An approach to solving systems of polynomials via modular arithmetics with applications

DOI10.1016/j.cam.2011.06.018zbMath1225.13029OpenAlexW2087304798MaRDI QIDQ645703

Valery G. Romanovski, Mateja Prešern

Publication date: 10 November 2011

Published in: Journal of Computational and Applied Mathematics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/j.cam.2011.06.018

zbMATH Keywords

decomposition of affine varietiespolynomial systems of ODEslinearizability of ODEs

Mathematics Subject Classification ID

Transformation and reduction of ordinary differential equations and systems, normal forms (34C20) Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) (13P10) Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations (34C07) Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.) (13P25) Solving polynomial systems; resultants (13P15) Numerical algorithms for computer arithmetic, etc. (65Y04)

Related Items (33)

The solution of the \(1:-3\) resonant center problem in the quadratic case ⋮ Centers for a class of generalized quintic polynomial differential systems ⋮ The center problem and composition condition for Abel differential equations ⋮ Nondegenerate centers and limit cycles of cubic Kolmogorov systems ⋮ Integrability conditions of a resonant saddle perturbed with homogeneous quintic nonlinearities ⋮ On integrability and linearizability of persistent \(p:- q\) resonant systems ⋮ Integrability of fractional order generalized systems with \(p:-q\) resonance ⋮ Bi-center problem for some classes of \(\mathbb{Z}_2\)-equivariant systems ⋮ On integrability conditions and limit cycle bifurcations for polynomial systems ⋮ Isochronicity of a \(\mathbb{Z}_2\)-equivariant quintic system ⋮ Rigid centres on the center manifold of tridimensional differential systems ⋮ Integrability of complex planar systems with homogeneous nonlinearities ⋮ The center problem for a \({1:-4}\) resonant quadratic system ⋮ Nondegenerate and nilpotent centers for a cubic system of differential equations ⋮ Liénard Equation and Its Generalizations ⋮ Integrability and linearizability of symmetric three-dimensional quadratic systems ⋮ The center problem for a \(2:-3\) resonant cubic Lotka-Volterra system ⋮ Center conditions for polynomial Liénard systems ⋮ Qualitative studies of some biochemical models ⋮ Centers and isochronous centers for generalized quintic systems ⋮ Center conditions for nilpotent cubic systems using the Cherkas method ⋮ Center problem in the center manifold for quadratic differential systems in \(\mathbb{R}^3\) ⋮ Center conditions for generalized polynomial Kukles systems ⋮ Isochronicity and linearizability of a planar cubic system ⋮ A hybrid symbolic-numerical approach to the center-focus problem ⋮ Linearizability and critical period bifurcations of a generalized Riccati system ⋮ Usage of modular techniques for efficient computation of ideal operations ⋮ Investigation of center manifolds of three-dimensional systems using computer algebra ⋮ Integrability of Lotka–Volterra Planar Complex Cubic Systems ⋮ Centers and Limit Cycles of a Generalized Cubic Riccati System ⋮ Isochronicity of bi-centers for symmetric quartic differential systems ⋮ Isochronicity and linearizability of planar polynomial Hamiltonian systems ⋮ Nondegenerate centers for Abel polynomial differential equations of second kind

Uses Software

• SINGULAR
• primdec

Cites Work

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• Localization and primary decomposition of polynomial ideals
• Darboux linearization and isochronous centers with a rational first integral
• Decomposition of algebraic sets and applications to weak centers of cubic systems
• Linearizability conditions of time-reversible quartic systems having homogeneous nonlinearities
• Gröbner bases and primary decomposition of polynomial ideals
• A p-adic approach to the computation of Gröbner bases
• Elimination methods
• On lucky ideals for Gröbner basis computations
• On lucky primes
• Modular algorithms for computing Gröbner bases.
• Normalizable, integrable, and linearizable saddle points for complex quadratic systems in \(\mathbb{C}^2\)
• \(1: - 3\) resonant centers on \(\mathbb{C}^2\) with homogeneous cubic nonlinearities
• Local bifurcations of critical periods in a generalized 2D LV system
• Bruno Buchberger's PhD thesis 1965: An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal. Translation from the German
• Some comments on the modular approach to Gröbner-bases
• Isochronicity of analytic systems via Urabe's criterion
• The Center and Cyclicity Problems
• P-adic reconstruction of rational numbers
• Linearizability of linear systems perturbed by fifth degree homogeneous polynomials
• Linearizability conditions for Lotka–Volterra planar complex cubic systems
• Exact Solutions of Linear Equations with Rational Coefficients by Congruence Techniques
• Nondegenerate linearizable centres of complex planar quadratic and symmetric cubic systems in \(\mathbb C^2\)

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