arXiv subjects
Publications and source records attributed to Luis Renato G. Dias.
We show that the variation of the topology at infinity of a two-variable polynomial function is localisable at a finite number of "atypical points" at infinity. We construct an effective algorithm with low complexity in order to detect sharply the bifurcation values of the polynomial function.
We obtain a structure theorem for the nonproperness set $S_f$ of a nonsingular polynomial mapping $f:\mathbb{C}^n \to \mathbb{C}^n$. Jelonek's results on $S_f$ and our result show that if $f$ is a counterexample to the Jacobian conjecture, then $S_f$ is a hypersurface such that $S_f\cap Z \neq \emptyset$, for any $Z\subset \mathbb{C}^n$ biregular to $\mathbb{C}^{n-1}$ and $Z = h^{-1}(0)$ for a polynomial submersion $h: \mathbb{C}^n \to \mathbb{C}$. Also, we present topological approaches to the Jacobian conjecture in $\mathbb{C}^n$.
We answer to a problem raised by recent work of Jelonek and Kurdyka: how can one detect by rational arcs the bifurcation locus of a polynomial map $\bR^n\to\bR^p$ in case $p>1$. We describe an effective estimation of the "nontrivial" part of the bifurcation locus.
We present a new approach for estimating the set of bifurcation values at infinity. This yields a significant shrinking of the number of coefficients in the recent algorithm introduced by Jelonek and Kurdyka for reaching critical values at infinity by rational arcs.