Related papers: Belyi map verification using certified path tracki…
We exhibit an algorithm that, given input a curve $X$ over a number field, computes as output the minimal degree of a Belyi map $X \to \mathbb{P}^1$.
Complex structure moduli of a Calabi-Yau threefold in $N=1$ supersymmetric heterotic compactifications can be stabilized by holomorphic vector bundles. The stabilized moduli are determined by a computation of Atiyah class. In this paper, we…
The focus of this paper is on reducing the complexity in verification by exploiting modularity at various levels: in specification, in verification, and structurally. For specifications, we use the modular language CSP-OZ-DC, which allows…
We establish interval arithmetic as a practical tool for certification in numerical algebraic geometry. Our software HomotopyContinuation.jl now has a built-in function certify, which proves the correctness of an isolated nonsingular…
Polynomial systems occur in many areas of science and engineering. Unlike general nonlinear systems, the algebraic structure enables to compute all solutions of a polynomial system. We describe our massive parallel predictor-corrector…
Training large language models (LLMs) at scale requires parallel execution across thousands of devices, incurring enormous computational costs. Yet, these costly distributed trainings are rarely verified, leaving them prone to silent errors…
We present all Belyi maps P^1(C) -> P^1(C) having almost simple primitive monodromy groups (not isomorphic to A_n or S_n) containing rigid and rational generating triples of degree between 50 and 250. This also leads to new polynomials…
The moduli space of regular stable maps with values in a complex manifold admits naturally the structure of a complex orbifold. Our proof uses the methods of differential geometry rather than algebraic geometry. It is based on Hardy…
Determining Fourier coefficients of modular forms of a finite index noncongruence subgroups of the modular group is still a non-trivial task. In this brief note we describe a new algorithm to reliably calculate an approximation for a…
A type system combining type application, constants as types, union types (associative, commutative and idempotent) and recursive types has recently been proposed for statically typing path polymorphism, the ability to define functions that…
Software correctness is ensured mathematically through formal verification, which involves the resources of generating formal requirement specifications and having an implementation that must be verified. Tools such as model-checkers and…
We prove an analogue of Belyi's theorem in characteristic two. Our proof consists of the following three steps. We first introduce a new notion called "pseudo-tame" for morphisms between curves over an algebraically closed field of…
In \cite{K-rig}, a map $\beta:\mathcal R\to\mathcal{B}el$ from the set $\mathcal R$ of equivalence classes of rigid germs of finite morphisms branched in germs of curves having $ADE$ singularity types onto the set $\mathcal{B}el$ of…
The article focuses on three different notions of polynomiality for maps of modules. In addition to the polynomial maps studied by Eilenberg and Mac Lane and the strict polynomial maps ("lois polynomes") considered by Roby, we introduce…
We implement a real polyhedral homotopy method using three functions. The first function provides a certificate that our real polyhedral homotopy is applicable to a given system; the second function generates binomial systems for a start…
Polynomial systems occur in many fields of science and engineering. Polynomial homotopy continuation methods apply symbolic-numeric algorithms to solve polynomial systems. We describe the design and implementation of our web interface and…
We use our recent implementation of a certified homotopy tracking algorithm to search for start systems that minimize the average complexity of finding all roots of a regular system of polynomial equations. While finding optimal start…
In this paper, we develop several tools to study the degree growth and stabilization of monomial maps. Using these tools, we can classify semisimple three dimensional monomial maps by their dynamical behavior.
Persistent homology encodes the evolution of homological features of a multifiltered cell complex in the form of a multigraded module over a polynomial ring, called a multiparameter persistence module, and quantifies it through invariants…
The possibility of errors in human-engineered formal verification software, such as model checkers, poses a serious threat to the purpose of these tools. An established approach to mitigate this problem are certificates -- lightweight,…