arXiv subjects
Publications and source records attributed to Mihai Tibar.
We introduce the sphere fibration for real map germs with radial discriminant and we address the problem of its equivalence with the Milnor-Hamm tube fibration. Under natural conditions, we prove the existence of open book structures with singularities and solve the equivalence problem.
We prove a precise version of a general conjecture on the polar degree stated by June Huh. We confirm Huh's conjectural list of all projective hypersurfaces with isolated singularities and polar degree equal to 2.
By combining analytic and geometric viewpoints on the concentration of the curvature of the Milnor fibre, we prove that Lipschitz homeomorphisms preserve the zones of multi-scale curvature concentration as well as the gradient canyon structure of holomorphic functions of two variables. This yields the first new Lipschitz invariants after those discovered by Henry and Parusinski in 2003.
We study the bifurcation values of real polynomial maps $f: \bR^{2n} \to \bR^2$ which reflect the lack of asymptotic regularity at infinity. We formulate real counterparts of some structure results which have been previously proved in case of complex polynomials by Kushnirenko, Némethi and Zaharia and other authors, emphasizing the typical real phenomena that occur.
We introduce a sectional criterion for testing if complex analytic germs $(X,0) \subset (\bC^n, 0)$ are Lipschitz non normally embedded.
The problem of detecting the bifurcation set of polynomial mappings $\mathbb{ C}^m \to \mathbb{ C}^k$, $m\ge 2$, $m\ge k\ge 1$, has been solved in the case $m=2$, $k=1$ only. Its solution, which goes back to the 1970s, involves the non-constancy of the Euler characteristic of fibres. We provide a complete answer to the general case $m= k+1 \ge 3$ in terms of the Betti numbers of fibres and of a vanishing phenomenon discovered in the late 1990s in the real setting.
We consider an equisingularity problem for polynomial families of affine hypersurfaces $X_τ\subset \mathbb C^n$ with (at worst) isolated singularities. We show that the constancy of the global polar invariants $γ^* (X_τ)$ is equivalent to the $t$-equisingularity at infinity, an asymptotic-type equisingularity that we introduce. We prove that $γ^*$-constancy implies C$^\infty$-triviality in the neighbourhood of infinity. We show how the invariants $γ^*$ enter in the description of a CW-complex model of a hypersurface $X_τ$ and therefore provide in particular new invariants at infinity for polynomial functions $f: \mathbb C^n \to \mathbb C$.
We introduce and study the vanishing homology of singular projective hypersurfaces. We prove its concentration in two levels in case of 1-dimensional singular locus $Σ$, and moreover determine the ranks of the nontrivial homology groups. These two groups depend on the monodromy at special points of $Σ$ and on the effect of the monodromy of the local system over its complement.
We study vanishing cycles naturally attached to a meromorphic function with isolated singularities, in both local and global settings.
We locate the Malgrange non-regular values of a given polynomial function $f:\bC^n \to \bC$ by using a series of affine polar curves. We moreover show that all non-trivial Malgrange non-regular values of $f$ are indicated by a single "super-polar curve" which we introduce here, providing also an effective algorithm of detection.
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.
In case of one-dimensional singular locus, we use deformations in order to get refined information about the Betti numbers of the Milnor fibre.
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.
Beyond normal surfaces there are several open questions concerning 2- dimensional spaces. We present some results and conjectures along this line.
We prove the minimality of the CW-complex structure for complements of hyperplane arrangements in $\mathbb C^n$ by using the theory of Lefschetz pencils and results on the variation maps within a pencil of hyperplanes. This also provides a method to compute the Betti numbers of complements of arrangements via global polar invariants.
We provide significant conditions under which we prove the existence of stable open book structures at infinity, i.e. on spheres $S^{m-1}_R$ of large enough radius $R$. We obtain new classes of real polynomial maps $\mathbb R^m \to \mathbb R^p$ which induce such structures.
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.
We prove extensions of Milnor's theorem for germs with nonisolated singularity and use them to find new classes of genuine real analytic mappings $ψ$ with positive dimensional singular locus $\Sing ψ\subset ψ^{-1}(0)$, for which the Milnor fibration exists and yields an open book structure with singular binding.