### Session S14 - Global Injectivity, Jacobian Conjecture, and Related Topics

## Talks

Monday, July 12, 12:00 ~ 12:50 UTC-3

## The Jacobian conjecture through the lens of formal inverse and combinatorics

### David Wright

#### Washington University in St. Louis, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

One approach to the celebrated Jacobian conjecture is to consider the formal power series of a map satisfying the Jacobian hypothesis and to try to show that this system of power series is actually a polynomial map, i.e., all but finitely many of its summands are zero. It generally begins by taking a map of cubic homogeneous type and studying the homogeneous summands of its formal inverse and the polynomial conditions that say the jacobian determinant is 1 (``Jacobian constraints"). The conjecture thereby becomes equivalent to a statement about ideal membership in a polynomial ring. Combinatorics enters the scene in understanding the terms of the formal inverse and the Jacobian constraints. This line of attack has yielded a number of partial results over the years. We will explain why the approach is tantalizingly compelling, but also has glaring weaknesses that leave us trying to prove, using formal inverse and combinatorics, known results that have been proved by other methods.

Monday, July 12, 12:50 ~ 13:40 UTC-3

## Geometric ideas on injectivity applied to discrete dynamics

### Eduardo C. Balreira

#### Trinity University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

A central question in discrete dynamics is to establish when a map has a globally asymptotically stable fixed point. This can be naturally framed as a question of injectivity. We will discuss how geometric methods to detect injectivity can be applied in this area. We will illustrate how new results on monotone maps in terms of preservation of normals to hypersurfaces generalizes the classical work for monotone maps in the plane. In addition, these ideas provide new insight on a geometric generalization of the Gale-Nikaido conjecture, at least for the planar case.

Monday, July 12, 14:00 ~ 14:50 UTC-3

## Real Jacobian mates

### Janusz Gwoździewicz

#### Pedagogical University of Cracow, Poland - This email address is being protected from spambots. You need JavaScript enabled to view it.

Let $p$ be a real polynomial in two variables. We say that a polynomial~$q$ is a real Jacobian mate of $p$ if the Jacobian determinant of the mapping $(p,q):\mathbb{R}^2\to\mathbb{R}^2$ vanishes nowhere.

We will present a class of polynomials that do not have real Jacobian mates. In particular every polynomial such that its Newton polygon has a specific edge belongs to this class. Then we will apply this result in order to simplify the proofs in two articles cited below:

[1] F. Braun, J.R. dos Santos Filho, The real jacobian conjecture on $\mathbb{R}^2$ is true when one of the components has degree $3$, Discrete Contin. Dyn. Syst. 26, (2010) 75--87.

[2] F. Braun, B. Oré-Okamoto, On polynomial submersions of degree $4$ and the real Jacobian conjecture in $\mathbb{R}^2$, J. Math. Anal. Appl 443, (2016) 688--706.

Monday, July 12, 14:50 ~ 15:40 UTC-3

## Global inversion for metrically regular mapping between Banach spaces

### Olivia Gutú

#### Universidad de Sonora, Mexico - This email address is being protected from spambots. You need JavaScript enabled to view it.

At the beginning of the last century, Hadamard introduced a global inversion criteria to ensure the existence and uniqueness of the nonlinear system given by a continuously differentiable local homeomorphism. From this beginning, seminal ideas have been developed over and over so far in different contexts but with four fundamental coincidence points: properness type conditions,path-lifting type conditions, metric-regularity asymptotic type conditions, Palais-Smale type conditions. In this talk we will present a recapitulation of global inverse theorems in the framework of metrically regular maps between Banach spaces. Metric regularity of a mapping at a point is a local concept involving certain rates and modulus, based on the Implicit Function Theorem, Banach Open Mapping Theorem and the Lyusternik-Graves Theorem. We consider this theoretical framework the ``definitive'' one since it encompasses the most relevant theorems in global inversion that have been documented in the literature in different contexts since Hadamard's original article of 1906: e.g. Gâteaux derivatives, generalized Jacobians, coderivatives, strict prederivatives, pseudo-Jacobians.

Wednesday, July 14, 12:00 ~ 12:50 UTC-3

## A sharp estimate on the size of the fiber of certain local biholomorphisms

### Xiaoyang Chen

#### Institute for Advanced Study Tongji University, China - This email address is being protected from spambots. You need JavaScript enabled to view it.

We will talk about a sharp estimate on the size of the fiber of certain local biholomorphisms, which is based on a joint work with Professor Frederico Xavier. Our work in particular implies that a local biholomorphism between $\mathbb{C}^n$ is invertible if and only if the pull-back of every complex line is $1$-connected.

Wednesday, July 14, 12:50 ~ 13:40 UTC-3

## Jacobian conjecture via intersection homology

### Anna Valette

#### Jagiellonian University, Poland - This email address is being protected from spambots. You need JavaScript enabled to view it.

The aim of this talk is to give an approach to the Jacobian conjecture using intersection homology. The main idea is to reduce the study of a given mapping $F:\mathbb{C}^n \to \mathbb{C}^n$ to the study of a singular semi-algebraic set. We will construct a pseudomanifold $N_F$ (i.e. a semi-algebraic subset of $\mathbb{R}^m$ whose singular locus is of codimension at least $2$ in itself and whose regular locus is dense in itself) associated to a given polynomial map $F:\mathbb{C}^n\to\mathbb{C}^n$. We will show that in the case $n=2$, the map $F$ with non-vanishing Jacobian is not proper iff the intersection homology of $N_F$ is nontrivial.

Wednesday, July 14, 14:00 ~ 14:50 UTC-3

## Some aspects of the complex Jacobian conjecture

### Nguyen Thi Bich Thuy

#### Universidade Estadual Paulista, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.

The complex Jacobian conjecture: ``If a polynomial mapping $F: \mathbb{C}^n \to \mathbb{C}^n$ satisfies the condition $$ \det JF(p) \neq 0, \quad \forall p \in \C^n, $$ where $JF(p)$ is the Jacobian matrix of $F$ at $p$, then $F$ is an automorphism'' originally stated by Keller in 1939, is still open in the $2$-dimensional case even though the real case was solved negatively by Pinchuk [7] in 1994. In this talk, after a short survey on principal results and approaches to the conjecture, we will explicit the Newton polygon approach firstly discovered by Abhyankar [1] and developed by many mathematicians, especially by Nagata in his nice paper [2], that gave almost principal results of the conjecture in the $2$-dimensional case. Posteriorly, we present an approach to the conjecture using intersection homology [3],[4], [8]. The intersection homology of a singular variety constructed in [8] associated to a Pinchuk's counter-example was calculated [5]. Finally, we introduce a new concept: pertinent variables [6] in the study of the Jacobian conjecture. We offer also some relations between pertinent variables and Newton polygon approaches.

[1] Abhyankar, S., Expansion Techniques in Algebraic Geometry, Tata Institute of fundamental Research, Tata Institute, 1977.

[2] Nagata, M., Some remarks on the two-dimensional Jacobian conjecture, China J. Math. 17, (1989), 1--20.

[3] Nguyen, T.B.T., Valette, A. and Valette, G., On a singular variety associated to a polynomial mapping, Journal of Singularities volume 7, 190--204, 2013.

[4] Nguyen, T.B.T. and Ruas, M.A.S., On singular varieties associated to a polynomial mapping from $\mathbb{C}^n$ to $\mathbb{C}^{n-1}$, Asian Journal of Mathematics, v.22, p.1157--1172, 2018.

[5] Nguyen, T.B.T., Geometry of singularities of a Pinchuk's map, arXiv: 1710.03318v2, 2018.

[6] Nguyen, T.B.T., The $2$-dimensional Complex Jacobian Conjecture under the viewpoint of ``pertinent variables'', ArXiv:1902.05923, 2019.

[7] Pinchuk, S., A counterexample to the strong real Jacobian conjecture, Math. Zeitschrift, 217, 1--4, (1994).

[8] Valette, A. and Valette, G., Geometry of polynomial mappings at infinity via intersection homology, Ann. I. Fourier vol. 64, fascicule 5, 2147--2163, 2014.

Wednesday, July 14, 14:50 ~ 15:40 UTC-3

## A note on the Jacobian conjecture

### Zbigniew Jelonek

#### Polska Akademia Nauk, Poland - This email address is being protected from spambots. You need JavaScript enabled to view it.

Let $F:\mathbb{C}^n\to\mathbb{C}^n$ be a polynomial mapping with a non-vanishing Jacobian. If the set $S_F$ of non-properness of $F$ is smooth, then $F$ is a surjective mapping. Moreover, if $S_F$ is connected, then $\chi(S_F)>0.$ Additionally, if $n=2$, then the set $S_F$ of non-properness of $F$ cannot be a curve without self-intersections.

Tuesday, July 20, 16:00 ~ 16:50 UTC-3

## On the Jacobian conjecture in $\mathbb{R}^2$ and its relations with the global centers of $\mathbb{R}^2$

### Jaume Llibre

#### Universitat Autònoma de Barcelona, Spain - This email address is being protected from spambots. You need JavaScript enabled to view it.

Let $F = (f,g):\mathbb{R}^2\to\mathbb{R}^2$ be a polynomial map such that the Jacobian $\det DF(p)$ is different from zero for all $p\in\mathbb{R}^2$. The real Jacobian conjecture is about the injectivity of $F$. While the Jacobian conjecture assumes that $\det DF(p)$ is a constant different from zero and claims that the map $F$ is injective. This talk presents a survey on these conjectures and their relations with the global centers in $\mathbb{R}^2$.

Tuesday, July 20, 16:50 ~ 17:40 UTC-3

## The Jacobian conjecture and the Mathieu subspaces

### Wenhua Zhao

#### Ilinois State University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

The notion of Mathieu subspaces was introduced by the speaker in 2009, which can be viewed as a natural but highly non-trivial generalization of the notion of ideals of associative algebras. In this talk we will discuss the connections of the Jacobian conjecture and the Mathieu subspaces. Furthermore, some examples and open problems as well as some latest developments of Mathieu subspaces will also be discussed. This talk will be introductory and accessible to the general audience.

Tuesday, July 20, 18:00 ~ 18:50 UTC-3

## A counterexample to a conjecture of Nollet and Xavier

### Jean Venato-Santos

#### Universidade Federal de Uberlândia, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this lecture we present a non-injective analytic local diffeomorphism of $\mathbb{R}^3$ such that the pre-image of every affine hyperplane is connected. This disproves a conjecture proposed by S. Nollet and F. Xavier in 2002. The talk is based on collaborations with F. Braun from Universidade Federal de São Carlos and L.R.G. Dias from Universidade Federal de Uberlândia.

Tuesday, July 20, 18:50 ~ 19:40 UTC-3

## On the lower bound for the degree of a hypothetical counterexample to the plane Jacobian conjecture

### Christian Valqui

#### Pontifical Catholic University of Peru, Peru - This email address is being protected from spambots. You need JavaScript enabled to view it.

We use the detailed description of the shape of the support of a hypothetical minimal counterexample $(P,Q)$ to the plane Jacobian conjecture in order to discard all cases of $(\deg(P),\deg(Q))$ satisfying $\max\{\deg(P),\deg(Q)\}< 125$, except the pair $(72,108)$ (and the symmetric pair $(108,72)$), thus increasing the lower bound of $100$ obtained by Moh up to $108$.

This is a joint work with Rodrigo Horruitiner, Juan José Guccione and Jorge Guccione.

## Posters

## Topological classification of submersion functions

### Ingrid S. Meza-Sarmiento

#### Universidade Federal de São Carlos, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.

Let $f, g : \mathbb{R}^2\to \mathbb{R}$ be two submersion functions and $\mathscr{F}(f)$ and $\mathscr{F}(g)$ be the regular foliations of $\mathbb{R}^2$ whose leaves are the connected components of the level sets of $f$ and $g$, respectively. We say that $f$ and $g$ are topologically equivalent (resp. o-topologically equivalent) if there exist homeomorphisms (resp. orientation preserving homeomorphisms) $h: \mathbb{R}^2\to \mathbb{R}^2$ and $\ell: \mathbb{R} \to \mathbb{R}$ such that $$ \ell \circ f = g \circ h. $$ The topological equivalence of $f$ and $g$ guarantees the topological equivalence of $\mathscr{F}(f)$ and $\mathscr{F}(g)$, but the converse is not true, in general. The main objective in this poster is to introduce a class of submersions, wide enough in order to contain non-trivial behaviors, Broughton example included, and to give necessary and sufficient conditions for that the converse implication be also valid inside this class. We also present a complete topological invariant for this class of submersions.