Lectures on Differential Galois Theory

Magid, Andy R.

doi:10.1090/ulect/007

Cited by 150 publications

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“…The K-algebra R is called the full universal solution algebra (see [9], Definition 2.12). By construction, it contains n solutions of equation (2.1) linearly independent over constants.…”

Section: Resultsmentioning

confidence: 99%

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Picard-Vessiot extensions for real fields

Sowa

2010

Proc. Amer. Math. Soc.

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“…The K-algebra R is called the full universal solution algebra (see [9], Definition 2.12). By construction, it contains n solutions of equation (2.1) linearly independent over constants.…”

Section: Resultsmentioning

confidence: 99%

“…It was made more accessible by the book of I. Kaplansky [7]. We also refer the reader to [4], [9] and [11] for the results of Picard-Vessiot theory used in this paper.…”

Section: L(y ) := Ymentioning

confidence: 99%

Picard-Vessiot extensions for real fields

Sowa

2010

Proc. Amer. Math. Soc.

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“…Let m ⊂ A be any maximal differential ideal. Since every maximal differential ideal over a field of zero characteristic is prime (see [17,Proposition 1.19]), A/m contains no zero divisors. Replacing A with A/m if necessary, we may assume that A is integral.…”

Section: Power Series Versionmentioning

confidence: 99%

A Differential Analog of the Noether Normalization Lemma

Pogudin¹

2016

Int Math Res Notices

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In this paper, we prove the following differential analog of the Noether normalization lemma: for every d-dimensional differential algebraic variety over differentially closed field of zero characteristic there exists a surjective map onto the d-dimensional affine space. Equivalently, for every integral differential algebra A over differential field of zero characteristic there exist differentially independent b1, . . . , b d such that A is differentially algebraic over subalgebra B differentially generated by b1, . . . , b d , and whenever p ⊂ B is a prime differential ideal, there exists a prime differential ideal q ⊂ A such that p = B ∩ q.We also prove the analogous theorem for differential algebraic varieties over the ring of formal power series over an algebraically closed differential field and present some applications to differential equations. MSC2010: 12H05.

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“…For a precise definition of the differential Galois group and general facts from differential algebra see [16,41,5,35,47]. We can consider G as a subgroup of GL(s, C) which acts on fundamental solutions of (2.2) and does not change polynomial relations among them.…”

Section: Theorymentioning

confidence: 99%

Differential Galois approach to the non-integrability of the heavy top problem

Maciejewski¹,

Przybylska²

2005

Annales De La Faculté Des Sciences De Toulouse Mathématiques

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We study integrability of the Euler-Poisson equations describing the motion of a rigid body with one fixed point in a constant gravity field. Using the Morales-Ramis theory and tools of differential algebra we prove that a symmetric heavy top is integrable only in the classical cases of Euler, Lagrange, and Kovalevskaya and is partially integrable only in the Goryachev-Chaplygin case. Our proof is alternative to that given by Ziglin (

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Lectures on Differential Galois Theory

Cited by 150 publications

References 0 publications

Picard-Vessiot extensions for real fields

Picard-Vessiot extensions for real fields

A Differential Analog of the Noether Normalization Lemma

Differential Galois approach to the non-integrability of the heavy top problem

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