Atypical points at infinity and algorithmic detection of the bifurcation locus of real polynomials

Bifurcation locus and branches at infinity of a polynomial f : C2 -> c

We show that the number of bifurcation points at infinity of a polynomial function f : C2 -> C is at most the number of branches at infinity of a generic fiber of f and that this upper bound can be diminished by one in certain cases.

Hodge loci and atypical intersections: conjectures

We present a conjecture on the geometry of the Hodge locus of a (graded polarizable, admissible) variation of mixed Hodge structure over a complex smooth quasi-projective base, generalizing to this context the Zilber-Pink Conjecture for mixed Shimura varieties (in particular the André-Oort conjecture).

Towards effective detection of the bifurcation locus of real polynomial maps

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.

有理函数空间中的抛物分歧轨迹

Parabolic bifurcation loci in the spaces of rational functions

本文给出了 $\mathbb{P}^1$ 上次数 $d>1$ 的有理函数空间 $\operatorname{Rat}_d$ 中抛物分歧轨迹的几何描述，推广了首一多项式情形的研究，且结果对二次有理函数同样具有创新性。

On Newton non-degeneracy of polynomial mappings

We introduce a new non-degeneracy condition at infinity for a real or a mixed polynomial mapping $F$ which allows us to approximate its bifurcation locus in terms of certain Newton polyhedra. We derive a sufficiency result for the Jacobian problem in the setting of mixed mappings.

Invertible polynomial mappings via Newton non-degeneracy

We prove a sufficient condition for the Jacobian problem in the setting of real, complex and mixed polynomial mappings. This follows from the study of the bifurcation locus of a mapping subject to a new Newton non-degeneracy condition.