A Remark on Determinantal Loci

Mount, Kenneth R.

doi:10.1112/jlms/s1-42.1.595

Cited by 17 publications

(6 citation statements)

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“…Proofs of Theorem 4.7 can be found in Northcott [34, Proposition 2, p. 1971 for the case in question and Mount [30] for the more general case of arbitrary sized minors of a rectangular matrix.' Theorems 4.6 and 4.7 then imply LEMMA 4.8.…”

Section: A Characterization Of Null Lagrangiansmentioning

confidence: 99%

Null Lagrangians, weak continuity, and variational problems of arbitrary order

Ball¹,

Currie²,

Olver

1981

Journal of Functional Analysis

293

247

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Section: A Characterization Of Null Lagrangiansmentioning

confidence: 99%

Null Lagrangians, weak continuity, and variational problems of arbitrary order

Ball¹,

Currie²,

Olver

1981

Journal of Functional Analysis

293

247

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“…(It is significant that, for polynomial functions, this transform is essentially equivalent to the powerful symbolic method of Aronhold in classical invariant theory.) The problem of classifying homogeneous null Lagrangians transforms into the problem of whether the determinantal ideal generated by the maximal minors of a matrix of independent variables is prime, a result proved by Northcott, [17], and Mount, [16], in the 1960's. The characterization of n-th order divergences as hyperjacobians rests on the deeper fact that the symbolic powers of this same determinantal ideal are the same as the actual powers of it.…”

Section: (Uyv XX + 2u X V Xy ) + D 3 Y(u X V Xx )mentioning

confidence: 99%

Hyperjacobians, determinantal ideals and weak solutions to variational problems

Olver¹

1983

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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SynopsisThe problem of classifying homogeneous null Lagrangians satisfying an nth order divergence identity is completely solved. All such differential polynomials are affine combinations of higher order Jacobian determinants, called hyperjacobians, which can be expressed as higher dimensional determinants of higher order Jacobian matrices. Special cases, called transvectants, are of importance in classical invariant theory. Transform techniques reduce this question to the characterization of the symbolic powers of certain determinantal ideals. Applications to the proof of existence of minimizers of certain quasi-convex variational problems with weakened growth conditions are discussed.

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“…According to a theorem of Northcott, [13], and Mount, [12], *A* is a prime ideal, and so by the Hilbert Nullstellensatz…”

Section: Q(^lzl> •••> Aj^r) -(A»i* 'Aj) • Q (Z^ Zj)mentioning

confidence: 99%