May 10,  2024

on-line on zoom

• 10:00-10:50 Carles Bivià-Ausina (Universitat Politècnica de València) Global multiplicity, special closure and non-degeneracy of polynomial maps

  Abstract:   Given a polynomial map \(F :\mathbb{C}^n\to\mathbb{C}^p\) with finite zero set, \(p\ge n\), we introduce the notion of global multiplicity associated to \(F\), which is analogous to the multiplicity of ideals in Noetherian local rings. This notion allows to characterize numerically the Newton nondegeneracy at infinity of \(F\). This fact motivates us to study a combinatorial inequality concerning the normalized volume of global Newton polyhedra and to characterize the corresponding equality by using the notion of special closure. We will also show a counterpart of the Rees’ multiplicity theorem in the context of polynomial maps. This is a joint work with J.A.C. Huarcaya (Universidad Nacional Mayor de San Marcos, Lima, Perú).

• 11:00-11:50  Boulos el Hilany (Technische Universität Braunschweig) The polyhedral type of a complex polynomial map on the plane

  Abstract: Two continuous maps \(f, g : \mathbb{C}^2\longrightarrow\mathbb{C}^2\) are said to be topologically equivalent if there are homeomorphisms \(\varphi,\psi:\mathbb{C}^2\longrightarrow\mathbb{C}^2\) satisfying \(\psi\circ f\circ\varphi = g\). It is known that there are at most finitely many topologically non-equivalent polynomial maps above with any given degree \(d\).The number of these topological types is known only whenever \(d=2\). Recently, we provided a description of several topological invariants for generic complex polynomial maps on the plane sharing a pair of Newton polytopes of a certain type. In this talk, I will present this description, together with the consequent tool for constructing topologically non-equivalent maps of degree \(d\). In turn, we obtain non-trivial lower bounds on their numbers. This is a joint work with Kemal Rose.