# 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…

## 7 Citations

Degree bound for separating invariants of abelian groups

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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…

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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

Rational invariants of a group action. Construction and rewriting

- MathematicsJ. Symb. Comput.
- 2007

Calculating Generators for Invariant Fields of Linear Algebraic Groups

- MathematicsAAECC
- 1999

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

- Mathematics
- 2007

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

- Mathematics
- 2007

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…

Homomorphisms, localizations and a new algorithm to construct invariant rings of finite groups

- Mathematics
- 2007

Rational invariants of scalings from Hermite normal forms

- MathematicsISSAC
- 2012

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

- Mathematics
- 2006

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

- MathematicsFound. Comput. Math.
- 2013

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

- MathematicsApplicable Algebra in Engineering, Communication and Computing
- 1999

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

- MathematicsNagoya 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…