7 papers
Root bounds of vertical systems using tropical geometry
Elisenda Feliu, Paul Alexander Helminck, Oskar Henriksson +3
Sparse polynomial systems with vertical coefficient dependencies arise naturally when describing the critical points of optimization problems and, when augmented with linear forms,…
Angular pair-of-pants decompositions of complex varieties
Yassine Elmaazouz, Paul Alexander Helminck
We define the notion of torically hyperbolic varieties and we construct pair-of-pants decompositions for these in terms of angle sets of essential projective hyperplane complements…
A symbolic algorithm for calculating power series expansions and dual intersection graphs of semistable models
Paul Alexander Helminck
In this paper we develop a symbolic algorithm to calculate multivariate power series expansions of univariate polynomials over general base rings. We use this to give a complete po…
Generic root counts and flatness in tropical geometry
Paul Alexander Helminck, Yue Ren
We use tropical and non-archimedean geometry to study the generic number of solutions of families of polynomial equations over a parameter space . In particular, we are interest…
A tropical method for solving parametrized polynomial systems
Paul Alexander Helminck, Oskar Henriksson, Yue Ren
We give a framework for constructing generically optimal homotopies for parametrized polynomial systems from tropical data. Here, generically optimal means that the number of paths…
On the topology of the moduli of tropical unramified p-covers
Yassine El Maazouz, Paul Alexander Helminck, Felix Röhrle +2
We study the topology of the moduli space of unramified -covers of tropical curves of genus , where is a prime number. We use recent techniques by Chan-…