252 citations · 961 across the 42 of their papers we have counts for
10 papers · 1 filter
A stratified polyhedral homotopy method for sampling positive-dimensional zero sets of polynomial systems
Tianran Chen
Numerical algebraic geometry revolves around the study of solutions to polynomial systems via numerical methods. The polyhedral homotopy of Huber and Sturmfels for computing isolat…
Numerical Nonlinear Algebra
Daniel J. Bates, Paul Breiding, Tianran Chen +3
Numerical nonlinear algebra is a computational paradigm that uses numerical analysis to study polynomial equations. Its origins were methods to solve systems of polynomial equation…
On the typical and atypical solutions to the Kuramoto equations
Tianran Chen, Evgeniia Korchevskaia, Julia Lindberg
The Kuramoto model is a dynamical system that models the interaction of coupled oscillators. There has been much work to effectively bound the number of equilibria to the Kuramoto…
GPU-accelerated path tracker for polyhedral homotopy
Tianran Chen
The polyhedral homotopy method of Huber and Sturmfels is a particularly efficient and robust numerical method for solving system of (Laurent) polynomial equations. A central compon…
On the root count of algebraic Kuramoto equations in cycle networks with uniform coupling
Tianran Chen, Evgeniia Korchevskaia
The Kuramoto model is a classical model used in the study of spontaneous synchronizations in networks of coupled oscillators. In this model, frequency synchronization configuration…
A geometric criterion on the equality between BKK bound and intersection index
Tianran Chen
The Bernshtein-Kushnirenko-Khovanskii theorem provides a generic root count for system of Laurent polynomials in terms of the mixed volume of their Newton polytopes (i.e., the BKK…
A toric deformation method for solving Kuramoto equations
Tianran Chen, Robert Davis
The study of frequency synchronization configurations in Kuramoto models is a ubiquitous mathematical problem that has found applications in many seemingly independent fields. In t…
Counting equilibria of the Kuramoto model using birationally invariant intersection index
Tianran Chen, Robert Davis, Dhagash Mehta
Synchronization in networks of interconnected oscillators is a fascinating phenomenon that appear naturally in many independent fields of science and engineering. A substantial amo…
Unmixing the mixed volume computation
Tianran Chen
Computing mixed volume of convex polytopes is an important problem in computational algebraic geometry. This paper establishes sufficient conditions under which the mixed volume of…
Parallel degree computation for solution space of binomial systems with an application to the master space of gauge theories
Tianran Chen, Dhagash Mehta
The problem of solving a system of polynomial equations is one of the most fundamental problems in applied mathematics. Among them, the problem of solving a system of binomial equa…