arXiv subjects

Paul Alexander Helminck

Publications and source records attributed to Paul Alexander Helminck.

18 recordsLinked to original sources

arXiv2026

Root bounds of vertical systems using tropical geometry

Sparse polynomial systems with vertical coefficient dependencies arise naturally when describing the critical points of optimization problems and, when augmented with linear forms, the steady states of chemical reaction networks. Moreover, any polynomial system is the specialization of such a parametrized system. We prove that the generic number of complex zeros of an augmented vertically parametrized system is the tropical intersection number of a tropical linear space and a classical linear space. In the special case when the matroid of the tropical linear space is cotransversal, we express this number as a mixed volume. We also obtain bounds on the maximal number of positive zeros, which is often the significant number in applications. We derive lower bounds from the number of intersections between positive tropicalizations, and when the positive zeros have toric structure, we provide upper bounds that are simpler and in some cases smaller than the generic root count. The resulting algorithms are implemented in Julia.

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arXiv2026

Angular pair-of-pants decompositions of complex varieties

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. This construction generalizes the classical pair-of-pants decomposition for hyperbolic Riemann surfaces. In our first main theorem, we prove that the natural angle map associated to an essential projective hyperplane complement is a homotopy equivalence, extending earlier work of Salvetti and Björner-Ziegler. By a topological argument, we further show that the angle map for a finite Kummer covering of an essential projective hyperplane complement is likewise a homotopy equivalence. We then explain how these local building blocks can be glued along the dual intersection complex of a semistable degeneration. Using the theory of Kato-Nakayama spaces, we prove that the resulting space is homotopy equivalent to the original algebraic variety. We make this explicit for complete intersections in projective space using techniques from tropical geometry.

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arXiv2025

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

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 power series algorithm to calculate the dual intersection graph of a semistable model of a curve over a non-archimedean field. We first study the problem of recovering the relative poset structure of a finite covering $X'\to X$ of normal, relatively unibranch, Noetherian connected schemes. We show that we can reconstruct the poset structure of $X'$ in terms of group-theoretic data over the base $X$. This group-theoretic data consists of glued double cosets, and we show how these can be interpreted in terms of glued power series approximations. We then show how our algorithms calculate these glued power series approximations, so that we can work with the branches of normalizations $X'\to X$ without calculating integral closures. These algorithms have been implemented in OSCAR. We give a detailed study of the key steps in these algorithms for coverings of semistable models, with various examples to illustrate the non-trivial gluing phenomena. We conclude by interpreting these techniques in the context of analytic spaces, with an eye towards future applications in $p$-adic integration theory.

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arXiv2025

Generic root counts and flatness in tropical geometry

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 interested in the choices of parameters for which the generic root count is attained. Our families are given as subschemes $X\subseteq T$ where $T$ is a relative torus over $Y$. We generalize Bernstein's theorem from an intersecting family of hypersurfaces $X=V(f_1)\cap\dots\cap V(f_n)$ to an intersecting family of higher-codimensional schemes $X=X_1\cap\dots\cap X_k$, replacing the mixed volume by a tropical intersection product. Central to our work is the notion of tropical flatness of $X$ around a point $P\in Y$, which allows us to transfer tropical properties of the fiber over $P$ to generic properties. We show that tropical flatness holds over a dense open subset of the Berkovich analytification $Y^\text{an}$, and that the tropical intersection number is attained as a root count at all $P\in Y^\text{an}$ around which the $X_i$'s are tropically flat and the tropical prevariety of the fibers $\bigcap_{i=1}^k\text{Trop}(X_{i,P})$ is bounded. We then study the generic root count of a wide class of parametrized square polynomial systems. This in particular gives tropical formulas for the volumes of Newton-Okounkov bodies, and the number of complex steady states of chemical reaction networks.

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arXiv2024

A tropical method for solving parametrized polynomial systems

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 tracked is equal to the generic number of solutions. We focus on two types of parametrized systems -- vertically parametrized and horizontally parametrized systems -- and discuss techniques for computing the tropical data efficiently. We end the paper with several case studies, where we analyze systems arising from chemical reaction networks, coupled oscillators, and rigid graphs.

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arXiv2024

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

We study the topology of the moduli space of unramified $\mathbb{Z}/p$-covers of tropical curves of genus $g \geq 2$, where $p$ is a prime number. We use recent techniques by Chan--Galatius--Payne to identify contractible subcomplexes of the moduli space. We then use this contractibility result to show that this moduli space is simply connected. In the case of genus 2, we determine the homotopy type of this moduli space for all primes $p$. This work is motivated by prospective applications to the top-weight cohomology of the space of prime cyclic étale covers of smooth algebraic curves.

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arXiv2024

Tropical invariants for binary quintics and reduction types of Picard curves

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 to group actions on arbitrary varieties. The problem of finding tropical invariants for binary forms fits in this general framework by mapping the space of binary forms to symmetrized versions of the Deligne-Mumford compactification $\overline{M}_{0,n}$.

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arXiv2024

Toric ranks and component groups of modular curves

Let $p\neq{2,3}$ be a prime number and let $Γ\subset \mathrm{SL}_{2}(\mathbb{Z})$ be a congruence subgroup with modular curve $X_Γ/K$ and Jacobian $J(X_Γ)$. In this paper we give an explicit group-theoretic description of the semistable toric rank and component group of $J(X_Γ)$ at the finite places of $K$ lying over $p$. We first produce a suitable deformation retract of the minimal Berkovich skeleton of $X_Γ$ in terms of Hecke-Iwahori double coset spaces. We call this deformation retract the pruned skeleton of the curve. Our description of this skeleton includes a group-theoretic formula for the edge lengths, allowing us to give the component group of the modular curve as the quotient of a lattice using the monodromy pairing. For $X_{0}(N)$, $X_{1}(N)$, $X_{sp}(N)$ and $X_{sp}^{+}(N)$, we explicitly determine the pruned skeleta using a set of coset schemes over $\mathbb{Z}$. This in particular recovers results by Deligne-Rapoport, Edixhoven, Coleman-McMurdy and Tsushima on the semistable reduction type of $X_{0}(p^{n})$ for $n\leq{4}$. Finally, we determine the geometric Tamagawa number and the prime-to-$2$ structure of the component group of $X_{0}(N)$ over the extension given by Krir's theorem.

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arXiv2021

A generalization of the Newton-Puiseux algorithm for semistable models

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 semistable reduction theorem}, for which we give a short proof. To use this theorem in practice, we show that we can find extensions of chains of prime ideals in normalizations using compatible power series. This allows us to reconstruct the skeleton of the covering. In studying the connections between power series and extensions of prime ideals, we obtain generalizations of classical theorems from number theory such as the Kummer-Dedekind theorem and Dedekind's theorem for cycles in Galois groups.

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arXiv2021

Tropical Igusa Invariants

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} gives a set of criteria on the Igusa invariants of the curve that determine the minimal Berkovich skeleton of $X$ together with its edge lengths and vertex weights. In this paper we use the theory of Berkovich spaces to give a new proof of this theorem that works for arbitrary complete non-archimedean fields. We furthermore interpret the final result in terms of tropical moduli spaces and tropical Igusa invariants. This reformulation shows that the abstract tropicalization map ${M}_{2}\to\mathrm{trop}(M_{2})$ factors through the tropicalization of a concrete embedding of ${M}_{2}$ into a weighted projective space.

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arXiv2021

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

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 simultaneous semistable reduction theorem for residually tame coverings. We then use this to construct a functor from the category of residually tame coverings of a marked curve $(X,D)$ to the category of tame coverings of a metrized complex $Σ$ associated to $(X,D)$. We enhance the latter category by adding a set of gluing data to every covering and we show that this yields an equivalence of categories. Using this equivalence, we then define filtrations of the fundamental group of the marked curve, giving for instance the absolute decomposition and inertia groups of the metrized complex. We then use the analytic slope formula to prove that the extensions that arise from the abelianizations of the decomposition and inertia quotients coincide with the extensions that arise from the toric and connected parts of the analytic Jacobian of the curve.

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arXiv2021

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

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 set of tropical invariants for $f(x)$ using symmetrized Plücker coordinates and we show that these invariants determine the tree associated to $f(x)$. We then prove that this tree completely determines the reduction type of $X$ for $n$ that are not divisible by the residue characteristic. The conditions on the tropical invariants that distinguish between the different types are given by half-spaces as in the elliptic curve case. These half-spaces arise naturally as the moduli spaces of certain Newton polygon configurations. We give a procedure to write down their equations and we illustrate this by giving the half-spaces for polynomials $f(x)$ of degree $d\leq{5}$.

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arXiv2020

Tropical superelliptic curves

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 that each one is realizable by an algebraic superelliptic curve when $n$ is prime. Lastly, we study the locus of superelliptic weighted metric graphs inside the moduli space of tropical curves of genus $g$.

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arXiv2019

Faithful tropicalizations of elliptic curves using minimal models and inflection points

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 $v(j(E))<0$ admits a tropicalization that contains a cycle of length $-v(j(E))$. We first define an adapted form of minimal models over non-discrete valuation rings and we recover several well-known theorems from the discrete case. Using these, we create an explicit family of marked elliptic curves $(E,P)$, where $E$ has multiplicative reduction and $P$ is an inflection point that reduces to the singular point on the reduction of $E$. We then follow the strategy as in \cite[Theorem 6.2]{BPR11} and construct an embedding such that its tropicalization contains a cycle of length $-v(j(E))$. We call this a numerically faithful tropicalization. A key difference between this approach and the approach in \cite{BPR11} is that we do not require any of the analytic theory on Berkovich spaces such as the {\it{Poincaré-Lelong formula}} or \cite[Theorem 5.25]{BPR11} to establish the numerical faithfulness of this tropicalization.

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arXiv2019

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

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 $x^2+11y^2$. We first use class field theory on the Galois closure of the number field corresponding to the polynomial $x^3-x^2-x-1$ to give the splitting behavior of primes in this number field. After that, we apply these results to the explicit exponential formula for $T_{p-1}$. We also give a connection between the Tribonacci numbers and the Fourier coefficients of the unique newform of weight $2$ and level $11$.

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arXiv2018

Semisimple pointed isogeny graphs for abelian varieties

In this paper we show that if $ϕ_{i}:A_{i}\rightarrow{A}$ is a semisimple pointed $K$-rational $\ell$-isogeny graph of order $n$ for a prime $\ell$, then the group of $\ell$-torsion points $A[\ell](\overline{K})$ contains a subspace of dimension $n$ generated by $K$-rational points. We also show that the same result is true for elliptic curves without the semisimplicity condition. Furthermore, we give an explicit counterexample for abelian varieties of higher dimension to show that the semisimplicity condition is indeed necessary.

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arXiv2018

Tropicalizing abelian covers of algebraic curves

In this thesis, we study the Berkovich skeleton of an algebraic curve over a discretely valued field $K$. We do this using coverings $C\rightarrow{\mathbb{P}^{1}}$ of the projective line. To study these coverings, we take the Galois closure of the corresponding injection of function fields $K(\mathbb{P}^{1})\rightarrow{K(C)}$, giving a Galois morphism $\overline{C}\rightarrow{\mathbb{P}^{1}}$. A theorem by Liu and Lorenzini tells us how to associate to this morphism a Galois morphism of semistable models $\mathcal{C}\rightarrow{\mathcal{D}}$. That is, we make the branch locus disjoint in the special fiber of $\mathcal{D}$ and remove any vertical ramification on the components of $\mathcal{D}_{s}$. This morphism $\mathcal{C}\rightarrow{\mathcal{D}}$ then gives rise to a morphism of intersection graphs $Σ(\mathcal{C})\rightarrow{Σ(\mathcal{D})}$. Our goal is to reconstruct $Σ(\mathcal{C})$ from $Σ(\mathcal{D})$ and we will do this by giving a set of covering and twisting data. These then give algorithms for finding the Berkovich skeleton of a curve $C$ whenever that curve has a morphism $\overline{C}\rightarrow{\mathbb{P}^{1}}$ with a solvable Galois group. In particular, this gives an algorithm for finding the Berkovich skeleton of any genus three curve. These coverings also give a new proof of a classical result on the semistable reduction type of an elliptic curve, saying that an elliptic curve has potential good reduction if and only if the valuation of the $j$-invariant is positive.

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arXiv2017

Tropicalizing tame degree three coverings of the projective line

In this paper, we study the problem of tropicalizing tame degree three coverings of the projective line. Given any degree three covering $C\longrightarrow{\mathbb{P}^{1}}$, we give an algorithm that produces the Berkovich skeleton of $C$. In particular, this gives an algorithm for finding the Berkovich skeleton of a genus $3$ curve. The algorithm uses a continuity statement for inertia groups of semistable Galois coverings, which we prove first. After that we give a formula for the decomposition group of an irreducible component $Γ\subset{\mathcal{C}_{s}}$ for a semistable Galois covering $\mathcal{C}\longrightarrow{\mathcal{D}}$. We conclude the paper with a simple application of these $S_{3}$-coverings to elliptic curves, giving another proof of the familiar semistability criterion for elliptic curves using a natural degree three morphism to $\mathbb{P}^{1}$ instead of the usual degree two morphism.

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