# 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} }

## 57 Citations

### Formal Algorithmic Elimination for PDEs

- MathematicsISSAC
- 2016

This tutorial discusses aspects of this correspondence between radical differential ideals and their analytic solution sets in differential algebra involving symbolic computation, and an introduction to the Thomas decomposition method is given.

### Singularities of algebraic differential equations

- Mathematics, Computer ScienceAdv. Appl. Math.
- 2021

### Strong Consistency and Thomas Decomposition of Finite Difference Approximations to Systems of Partial Differential Equations

- Mathematics, Computer ScienceArXiv
- 2020

An algorithmic approach that combines differential and difference algebra to analyze s(trong)-consistency of finite difference approximations of regular solution grids is suggested, which generalizes the definition given earlier for Cartesian grids.

### Applying Thomas decomposition and algebraic analysis to certain nonlinear PDE systems

- Mathematics
- 2013

This talk is about work in progress in collaboration with Thomas Cluzeau, Universite de Limoges, and Alban Quadrat, Inria Saclay. We report on first steps of a study of certain systems of nonlinear…

### Counting solutions of algebraic systems via triangular decomposition

- Mathematics, Computer Science
- 2014

The goal of this thesis is to analyze the solution sets of systems of polynomial equations and inequations over algebraically closed fields algorithmically, and defines a polynometric, which is used as a generalization of the cardinality of a finite set, to give a measure for the size of such a set.

### Algebraic and Geometric Analysis of Singularities of Implicit Differential Equations (Invited Talk)

- MathematicsCASC
- 2020

An algorithm for detecting all singularities of an algebraic differential equation over the complex numbers and how geometric methods allow us to determine the local solution behaviour in the neighbourhood of a singularity including the regularity of the solution is presented.

### Rational general solutions of systems of first-order algebraic partial differential equations

- MathematicsJ. Comput. Appl. Math.
- 2018

### Algebraic and Puiseux series solutions of systems of autonomous algebraic ODEs of dimension one in several variables

- Mathematics, Computer ScienceJ. Symb. Comput.
- 2022

### Thomas Decomposition and Nonlinear Control Systems

- Mathematics
- 2020

The paper gives an introduction to the Thomas decomposition method and shows how notions such as invertibility, observability and flat outputs can be studied.

### The MAPLE package TDDS for computing Thomas decompositions of systems of nonlinear PDEs

- MathematicsComput. Phys. Commun.
- 2019

## References

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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…

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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.

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This paper studies triangular differential systems arising from various decompositions of partial differential polynomial systems. In theoretical aspects, we emphasizeon translating differential…

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. 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…

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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…

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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…

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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.

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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.