Using Algebraic Geometry

Cox, David A.; Little, John B.; O’Shea, Donal

doi:10.1007/978-1-4757-6911-1

Cited by 1,110 publications

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“…If k is a differential field and Σ is a finite system of differential equations and inequations over k such that Σ has a solution in some l ⊃ k, then Σ has a solution in any differentially closed K ⊃ k. Remark 2. Let us remind an important property of the resultant of univariate polynomials which we will use several times throughout the paper (for details see [9…”

Section: ])mentioning

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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Section: ])mentioning

confidence: 99%

A Differential Analog of the Noether Normalization Lemma

Pogudin¹

2016

Int Math Res Notices

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“…According to algebraic geometry [5], the resultant of (9) is a one-dimensional polynomial φ of λ. We call φ(λ) the M-characteristic polynomial of A .…”

Section: M-eigenvalues Of a Fourth-order Partially Symmetric Tensormentioning

confidence: 99%

“…Proof According to the resultant theory [5], (9) has a nonzero complex solution (x, y) if and only if λ is a root of the resultant. The conclusion follows.…”

Section: Theorem 2 An M-eigenvalue Of a Is Always A Real Root Of The mentioning

confidence: 99%

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Conditions for strong ellipticity and M-eigenvalues

Dai

Han

2009

Front. Math. China

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The strong ellipticity condition plays an important role in nonlinear elasticity and in materials. In this paper, we define M-eigenvalues for an elasticity tensor. The strong ellipticity condition holds if and only if the smallest M-eigenvalue of the elasticity tensor is positive. If the strong ellipticity condition holds, then the elasticity tensor is rank-one positive definite. The elasticity tensor is rank-one positive definite if and only if the smallest Z-eigenvalue of the elasticity tensor is positive. A Z-eigenvalue of the elasticity tensor is an M-eigenvalue but not vice versa. If the elasticity tensor is second-order positive definite, then the strong ellipticity condition holds. The converse conclusion is not right. Computational methods for finding M-eigenvalues are presented.

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