arXiv subjects

J. Gwoździewicz

Publications and source records attributed to J. Gwoździewicz.

2 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↗

arXiv2015

Łojasiewicz exponents and Farey sequences

\noindent Let $I$ be an ideal of the ring of formal power series $\bK[[x,y]]$ with coefficients in an algebraically closed field $\bK$ of arbitrary characteristic. Let $Φ$ denote the set of all parametrizations $φ=(φ_1,φ_2)\in \bK[[t]]^2$, where $φ\neq (0,0)$ and $φ(0,0)=(0,0)$. The purpose of this paper is to investigate the invariant \[ \Lo(I)=\sup_{φ\in Φ}\left(\inf_{f\in I} \frac{\ord f \circ φ}{\ord φ}\right) \] \noindent called the {\it Łojasiewicz exponent} of $I$. Our main result states that for the ideals $I$ of finite codimension the Łojasiewicz exponent $\Lo(I)$ is a Farey number i.e. an integer or a rational number of the form $N+\frac{b}{a}$, where $a,b,N$ are integers such that $0<b<a<N$.

math.AG↗