72 research outputs found
Efficient Algorithms for Computing Rational First Integrals and Darboux Polynomials of Planar Polynomial Vector Fields
International audienceWe present fast algorithms for computing rational first integrals with bounded degree of a planar polynomial vector field. Our approach builds upon a method proposed by Ferragut and Giacomini, whose main ingredients are the calculation of a power series solution of a first order differential equation and the reconstruction of a bivariate polynomial annihilating this power series. We provide explicit bounds on the number of terms needed in the power series. This enables us to transform their method into a certified algorithm computing rational first integrals via systems of linear equations. We then significantly improve upon this first algorithm by building a probabilistic algorithm with arithmetic complexity and a deterministic algorithm solving the problem in at most arithmetic operations, where~ denotes the given bound for the degree of the rational first integral, and where is the degree of the vector field, and the exponent of linear algebra. We also provide a fast heuristic variant which computes a rational first integral, or fails, in arithmetic operations. By comparison, the best previous algorithm uses at least arithmetic operations. We then show how to apply a similar method to the computation of Darboux polynomials. The algorithms are implemented in a Maple package RationalFirstIntegrals which is available to interested readers with examples showing its efficiency
Bounding the number of remarkable values via Jouanolou's theorem
International audienceIn this article we bound the number of remarkable values of a polynomial vector field. The proof is short and based on Jouanolou's theorem about rational first integrals of planar polynomial derivations. Our bound is given in term of the size of a Newton polygon associated to the vector field. We prove that this bound is almost reached
A generalization of the S-function method applied to a Duffing-Van der Pol forced oscillator
In [1,2] we have developed a method (we call it the S-function method) that
is successful in treating certain classes of rational second order ordinary
differential equations (rational 2ODEs) that are particularly `resistant' to
canonical Lie methods and to Darbouxian approaches. In this present paper, we
generalize the S-function method making it capable of dealing with a class of
elementary 2ODEs presenting elementary functions. Then, we apply this method to
a Duffing-Van der Pol forced oscillator, obtaining an entire class of first
integrals
On a Quantization of the Classical -Functions
The Jacobi theta-functions admit a definition through the autonomous
differential equations (dynamical system); not only through the famous Fourier
theta-series. We study this system in the framework of Hamiltonian dynamics and
find corresponding Poisson brackets. Availability of these ingredients allows
us to state the problem of a canonical quantization to these equations and
disclose some important problems. In a particular case the problem is
completely solvable in the sense that spectrum of the Hamiltonian can be found.
The spectrum is continuous, has a band structure with infinite number of
lacunae, and is determined by the Mathieu equation: the Schr\"odinger equation
with a periodic cos-type potential
On planar polynomial vector fields with elementary first integrals
© 2019 Elsevier Inc. We show that under rather general conditions a polynomial differential system having an elementary first integral already must admit a Darboux first integral, and we explicitly characterize the vector fields in this class. We also investigate some exceptional cases, i.e. equations admitting an elementary first integral but not a Darboux first integral. In particular we provide a rather detailed discussion of exceptional elementary first integrals built from algebraic functions of prime degree
Darboux evaluations of algebraic Gauss hypergeometric functions
This paper presents explicit expressions for algebraic Gauss hypergeometric
functions. We consider solutions of hypergeometric equations with the
tetrahedral, octahedral and icosahedral monodromy groups. Conceptually, we
pull-back such a hypergeometric equation onto its Darboux curve so that the
pull-backed equation has a cyclic monodromy group. Minimal degree of the
pull-back coverings is 4, 6 or 12 (for the three monodromy groups,
respectively). In explicit terms, we replace the independent variable by a
rational function of degree 4, 6 or 12, and transform hypergeometric functions
to radical functions.Comment: The list of seed hypergeometric evaluations (in Section 2) reduced by
half; uniqueness claims explained; 34 pages; Kyushu Journal of Mathematics,
201
Algebraic Integrability of Foliations of the Plane
We give an algorithm to decide whether an algebraic plane foliation F has a
rational first integral and to compute it in the affirmative case. The
algorithm runs whenever we assume the polyhedrality of the cone of curves of
the surface obtained after blowing-up the set B_F of infinitely near points
needed to get the dicritical exceptional divisors of a minimal resolution of
the singularities of F. This condition can be detected in several ways, one of
them from the proximity relations in B_F and, as a particular case, it holds
when the cardinality of B_F is less than 9
О применении метода М.Н. Лагутинского к интегрированию дифференциальных уравнений 1-го порядка. Часть 1. Отыскание алгебраических интегралов
The method of M.N. Lagutinski (1871-1915) allows to find rational integrals and Darboux polynomials for given differential ring and thus can be used for integration of ordinary differential equations in symbolic form. A realization of Lagutinski method was made under free opensource mathematics software system Sage and will be presented in this article with application for symbolic integration of 1st order differential equations. In the first part of the article basic concepts of the Lagutinski method is briefly stated for polynomials rings. Then this method is applied to search of algebraic integrated curves for given ordinary differential equations of the form d + d withМетод М.Н. Лагутинского (1871-1915) позволяет искать рациональные интегралы и многочлены Дарбу заданного дифференциального кольца и поэтому может быть использован при интегрировании обыкновенных дифференциальных уравнений в символьном виде. В настоящей статье представлена реализация метода Лагутинского, выполненная в свободной системой компьютерной алгебры Sage, и дан обзор её возможностей по интегрированию дифференциальных уравнений 1-го порядка в символьном виде. В первой части статьи кратко изложены основные понятия метода Лагутинского для полиномиальных колец, затем этот метод приложен к отысканию алгебраических интегральных кривых дифференциальных уравнений вида d +
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