgebraic sets with piecewise linear boundary commented

338


above. Articles [7, 10] are of di↵erent nature because we find in them new obstructionsfor a subset of Rm to be either a polynomial or a regular image of Rn. In the first onewe found some properties of the di↵erence Cl(S) \ S while in the second it is shownthat the set of points at infinite of a polynomial image of Rn is a connected set.

The constructive solution to the open quadrant problem provided in [4] involvesquite complicated computer calculations that the third author never liked. In facthe provided in his Ph.D. Thesis a di↵erent topological proof for the map proposedin [4], together with an algebraic proof involving a di↵erent polynomial map. Thismap has inspired the first and third authors for a short algebraic proof of the openquadrant problem involving a new polynomial map [11] and has led us to look for apolynomial map with optimal algebraic structure whose image is the open quadrant.It is important to establish clearly the meaning of ‘optimal algebraic structure’ [11,§3(A)]. It is natural to wonder how a polynomial map looks like when completelyexpanded and how it compares with other polynomial maps. We care about the totaldegree of the involved polynomial map (the sum of the degrees of its components)and its total number of (non-zero) monomials. We would like to find a polynomialmap with the less possible total degree and the less possible number of monomials.The example in [4] has total degree 56 and its total number of monomials is 168. Thepolynomial map in [11] has total degree 72 and its total number of monomials is 350.In this work we will prove:

Theorem 1.3 The open quadrant Q is the image of the polynomial map

f : R2! R2, (x, y) 7! ((x2y4+x4y2�y2�1)2+x6y4, (x6y2+x2y2�x2

�1)2+x6y4).

This polynomial map has total degree 28 and its total number of monomials is 22,which certainly improves the already known explicit solutions to the open quadrantproblem. It has been constructed following a similar strategy to that in [4, §3]. Ourexperience approaching this problem suggests us that this map is surely close to havethe optimal desired algebraic structure.

The article is organized as follows. In Section 2 we present all basic notions andtopological preliminaries used in Section 3 to prove Theorem 1.3.

2. Topological preliminaries

Denote the closed disc of center the origin and radius A > 0 of the plane R2 with DA.A warped disc is a subset DA,⇠ := {z = ⇠(x, y), x2+y2 A2

} ⇢ R3 where ⇠ : R2! R

is a continuous function. Consider the homeomorphism

⇣ : R3! R3, (x, y, z) 7! (x, y, z � ⇠(x, y))

that maps DA,⇠ onto DA ⇥ {0}. The image of DA,⇠ under a permutation of thevariables of R3 will be also called a warped disc.

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⇣

Figure 1: The homeomorphism ⇣ for ⇠(x, y) :=pB2

�min(y2, B2) acting on R3.

For each " > 0 consider the open neighborhood

DA(") := {x2 + y2 < (A+ ")2}⇥ (�", ") ⇢ R3

of DA. Clearly, DA,⇠(") := ⇣�1(DA(")) is an open neighborhood of DA,⇠ in R3.

Definition 2.1 A (continuous) path ↵ : [a, b] ! R3 meets transversally once thewarped disc DA,⇠ if there exist s0 2 (a, b) and " > 0 such that J := ↵�1(DA,⇠(")) =(s0 � ", s0 + ") is an open subinterval of [a, b] and (⇣ � ↵)|J(t) = (0, 0, t� s0).

Remark 2.2 If the path ↵ : [a, b] ! R3 meets transversally once the warped discDA,⇠, then ↵([a, b]) \ @DA,⇠ = ?.

Let C be a topological space homeomorphic to a closed disc and let � : C ! R3

be a continuous map. The restriction @� := �|@C is called the boundary map of �.We say that the boundary map @� meets transversally once a warped disc DA,⇠ ⇢ R3

if there exists a parameterization � of @C such that ↵ := � � � meets transversallyonce the warped disc DA,⇠.

340


Given a path-connected topological space X and a point x0 2 X we denote thefundamental group of X at the base point x0 with ⇡1(X,x0). Each path ↵ startingand ending at x0 is called a loop with base point x0 and represents an element of⇡1(X,x0), that we denote with [↵].

Lemma 2.3 Let DA,⇠ be a warped disc of R3 and let X := R3\ @DA,⇠. Let ↵ :

[a, b] ! X be a loop with base point x0 2 X that meets transversally once DA,⇠. Then[↵] is a generator of ⇡1(X,x0) ⇠= Z.

Proof. Keep the notations introduced above. Let s0 2 (a, b) and " > 0 be such that

J := ↵�1(DA,⇠(")) = (s0 � ", s0 + ")

is an open subinterval of [a, b] and (⇣ � ↵)|J(t) = (0, 0, t� s0). After a reparameteri-zation of ↵ we may assume s0 = 0.

As ⇣ is a homeomorphism of R3, we will prove the statement for � := ⇣ � ↵,Y := R3

\ @DA and the base point y0 := �(�") = (0, 0,�"). Consider the path� : [0, 1] ! R3 given by

�(t) :=

8><

>:

(3(A+ ")t, 0, ") if 0 t 13 ,

(A+ ", 0, "� (t� 13 )6") if 1

3 < t 23 ,

(A+ "� 3(A+ ")(t� 23 ), 0,�") if 2

3 < t 1.

y

x

z

DA

y0

DA(✏)

�

�

Figure 2: The path � meets transversally once the disk DA.

341


Write �0 := �|J and �1 := �|[",b] ⇤ �|[a,�"]. We claim:

[�] = [�0 ⇤ �1] = [�0 ⇤ �] · [��1

⇤ �1] = g · e = g,

where e and g are respectively the identity element and a generator of ⇡1(Y, y0) ⇠= Z.The loop ��1

⇤ �1 with base point y0 is contained in R3\ DA, which is a simply

connected space. Consequently, [��1⇤ �1] = e in ⇡1(Y, y0).

The class [�0 ⇤ �] generates ⇡1(Y, y0). Indeed, Y has as deformation retract theset Z := @DA(") [ I" where I" := {(0, 0)} ⇥ {�" z "}. It is an exercise ofalgebraic topology to show that [�0 ⇤ �] is a generator of ⇡1(Z, y0) ⇠= ⇡1(Y, y0) ⇠= Z,as required. ⇤

Lemma 2.4 Let � : C ! X be a continuous map and assume that C is homeomor-phic to a closed disc. Let � : [a, b] ! @C be a parameterization starting and endingat z0 2 @C. Then [� � �] is the identity element of ⇡1(X,�(z0)).

Proof. Let : C ! {x2 + y2 1} be a homeomorphism. The continuous map

H : [0, 1]⇥ [a, b] ! X, (⇢, t) 7! (� � �1)(⇢ · ( � �)(t) + (1� ⇢) · (z0)))

is a homotopy map between � � � and the constant path, as required. ⇤

Proposition 2.5 Let C be a topological space homeomorphic to a closed disc and� : C ! R3 a continuous map. Assume @� : @C ! R3 meets transversally once awarped disc D ⇢ R3. Then @D \ �(Int(C)) 6= ?.

Proof. Assume by contradiction @D \ �(Int(C)) = ?. As @� meets transversallyonce D, the image �(@C) does not intersect @D by Remark 2.2. Thus, �(C) ⇢ X :=R3

\ @D. Let � : [a, b] ! @C be a parameterization starting and ending at z0 2 @Csuch that � � � meets transversally once D. By Lemma 2.4 the class [� � �] is theidentity element of ⇡1(X,�(z0)). However, by Lemma 2.3 the class [��] is a generatorof ⇡1(X,�(z0)) ⇠= Z, which is a contradiction. Consequently, @D \ �(Int(C)) 6= ?, asrequired. ⇤

3. Proof of Theorem 1.3

Observe first that the map f in the statement of Theorem 1.3 is the compositionf2 � f1 of the polynomial maps

f1 : R2! R2, (x, y) 7! (x2, y2),

f2 : R2! R2, (x, y) 7! ((xy2 + x2y � y � 1)2 + x3y2, (x3y + xy � x� 1)2 + x3y2).

As f1(R2) is the closed quadrant Q := {x � 0, y � 0}, we have to prove the equality

f2(Q) = Q. (3.1)

342


The inclusion f2(Q) ⇢ Q is straightforward because both components of f2 are strictlypositive on Q. It only remains to show the inclusion

Q ⇢ f2(Q). (3.2)

3.1. Reduction of the proof of inclusion (3.2)

Consider the (continuous) semialgebraic maps

g : Q ! R3, (x, y) 7! (xy2 + x2y � y � 1, x3/2y, x3y + xy � x� 1)

h : R3! R2, (x, y, z) 7! (x2 + y2, y2 + z2).

As f2 = h�g, we have to show that for each tuple (A2, B2) 2 Q there exists (x0, y0) 2 Qsuch that (h � g)(x0, y0) = (A2, B2). This is equivalent to check that the intersectionh�1({(A2, B2)}) \ g(Q) is non-empty.

Denote S := g(Q) and fix values B � A > 0. It holds that sets

h�1({(A2, B2)}) = {x2 + y2 = A2, y2 + z2 = B2},

h�1({(B2, A2)}) = {y2 + z2 = A2, x2 + y2 = B2}

contain respectively the boundaries of the warped discs

D1 : z = ⇠1(x, y), x2 + y2 A2, (3.3)

D2 : x = ⇠2(y, z), y2 + z2 A2, (3.4)

for the (continuous) semialgebraic functions

⇠1 : R2! R, (x, y) 7!

pB2

�min{y2, B2}, (3.5)

⇠2 : R2! R, (y, z) 7!

pB2

�min{y2, B2}. (3.6)

Consequently, we are reduced to prove:

3.1.1. For fixed values B � A > 0 the intersections @D1 \ S and @D2 \ S arenon-empty.

3.2. Proof of Statement 3.1.1

Write R := [0,+1) ⇥ (0, p2 ) and R := [0,+1) ⇥ [0, p2 ]. Consider the map � :=(�1,�2,�3) : R2

! R3 where

�1(⇢, ✓) := cos ✓ sin ✓(cos ✓ � sin ✓)2

+ ⇢(2 cos4 ✓ sin ✓ + cos ✓ sin4 ✓ + cos5 ✓) + ⇢2 cos5 ✓ sin ✓,

�2(⇢, ✓) :=p

cos ✓ sin ✓(cos ✓ + sin ✓ + ⇢ cos ✓ sin ✓),

�3(⇢, ✓) := ⇢ sin ✓.

Let us prove now some properties of the map � and the sets R and R:

343


3.2.1. �(R) ⇢ S.

Proof. The analytic map

: R ! Q, (⇢, ✓) 7!

✓sin ✓

cos ✓,(cos ✓ + sin ✓ + ⇢ cos ✓ sin ✓) cos2 ✓

sin ✓

◆,

satisfies (R) ⇢ Q and g � = �|R. Consequently, �(R) ⇢ S, as required. ⇤

3.2.2. The inequality �21(⇢, ✓)+�23(⇢, ✓) �

⇢2

4 holds for each (⇢, ✓) 2 R. Consequently,

dist(�(⇢, ✓),0) �⇢

2(3.7)

for each (⇢, ✓) 2 R.

Proof. As ⇢, cos ✓, sin ✓ are � 0 on R, we have

�1(⇢, ✓) � ⇢ cos ✓(cos4 ✓ + sin4 ✓) = ⇢ cos ✓(1� 2 cos2 ✓ sin2 ✓)

= ⇢ cos ✓

✓1�

sin2(2✓)

2

◆�

⇢

2cos ✓.

In addition, �3(⇢, ✓) = ⇢ sin ✓ � ⇢2 sin ✓, so

�21(⇢, ✓) + �23(⇢, ✓) �⇢2

4cos2 ✓ +

⇢2

4sin2 ✓ =

⇢2

4,

as required. ⇤

3.2.3. The map � satisfies �(0, ✓) = �(0, p2 � ✓) for ✓ 2 [0, p2 ]. Fix M > 0 and

consider the rectangle RM := [0,M ]⇥ [0, p2 ]. Denote �M := �|RM. Identify the points

(0, ✓) and (0, p2 � ✓) for ✓ 2 [0, p2 ] and endow the quotient space RM with the quotient

topology. Observe that the interior Int(RM ) of RM as a topological manifold withboundary is the quotient space RM obtained identifying the points (0, ✓) and (0, p2�✓)of RM := [0,M)⇥ (0, p2 ), where ✓ 2 (0, p2 ).

The canonical projection ⇡M : RM ! RM is continuous. As �M is compatiblewith ⇡M , there exists a continuous map �M : RM ! R3 such that the followingdiagram is commutative. In addition, �M (RM ) = �(RM ) ⇢ S.

RM RM

RM RM R3

�M⇡M⇡M |RM

�M

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Figure 3: Left and right views of �M (RM ) ⇢ S.

3.2.4. RM is homeomorphic to a disc and its boundary is the set

⇡M ({⇢ = M} [ {✓ = 0} [ {✓ = p2}).

Proof. Identify R2 with C (interchanging the order of the variables (⇢, ✓) (✓, ⇢))and consider the continuous map

µ : C ! C, z := ✓ +p

�1⇢ 7! w := u+p

�1v = ( 4pz � 1)2.

The restriction µ|{⇢>0} : {⇢ > 0} ! C \ ([0,+1)⇥ {0}) is a homeomorphism and the

image of RM \ {⇢ = 0} is

TM :=�(u, v) 2 R2 : ( ⇡v

8M )2 � ( 4Mp )2 u 1� ( v2 )2 \ ([0, 1]⇥ {0}).

The closure TM of TM is a compact convex set (as it is a closed bounded intersectionof two convex sets). By [1, Cor.11.3.4] TM is homeomorphic to a closed disc. Inaddition

µ|{⇢=0} : {⇢ = 0} ! [0,+1)⇥ {0}, ✓ 7! ( 4p✓ � 1)2

transforms the segment [0, p2 ]⇥{0} onto the interval [0, 1]. The preimage of t0 2 [0, 1]under µ|{⇢=0} is

{✓1 := p4 (1 +

p

t0), ✓2 := p4 (1�

p

t0)}.

As ✓1 = p2 � ✓2, the map � := µ|RM

: RM ! TM factors through RM and

there exists a continuous map � : RM ! TM such that the following diagram iscommutative.

345


10

TM

• •

µ

0p2 ✓

p4

M

RM⇢

Figure 4: Behavior of the map µ : RM ! TM .

RM RM

RM RM SM

�⇡M⇡M |RM

�

The map � is continuous and bijective and it maps the compact set RM onto the

Hausdor↵ space TM , so it is a homeomorphism. Consequently, RM is homeomorphicto a disc and its boundary is ⇡M ({⇢ = M} [ {✓ = 0} [ {✓ = p

2}), as required. ⇤

3.2.5. Fix B � A > 0 and consider the warped discs D1 and D2 introduced in (3.3)

and (3.4). Then there exists M > 0 such that the boundary map @�M : @RM ! R3

meets transversally once both discs D1 and D2.

Proof. As D1 and D2 are bounded set, there exists M0 > 0 such that D1 [D2 ⇢

{k(x, y, z)k < M0}. Take M := 4M0 and consider the set RM and the continuousmap �M introduced in paragraph 3.2.3.

We claim: the boundary map @�M : @RM ! R3 meets transversally once D1.

346


Figure 5: The boundary map @�M : @RM ! R3 meets transversally once D1.

Consider the parameterization of @RM given by

�1(t) :=

8><

>:

⇡M (t, p2 ), if 0 t M,

⇡M (M,M + p2 � t), if M < t M + p

2 ,

⇡M (2M + p2 � t, 0), if M + p

2 < t 2M + p2 .

We have

↵1(t) := �M � �1(t) =

8><

>:

�(t, p2 ), if 0 t M,

�(M,M + p2 � t), if M < t M + p

2 ,

�(2M + p2 � t, 0), if M + p

2 < t 2M + p2 .

Choose 0 < " < min{B,M0 �B} and consider the homeomorphism

⇣1 : R3! R3, (x, y, z) 7! (x, y, z � ⇠1(x, y)),

where ⇠1 is the (continuous) semialgebraic function introduced in (3.5). DenoteD1(") := ⇣�1

1 (DA(")). It is enough to check:

↵�11 (D1(")) = (B � ", B + ").

Pick p0 := ↵1(t0) 2 Im(↵1). We distinguish three cases:

(i) If 0 t0 M , then ⇣1(p0) = (⇣1 � �)(t0, 0) = (0, 0, t0 � B). Consequently,⇣1(p0) 2 DA(") if and only if �B < �" < t0 �B < " < M �B.

347


0

✓

p2

⇢ M•

•

identified!

�1(t)

0

✓

p2

⇢ M•

•

identified!

�2(t)

Figure 6: Behavior of the paths �1 and �2.

(ii) If M < t0 M + p2 , we have by (3.7)

dist(p0,0) �M2 = 2M0 >

p

2M0 > dist(q,0)

for each q 2 D1("). Therefore p0 /2 D1(").

(iii) If M + p2 < t0 2M + p

2 , then

p0 = ↵1(t0) = �(2M + p2 � t0, 0) = (2M + p

2 � t0, 0, 0),

so ⇣1(p0) = (2M + p2 � t0, 0,�B). As " < B, it holds ⇣1(p0) 62 DA("), so

p0 62 D1(").

We conclude ↵�11 (D1(")) = (B � ", B + "), so ↵1 meets transversally once D1.

Analogously one shows: the boundary map @�M : @RM ! R3 meets transversallyonce D2.

Consider in this case the parameterization of @RM given by

�2(t) :=

8><

>:

⇡M (t, 0), if 0 t M,

⇡M (M, t�M), if M < t M + p2 ,

⇡M (2M + p2 � t, p2 ), if M + p

2 < t 2M + p2 .

We have

↵2(t) := �M � �2(t) =

8><

>:

�(t, 0), if 0 t M,

�(M, t�M), if M < t M + p2 ,

�(2M + p2 � t, p2 ), if M + p

2 < t 2M + p2 .

348


Figure 7: The boundary map @�M : @RM ! R3 meets transversally once D2.

Proceed as above keeping the same values for A and " and using in this case thehomeomorphism

⇣2 : R3! R3, (x, y, z) 7! (z, y, x� ⇠2(z, y)),

where ⇠2 is the (continuous) semialgebraic function introduced in (3.6), to prove that↵2 meets transversally once the warped disk D2. ⇤

3.2.6. By 3.2.4 RM is homeomorphic to a closed disc. By Proposition 2.5 applied

to the continuous map �M : RM ! R3 and 3.2.5, we deduce that the boundaries ofboth warped discs D1 and D2 meet �M (RM ) ⇢ S. Thus, 3.1.1 holds, as required. ⇤

References

[1] M. Berger: Geometry, I. Universitext, Springer-Verlag, Berlin 1987.

[2] J. F. Fernando: Polynomial images of Rn. In Reelle algebraische und analytische Ge-ometrie, March 17– 23 (2002) Oberwolfach.

[3] J. F. Fernando: On the one-dimensional polynomial and regular images of Rn. J. PureAppl. Algebra 218, 9(2014) 1745–1753.

[4] J. F. Fernando, J. M. Gamboa: Polynomial images of Rn. J. Pure Appl. Algebra 179,3 (2003) 241–254.

[5] J. F. Fernando, J. M. Gamboa: Polynomial and regular images of Rn. Israel J. Math.153 (2006) 61–92.

349


[6] J. F. Fernando, J. M. Gamboa, C. Ueno: On convex polyhedra as regular images of Rn.Proc. London Math. Soc. (3) 103 (2011) 847–878.

[7] J. F. Fernando, J. M. Gamboa, C. Ueno: Sobre las propiedades de la frontera exterior delas imagenes polinomicas y regulares de Rn. In Contribuciones Matematicas en homenajea Juan Tarres, 159–178, UCM 2012.

[8] J. F. Fernando, C. Ueno: On complements of convex polyhedra as polynomial and regularimages of Rn. Int. Math. Res. Not. 18 (2014) 5084–5123.

[9] J. F. Fernando, C. Ueno: On the complements of 3-dimensional convex polyhedra aspolynomial images of R3. Internat. J. Math. 25, 7 (2014) 1450071, 18pp.

[10] J. F. Fernando, C. Ueno: On the set of points at infinity of a polynomial image of Rn.Discrete Comput. Geom. 52, 4 (2014) 583–611.

[11] J. F. Fernando, C. Ueno: A short proof for the open quadrant problem. Preprint RAAG(2014, submitted to MEGA 2015) 8 pp.

[12] J. M. Gamboa: Algebraic images of the real plane. In Reelle algebraische Geometrie,June 10-16 (1990) Oberwolfach.

[13] A. Harnack: Uber die Vielheiligkeit der ebenen algebraischen Kurven. Math. Ann. 10(1876) 189–198.

[14] S. Pinchuk: A counterexample to the real Jacobian Conjecture. Math. Z. 217 (1994)1–4.

[15] A. Tarski: A decision method for elementary algebra and geometry. Prepared for pub-lication by J.C.C. Mac Kinsey, Berkeley 1951.

[16] C. Ueno: Imagenes polinomicas y regulares de espacios euclıdeos. Ph.D. Thesis UCM2012.

[17] C. Ueno: A note on boundaries of open polynomial images of R2. Rev. Mat. Iberoam.24, 3 (2008) 981–988.

[18] C. Ueno: On convex polygons and their complements as images of regular and polynomialmaps of R2. J. Pure Appl. Algebra 216, 11 (2012) 2436–2448.

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