arXiv subjects

Emil Graf

Publications and source records attributed to Emil Graf.

5 recordsLinked to original sources

arXiv2026

Blow-up Parameter Landscapes for Polynomial Dynamical Systems

Finite-time blow-up is one of the ways in which a dynamical model can become singular, often signaling the breakdown of either the modeled physical system or the model itself. Determining whether blow-up occurs, and for which parameter values and initial conditions, is therefore a fundamental problem in the analysis of nonlinear dynamical systems. We develop a numerical framework for identifying regions of parameter space in which a dynamical system governed by a system of first-order ordinary differential equations with polynomial right-hand sides exhibits finite-time blow-up for at least one initial condition. The approach combines compactification of the phase space with computational algebraic techniques, producing partitioned parameter landscapes that reveal blow-up and non-blow-up regimes. Through several examples, we show that the method replaces problem-specific hand calculations with an automated computational tool for analyzing blow-up regions in parameter-dependent dynamical systems.

math.DS↗

arXiv2025

A Hidden Variable Resultant Method for the Polynomial Multiparameter Eigenvalue Problem

We present a novel, global algorithm for solving polynomial multiparameter eigenvalue problems (PMEPs) by leveraging a hidden variable tensor Dixon resultant framework. Our method transforms a PMEP into one or more univariate polynomial eigenvalue problems, which are solved as generalized eigenvalue problems. Our general approach avoids the need for custom linearizations of PMEPs. We provide rigorous theoretical guarantees for generic PMEPs and give practical strategies for nongeneric systems. Benchmarking on applications from aeroelastic flutter and leaky wave propagation confirms that our algorithm attains high accuracy and robustness while being broadly applicable to many PMEPs.

math.NA↗

arXiv2024

Numerical Instability of Algebraic Rootfinding Methods

We demonstrate that the most popular variants of all common algebraic multidimensional rootfinding algorithms are unstable by analyzing the conditioning of subproblems that are constructed at intermediate steps. In particular, we give multidimensional polynomial systems for which the conditioning of a subproblem can be worse than the conditioning of the original problem by a factor that grows exponentially with the number of variables.

math.NA↗

arXiv2019

Cardinalities of Prime Spectra of Precompletions

Given a complete local (Noetherian) ring $T$, we find necessary and sufficient conditions on $T$ such that there exists a local domain $A$ with $|A| < |T|$ and $\widehat{A} = T$, where $\widehat{A}$ denotes the completion of $A$ with respect to its maximal ideal. We then find necessary and sufficient conditions on $T$ such that there exists a domain $A$ with $\widehat{A} = T$ and $|\mbox{Spec}(A)| < |\mbox{Spec}(T)|$. Finally, we use "partial completions" to create local rings $A$ with $\widehat{A} = T$ such that $\mbox{Spec}(A)$ has varying cardinality in different varieties.

math.AC↗

arXiv2019

Structure of Spectra of Precompletions

Let T be a complete local (Noetherian) ring and let A be a local subring of T such that the completion of A with respect to its maximal ideal is T. We investigate the possible structures of the partially ordered set Spec(A). Specifically, we explore the minimal prime ideals of A and their formal fibers, the maximal chains of prime ideals in A, and the number of prime ideals in A containing combinations of minimal prime ideals of A.

math.AC↗