Rational Solutions of Riccati-like Partial Differential Equations

@article{Li2001RationalSO,
  title={Rational Solutions of Riccati-like Partial Differential Equations},
  author={Ziming Li and Fritz Schwarz},
  journal={J. Symb. Comput.},
  year={2001},
  volume={31},
  pages={691-716}
}
When factoring linear partial differential systems with a finite-dimensional solution space or analysing symmetries of nonlinear ODEs, we need to look for rational solutions of certain nonlinear PDEs. The nonlinear PDEs are called Riccati-like because they arise in a similar way as Riccati ODEs. In this paper we describe the structure of rational solutions of a Riccati-like system, and an algorithm for computing them. The algorithm is also applicable to finding all rational solutions of Lie?s… 
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References

SHOWING 1-10 OF 38 REFERENCES
Linear ordinary differential equations: breaking through the order 2 barrier
TLDR
This work presents an efficient algorithm for this subproblem, which has been implemented in the AXIOM computer algebra system for equations of arbitrary order over arbitrary fields of characteristic O, which never needs to compute with the individual complex singularities of the equation, and algebraic numbers are added only when they appear in the potential solutions.
Liouvillian Solutions of Linear Differential Equations with Liouvillian Coefficients
COHERENT, REGULAR AND SIMPLE SYSTEMS IN ZERO DECOMPOSITIONS OF PARTIAL DIFFERENTIAL SYSTEMS
This paper studies triangular differential systems arising from various decompositions of partial differential polynomial systems. In theoretical aspects, we emphasizeon translating differential
Specializations in differential algebra
1. Objectives and summary. Much of elementary differential algebra can be regarded as a generalization of the algebraic geometry of polynomial rings over a field to an analogous theory for rings of
Polynomial-Time Reductions from Multivariate to Bi- and Univariate Integral Polynomial Factorization
TLDR
An algorithm is presented which reduces the problem of finding the irreducible factors of f in polynomial-time in the total degree of f and the coefficient lengths of f to factoring a univariate integral polynomials, which implies the following theorem.
Algebraic factoring and rational function integration
TLDR
A new, simple, and efficient algorithm for factoring polynomials in several variables over an algebraic number field is presented and a constructive procedure is given for finding the least degree extension field in which the integral can be expressed.
ON THE FOUNDATION OF ALGEBRAIC DIFFERENTIAL GEOMETRY
An algebraic differential variety is defined as the zero-set of a differentialpolynomial set,and algebraic differential geometry is devoted to the study of such varieties.We give various
Chebyshev and Legendre Spectral Methods in Algebraic Manipulation Languages
  • J. Boyd
  • Computer Science
    J. Symb. Comput.
  • 1993
TLDR
This tutorial shows how to combine Galerkin and collocation methods with algebraic manipulation languages in REDUCE and Maple, and offers many guidelines and suggestions.
Factoring Rational Polynomials Over the Complex Numbers
TLDR
After reducing to the two-variable, square-free case, the classical algebraic geometry fact that the absolute irreducible factors of P(z_1 ,z_2 ) = 0) correspond to the connected components of the real surface (or complex curve) minus its singular points is applied.
...
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