Algorithmic Thomas decomposition of algebraic and differential systems

@article{Bchler2012AlgorithmicTD,
  title={Algorithmic Thomas decomposition of algebraic and differential systems},
  author={Thomas B{\"a}chler and Vladimir P. Gerdt and Markus Lange-Hegermann and Daniel Robertz},
  journal={J. Symb. Comput.},
  year={2012},
  volume={47},
  pages={1233-1266}
}

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References

SHOWING 1-10 OF 66 REFERENCES

Thomas Decomposition of Algebraic and Differential Systems

In this paper we consider disjoint decomposition of algebraic and non-linear partial differential systems of equations and inequations into so-called simple subsystems. We exploit THOMAS

On Decomposition of Algebraic PDE Systems into Simple Subsystems

  • V. Gerdt
  • Mathematics, Computer Science
  • 2008
An algorithmization of the Thomas method for splitting a system of partial differential equations and (possibly) inequalities into triangular subsystems whose Thomas called simple is presented.

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

Completion of Linear Differential Systems to Involution

This paper considers posing of an initial value problem for a linear differential system providing uniqueness of its solution and Lie symmetry analysis of nonlinear differential equations to determine the structure of arbitrariness in general solution of linear systems and thereby to find the size of symmetry group.

Completion of Linear Differential Systems to Involution

. In this paper we generalize the involutive methods and algorithms devised for polynomial ideals to differential ones generated by a finite set of linear differential polynomials in the differential

Involution - The Formal Theory of Differential Equations and its Applications in Computer Algebra

  • W. Seiler
  • Mathematics
    Algorithms and computation in mathematics
  • 2010
The book provides a self-contained account of the formal theory of general, i.e. also under- and overdetermined, systems of differential equations which in its central notion of involution combines

Gauss-Bruhat decomposition as an example of Thomas decomposition

Abstract.Both the Gauss-Bruhat decomposition and the LU-decomposition of the general linear group over a field are examples of a Thomas decomposition of systems of polynomial equations and

Computing representations for radicals of finitely generated differential ideals

This paper deals with systems of polynomial differential equations, ordinary or with partial derivatives. The embedding theory is the differential algebra of Ritt and Kolchin. We describe an

Unmixed-dimensional Decomposition of a Finitely Generated Perfect Differential Ideal

Some of the main problems in polynomial ideal theory can be solved by means of this decomposition: it is shown how the radical membership can be decided, a characteristic set of a prime differential ideal can be selected, and the differential dimension with a parametricSet of a differential Ideal can be read.

Notes on Triangular Sets and Triangulation-Decomposition Algorithms II: Differential Systems

  • E. Hubert
  • Mathematics, Computer Science
    SNSC
  • 2001
This is the second in a series of two tutorial articles devoted to triangulation-decomposition algorithms and uses results presented in the first article on polynomial systems but can be read independently.
...