P. A. Helminck

activity

20162026

most citedTropicalizing abelian covers of algebraic curves

9 citations · 21 across the 18 of their papers we have counts for

collaborators

19 papers

math.AG2026

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,…

math.AG2026

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…

math.AG2024

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…

math.NT2024

Toric ranks and component groups of modular curves

Paul Alexander Helminck

Let $p \neq = 2, 3$ be a prime number and let $Γ \subset SL_{2} (Z)$ be a congruence subgroup with modular curve $X_{Γ} / K$ and Jacobian $J (X_{Γ})$ . In this paper we give a…

math.AG2024★ 1 cited

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 $Z / p$ -covers of tropical curves of genus $g \geq 2$ , where $p$ is a prime number. We use recent techniques by Chan-…

math.AG2023

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…

math.AG2022

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 $Y$ . In particular, we are interest…

math.AG2022★ 1 cited

Tropical invariants for binary quintics and reduction types of Picard curves

Paul Alexander Helminck, Yassine El Maazouz, Enis Kaya

We express the reduction types of Picard curves in terms of tropical invariants associated to binary quintics. We also give a general framework for tropical invariants associated t…

math.AG2021

Invariants for trees of non-archimedean polynomials and skeleta of superelliptic curves

Paul Alexander Helminck

In this paper we generalize the $j$ -invariant criterion for the semistable reduction type of an elliptic curve to superelliptic curves $X$ given by $y^{n} = f (x)$ . We first define a…

math.AG2020★ 1 cited

A generalization of the Newton-Puiseux algorithm for semistable models

Paul Alexander Helminck

In this paper we give an algorithm that calculates the skeleton of a tame covering of curves over a complete discretely valued field. The algorithm relies on the {tame simultaneous…

math.AG2018★ 1 cited

Faithful tropicalizations of elliptic curves using minimal models and inflection points

Paul Alexander Helminck

We give an elementary proof of the fact that any elliptic curve $E$ over an algebraically closed non-archimedean field $K$ with residue characteristic $\neq = 2, 3$ and with $v(j(E))…

math.AG2018

Skeletal filtrations of the fundamental group of a non-archimedean curve

Paul Alexander Helminck

In this paper we study skeleta of residually tame coverings of a marked curve over a non-archimedean field. We first generalize a result by Liu and Lorenzini by proving a simultane…

math.AG2018

Semisimple pointed isogeny graphs for abelian varieties

Paul Alexander Helminck

In this paper we show that if $ϕ_{i} : A_{i} \to A$ is a semisimple pointed $K$ -rational $ℓ$ -isogeny graph of order $n$ for a prime $ℓ$ , then the group of $ℓ$ -torsio…

math.AG2018

Torsion points on elliptic curves and tame semistable coverings

P. A. Helminck

In this paper, we study tame Galois coverings of semistable models that arise from torsion points on elliptic curves. These coverings induce Galois morphisms of intersection graphs…

math.NT2018★ 4 cited

Tribonacci numbers and primes of the form $p = x^{2} + 11 y^{2}$

Tim Evink, Paul Alexander Helminck

In this paper we show that for any prime number $p$ not equal to $11$ or $19$ , the Tribonacci number $T_{p - 1}$ is divisible by $p$ if and only if $p$ is of the form $x^{2} + 11 y^{2}$ . We…

math.AG2017★ 4 cited

Tropicalizing tame degree three coverings of the projective line

Paul Alexander Helminck

In this paper, we study the problem of tropicalizing tame degree three coverings of the projective line. Given any degree three covering $C ⟶ P^{1}$ , we give…

math.AG2017

Tropical superelliptic curves

Madeline Brandt, Paul Alexander Helminck

We present an algorithm for computing the Berkovich skeleton of a superelliptic curve $y^{n} = f (x)$ over a valued field. After defining superelliptic weighted metric graphs, we show t…

math.AG2017★ 9 cited

Tropicalizing abelian covers of algebraic curves

Paul Alexander Helminck

In this thesis, we study the Berkovich skeleton of an algebraic curve over a discretely valued field $K$ . We do this using coverings $C \to P^{1}$ of the projectiv…

math.AG2016

Tropical Igusa Invariants

Paul Alexander Helminck

Let $X$ be a smooth geometrically connected projective curve of genus two over a complete non-archimedean field $K$ . For discretely valued $K$ , the first main theorem in \cite{liu}…

Root bounds of vertical systems using tropical geometry

Angular pair-of-pants decompositions of complex varieties

A tropical method for solving parametrized polynomial systems

Toric ranks and component groups of modular curves

On the topology of the moduli of tropical unramified p-covers

A symbolic algorithm for calculating power series expansions and dual intersection graphs of semistable models

Generic root counts and flatness in tropical geometry

Tropical invariants for binary quintics and reduction types of Picard curves

Invariants for trees of non-archimedean polynomials and skeleta of superelliptic curves

A generalization of the Newton-Puiseux algorithm for semistable models

Faithful tropicalizations of elliptic curves using minimal models and inflection points

Skeletal filtrations of the fundamental group of a non-archimedean curve

Semisimple pointed isogeny graphs for abelian varieties

Torsion points on elliptic curves and tame semistable coverings

Tribonacci numbers and primes of the form p=x2+11y2

Tropicalizing tame degree three coverings of the projective line

Tropical superelliptic curves

Tropicalizing abelian covers of algebraic curves

Tropical Igusa Invariants

Tribonacci numbers and primes of the form $p = x^{2} + 11 y^{2}$