Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Surjectivity of linear operators and semialgebraic global diffeomorphisms

Francisco Braun, Luis Renato Gonçalves Dias, Jean Venato Santos

Abstract

We prove that a $C^{\infty}$ semialgebraic local diffeomorphism of $\mathbb{R}^n$ with non-properness set having codimension greater than or equal to $2$ is a global diffeomorphism if $n-1$ suitable linear partial differential operators are surjective. Then we state a new analytic conjecture for a polynomial local diffeomorphism of $\mathbb{R}^n$. Our conjecture implies a very known conjecture of Z. Jelonek. We further relate the surjectivity of these operators with the fibration concept and state a general global injectivity theorem for semialgebraic mappings which turns out to unify and generalize previous results of the literature.

Surjectivity of linear operators and semialgebraic global diffeomorphisms

Abstract

We prove that a semialgebraic local diffeomorphism of with non-properness set having codimension greater than or equal to is a global diffeomorphism if suitable linear partial differential operators are surjective. Then we state a new analytic conjecture for a polynomial local diffeomorphism of . Our conjecture implies a very known conjecture of Z. Jelonek. We further relate the surjectivity of these operators with the fibration concept and state a general global injectivity theorem for semialgebraic mappings which turns out to unify and generalize previous results of the literature.

Paper Structure

This paper contains 4 sections, 12 theorems, 13 equations, 1 table.

Table of Contents

  1. Introduction
  2. Proof of theorems \ref{['t:main']} and \ref{['t:gs-geome']}
  3. A semialgebraic counterexample
  4. Fibrations and geometric properties of the operators $\Delta_i^F$

Key Result

Theorem \oldthetheorem

Let $F: \mathbb R^n \to \mathbb R^n$ be a $C^{\infty}$ semialgebraic mapping with nowhere vanishing $J(F)$ and such that $\operatorname{{codim}} S_F\geq 2$. Then $F$ is a bijective mapping if and only if $\Delta_i^F\left(C^{\infty}(\mathbb R^n)\right) = C^{\infty}(\mathbb R^n)$ for $n-1$ indices $i\

Theorems & Definitions (23)

  • Conjecture \oldthetheorem: Jelonek's conjecture
  • Conjecture \oldthetheorem
  • Theorem \oldthetheorem
  • Conjecture \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Corollary \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem: Part of Theorem 6.4.2 of DH
  • proof : Proof of Statement \ref{['1t']} of Theorem \ref{['t:gs-geome']}
  • ...and 13 more