## Algebraic Theory of Differential Equations |

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### Contents

Weil | 83 |

Factorization of Linear Systems Sergey P Tsarev | 111 |

Introduction to Dmodules Anton Leykin | 132 |

Symbolic representation and classiﬁcation of integrable | 156 |

Searching for integrable PDEs Jarmo Hietarinta | 217 |

Around Diﬀerential Galois Theory Anand Pillay | 232 |

### Common terms and phrases

algebraically closed algorithm analytic approximate symmetries Bernstein-Sato polynomial closed ﬁeld coeﬃcients corresponding D-modules deﬁned deﬁnition derivation diﬀerent differential diﬀerential algebraic diﬀerential ﬁeld diﬀerential Galois group diﬀerential Galois theory diﬀerential modules diﬀerential operators diﬀerential polynomials dimensional elements equa equation L(y evolutionary equations example existence factorization ﬁnd ﬁnite ﬁrst formal recursion operator functions G-torsor Galois theory Gršobner basis group G holonomic homogeneous ideal implies integrable equations integrable systems irreducible isomorphism Kolchin Laplace Lie algebra linear algebraic group linear diﬀerential equations liouvillian liouvillian solution LODO Math Mathematics model theory monodromy monomial order equations partial diﬀerential equations Picard-Vessiot extension Picard-Vessiot theory PPV-group problem proof Proposition rational solutions regular singular points ring satisﬁes scalar second order soliton solution space solvable solve subgroup subspace Symbolic Computation symbolic representation symmetric power Theorem tion torsor transformations variables vector Weyl algebra Zariski Zariski dense