arXiv subjects

B. Oréfice-Okamoto

Publications and source records attributed to B. Oréfice-Okamoto.

4 recordsLinked to original sources

arXiv2026

A sharp degree bound in the real Jacobian conjecture

Let $F=(p,q):\mathbb R^2\to \mathbb R^2$ be a polynomial map with nowhere zero Jacobian determinant. A long-standing problem is to determine the largest integer $k$ such that the condition $°p\le k$ guarantees the global injectivity of $F$. Although several partial results have been obtained over the past $30$ years, the sharp degree bound has remained unknown. In this paper, we prove that $F$ is injective whenever $°p=6$. On the other hand, we construct a non-injective polynomial map with nowhere vanishing Jacobian determinant for which $°p=7$. Combined with the previously known injectivity results for $°p\le 5$, our results completely settle the problem and establish the optimal degree bound. More precisely, we show that $7$ is the minimal degree for which non-injective examples can occur.

math.AG↗

arXiv2016

Equisingularity of map germs from a surface to the plane

Let $(X,0)$ be an ICIS of dimension 2 and let $f:(X,0)\to (\C^2,0)$ be a map germ with an isolated instability. We look at the invariants that appear when $X_s$ is a smoothing of $(X,0)$ and $f_s:X_s\to B_ε$ is a stabilization of $f$. We find relations between these invariants and also give necessary and sufficient conditions for a $1$-parameter family to be Whitney equisingular. As an application, we show that a family $(X_t,0)$ is Zariski equisingular if and only if it is Whitney equisingular and the numbers of cusps and double folds of a generic linear projection are constant on $t$.

math.AG↗

arXiv2014

On polynomial submersions of degree $4$ and the real Jacobian conjecture in $\R^2$

The main result of this paper is the following version of the real Jacobian conjecture: "Let $F=(p,q):\R^2\to\R^2$ be a polynomial map with nowhere zero Jacobian determinant. If the degree of $p$ is less than or equal to $4$, then $F$ is injective". Assume that two polynomial maps from $\R^2$ to $\R$ are equivalent when they are the same up to affine changes of coordinates in the source and in the target. We completely classify the polynomial submersions of degree $4$ with at least one disconnected level set up to this equivalence, obtaining four classes. Then, analyzing the half-Reeb components of the foliation induced by a representative $p$ of each of these classes, we prove there is not a polynomial $q$ such that the Jacobian determinant of the map $(p,q)$ is nowhere zero. Recalling that the real Jacobian conjecture is true for maps $F=(p,q)$ when all the level sets of $p$ are connected, we conclude the proof of the main result.

math.DS↗

arXiv2014

Equisingularity of families of isolated determinantal singularities

We study the topological triviality and the Whitney equisingularity of a family of isolated determinantal singularities. On one hand, we give a Lê-Ramanujam type theorem for this kind of singularities by using the vanishing Euler characteristic. On the other hand, we extend the results of Teissier and Gaffney about the Whitney equisingularity of hypersurfaces and complete intersections, respectively, in terms of the constancy of the polar multiplicities.

math.AG↗