Computing quotients by connected solvable groups

Gregor Kemper

doi:10.1016/j.jsc.2020.07.014

Computing quotients by connected solvable groups

Gregor Kemper

Chair of Algorithmic Algebra

Research output: Contribution to journal › Article › peer-review

Abstract

Consider an action of a connected solvable group G on an affine variety X. This paper presents an algorithm that constructs a semi-invariant f∈K[X]=:R and computes the invariant ring (R_f)^G together with a presentation. The morphism X_f→Spec((R_f)^G) obtained from the algorithm is a universal geometric quotient. In fact, it is even better than that: a so-called excellent quotient. If R is a polynomial ring, the algorithm requires no Gröbner basis computations. If R is a complete intersection, then so is (R_f)^G.

Original language	English
Pages (from-to)	426-440
Number of pages	15
Journal	Journal of Symbolic Computation
Volume	109
DOIs	https://doi.org/10.1016/j.jsc.2020.07.014
State	Published - 1 Mar 2022

Keywords

Algorithmic invariant theory
Geometric invariant theory
Geometric quotient
Solvable groups

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10.1016/j.jsc.2020.07.014

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abstract = "Consider an action of a connected solvable group G on an affine variety X. This paper presents an algorithm that constructs a semi-invariant f∈K[X]=:R and computes the invariant ring (Rf)G together with a presentation. The morphism Xf→Spec((Rf)G) obtained from the algorithm is a universal geometric quotient. In fact, it is even better than that: a so-called excellent quotient. If R is a polynomial ring, the algorithm requires no Gr{\"o}bner basis computations. If R is a complete intersection, then so is (Rf)G.",

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N2 - Consider an action of a connected solvable group G on an affine variety X. This paper presents an algorithm that constructs a semi-invariant f∈K[X]=:R and computes the invariant ring (Rf)G together with a presentation. The morphism Xf→Spec((Rf)G) obtained from the algorithm is a universal geometric quotient. In fact, it is even better than that: a so-called excellent quotient. If R is a polynomial ring, the algorithm requires no Gröbner basis computations. If R is a complete intersection, then so is (Rf)G.

AB - Consider an action of a connected solvable group G on an affine variety X. This paper presents an algorithm that constructs a semi-invariant f∈K[X]=:R and computes the invariant ring (Rf)G together with a presentation. The morphism Xf→Spec((Rf)G) obtained from the algorithm is a universal geometric quotient. In fact, it is even better than that: a so-called excellent quotient. If R is a polynomial ring, the algorithm requires no Gröbner basis computations. If R is a complete intersection, then so is (Rf)G.

KW - Algorithmic invariant theory

KW - Geometric invariant theory

KW - Geometric quotient

KW - Solvable groups

UR - https://www.scopus.com/pages/publications/85087978093

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DO - 10.1016/j.jsc.2020.07.014

M3 - Article

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SN - 0747-7171

VL - 109

SP - 426

EP - 440

JO - Journal of Symbolic Computation

JF - Journal of Symbolic Computation

ER -

Computing quotients by connected solvable groups

Abstract

Keywords

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