Rational Invariants of a Group Action

Hubert, Evelyne

doi:10.5802/ccirm.19

“…The class of rational invariants can furthermore address a wider class of nonlinear actions, such as those central in differential geometry 2 and algebraically characterize classical differential invariants [13,10]. General algorithms to compute rational invariants of (rational) action of algebraic groups [11,12,16,17,19] rely on Gröbner bases computations. It is remarkable how much simpler and more effective the present approach is for use with abelian groups.…”

Section: Introductionmentioning

confidence: 99%

Computation of invariants of finite abelian groups

Hubert

¹

,

Labahn

²

2016

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We investigate the field of rational invariants of the linear action of a finite abelian group in the non modular case. By diagonalization, the group is accurately described by an integer matrix of exponents. We make use of linear algebra to compute a minimal generating set of invariants and the substitution to rewrite any invariant in terms of this generating set. We show that the generating set can be chosen to consist of polynomial invariants. As an application, we provide a symmetry reduction scheme for polynomial systems the solution set of which are invariant by the group action. We furthermore provide an algorithm to find such symmetries given a polynomial system.

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Rational invariants of finite abelian groups

Labahn

¹

2015

ACM Commun. Comput. Algebra

0

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We investigate the field of rational invariants of the linear action of a finite abelian group in the non modular case. By diagonalization, the group is accurately described by an integer matrix of exponents. We make use of linear algebra to compute a minimal generating set of invariants and the substitution to rewrite any invariant in terms of this generating set. We show that the generating set can be chosen to consist of polynomial invariants. As an application, we provide a symmetry reduction scheme for polynomial systems the solution set of which are invariant by the group action. We furthermore provide an algorithm to find such symmetries given a polynomial system.

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Dimensional Analysis Using Toric Ideals: Primitive Invariants

Atherton

¹

,

Bates

²

,

Wynn

³

2014

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Classical dimensional analysis in its original form starts by expressing the units for derived quantities, such as force, in terms of power products of basic units etc. This suggests the use of toric ideal theory from algebraic geometry. Within this the Graver basis provides a unique primitive basis in a well-defined sense, which typically has more terms than the standard Buckingham approach. Some textbook examples are revisited and the full set of primitive invariants found. First, a worked example based on convection is introduced to recall the Buckingham method, but using computer algebra to obtain an integer matrix from the initial integer matrix holding the exponents for the derived quantities. The matrix defines the dimensionless variables. But, rather than this integer linear algebra approach it is shown how, by staying with the power product representation, the full set of invariants (dimensionless groups) is obtained directly from the toric ideal defined by . One candidate for the set of invariants is a simple basis of the toric ideal. This, although larger than the rank of , is typically not unique. However, the alternative Graver basis is unique and defines a maximal set of invariants, which are primitive in a simple sense. In addition to the running example four examples are taken from: a windmill, convection, electrodynamics and the hydrogen atom. The method reveals some named invariants. A selection of computer algebra packages is used to show the considerable ease with which both a simple basis and a Graver basis can be found.

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Rational Invariants of a Group Action

Cited by 3 publications

References 17 publications

Computation of invariants of finite abelian groups

Computation of invariants of finite abelian groups

Rational invariants of finite abelian groups

Dimensional Analysis Using Toric Ideals: Primitive Invariants

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