Computation of invariants of finite abelian groups

@article{Hubert2016ComputationOI,
  title={Computation of invariants of finite abelian groups},
  author={Evelyne Hubert and George Labahn},
  journal={Math. Comput.},
  year={2016},
  volume={85},
  pages={3029-3050}
}
We investigate the computation and applications of rational invariants of the linear action of a finite abelian group in the non-modular case. By diagonalization, the group action 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 how to compute a minimal generating set that consists of polynomial invariants. As an… 
Degree bound for separating invariants of abelian groups
It is proved that the universal degree bound for separating polynomial invariants of a finite abelian group (in non-modular characteristic) is strictly smaller than the universal degree bound for
Minimal degrees of invariants of (super)groups – a connection to cryptology
We investigate questions related to the minimal degree of invariants of finitely generated diagonalizable groups. These questions were raised in connection to security of a public key cryptosystem
Rational Invariants of Even Ternary Forms Under the Orthogonal Group
TLDR
A generating set of rational invariants of minimal cardinality for the action of the orthogonal group O3 on the space of the finite subgroup B3 of signed permutations and efficient algorithms for their evaluation and rewriting are determined.
Multivariate interpolation: Preserving and exploiting symmetry
Symmetry Preserving Interpolation
TLDR
The article shows how to exactly preserve symmetry in multivariate interpolation while exploiting it to alleviate the computational cost and revisit minimal degree and least interpolation with symmetry adapted bases, rather than monomial bases.
Ideal Interpolation, H-bases and symmetry
TLDR
For an ideal interpolation problem with symmetry, this work addresses the simultaneous computation of a symmetry adapted basis of the least interpolation space and the symmetry adapted H-basis of the ideal.
A fast algorithm for computing the Smith normal form with multipliers for a nonsingular integer matrix
A Las Vegas randomized algorithm is given to compute the Smith multipliers for a nonsingular integer matrix A, that is, unimodular matrices U and V such that AV = US , with S the Smith normal form of

References

SHOWING 1-10 OF 57 REFERENCES
Calculating Generators for Invariant Fields of Linear Algebraic Groups
TLDR
An algorithm to calculate generators for the invariant field k(x)G of a linear algebraic group G from the defining equations of G by exploiting a field-ideal-correspondence which has been applied to the decomposition of rational mappings before.
The Computation of Invariant Fields and a Constructive Version of a Theorem by Rosenlicht
Let G be an algebraic group acting on an irreducible variety X. We present an algorithm for computing the invariant field k(X)G. Moreover, we give a constructive version of a theorem of Rosenlicht,
INVARIANT FIELDS AND LOCALIZED INVARIANT RINGS OF p-GROUPS
It is well known that for a p-group, the invariant field is purely transcendental (T. Miyata, Invariants of certain groups I, Nagoya Math. J. 41 (1971), 69?73). In this note, we show that a minimal
Rational invariants of scalings from Hermite normal forms
TLDR
A complete solution to the scaling symmetry reduction of a polynomial system is presented and their unimodular multipliers are presented.
The Computation of Invariant Fields and a new Proof of a Theorem by Rosenlicht
Let G be an algebraic group acting on an irreducible variety X. We present an algorithm for computing the invariant field k(X). This algorithm leads to a new, constructive proof of a theorem of
Scaling Invariants and Symmetry Reduction of Dynamical Systems
Scalings form a class of group actions that have theoretical and practical importance. A scaling is accurately described by a matrix of integers. Tools from linear algebra over the integers are
Gröbner Bases for Ideals in Laurent Polynomial Rings and their Application to Systems of Difference Equations
TLDR
A basic theory of Gröbner bases for ideals in the algebra of Laurent polynomials (and, more generally, in its monomial subalgebras) is developed and a method to compute the intersection of an ideal with the subalgebra of all polynOMials is presented.
Invariants of Certain Groups I
  • T. Miyata
  • Mathematics
    Nagoya Mathematical Journal
  • 1971
Let G be a group and let k be a field. A K-representation ρ of G is a homomorphism of G into the group of non-singular linear transformations of some finite-dimensional vector space V over k. Let K
...
...