×
Hubert, Evelyne; Kogan, Irina A.

Rational invariants of a group action. Construction and rewriting. (English) Zbl 1121.13010

J. Symb. Comput. 42, No. 1-2, 203-217 (2007).
Summary: Geometric constructions applied to a rational action of an algebraic group lead to a new algorithm for computing rational invariants. A finite generating set of invariants appears as the coefficients of a reduced Gröbner basis. The algorithm comes in two variants. In the first construction the ideal of the graph of the action is considered. In the second one the ideal of a cross-section is added to the ideal of the graph. Zero-dimensionality of the resulting ideal brings a computational advantage. In both cases, reduction with respect to the computed Gröbner basis allows us to express any rational invariant in terms of the generators.

Cited in 25 Documents

MSC:

13A50 Actions of groups on commutative rings; invariant theory
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
68W30 Symbolic computation and algebraic computation

Keywords:

rational invariants; algebraic group actions; cross-section; Gröbner basis; differential invariants; moving frame

Software:

SINGULAR
Cite Review PDF
Full Text: DOI

References:

[1] Becker, T.; Weispfenning, V., Gröbner Bases—A Computational Approach to Commutative Algebra (1993), Springer-Verlag: Springer-Verlag New York · Zbl 0772.13010
[2] Cox, D.; Little, J.; O��Shea, D., Ideals, Varieties, and Algorithms (1992), Springer-Verlag · Zbl 0756.13017
[3] Derksen, H., Computation of invariants for reductive groups, Adv. Math., 141, 2, 366-384 (1999) · Zbl 0927.13007
[4] Derksen, H.; Kemper, G., Computational invariant theory, (Invariant Theory and Algebraic Transformation Groups, I. Invariant Theory and Algebraic Transformation Groups, I, Encyclopaedia of Mathematical Sciences, vol. 130 (2002), Springer-Verlag: Springer-Verlag Berlin) · Zbl 1011.13003
[5] Eisenbud, D., Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics (1994), Springer-Verlag: Springer-Verlag New York · Zbl 0801.19001
[6] Fels, M.; Olver, P. J., Moving coframes. II. Regularization and theoretical foundations, Acta Appl. Math., 55, 2, 127-208 (1999) · Zbl 0937.53013
[7] Giusti, M.; Heintz, J., La détermination des points isolés et de la dimension d’une variété algébrique peut se faire en temps polynomial, (Computational algebraic geometry and commutative algebra (Cortona, 1991). Computational algebraic geometry and commutative algebra (Cortona, 1991), Sympos. Math., vol. XXXIV (1993), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 216-256 · Zbl 0829.14029
[8] Giusti, M.; Heintz, J.; Sabia, J., On the efficiency of effective Nullstellensätze, Comput. Complexity, 3, 1, 56-95 (1993) · Zbl 0824.68051
[9] Greuel, G.-M.; Pfister, G., A Singular Introduction to Commutative Algebra (2002), Springer-Verlag: Springer-Verlag Berlin · Zbl 1023.13001
[10] Hubert, E., Kogan, I.A., 2006. Smooth and algebraic invariants of a group action. Local and global construction (submitted for publication); Hubert, E., Kogan, I.A., 2006. Smooth and algebraic invariants of a group action. Local and global construction (submitted for publication)
[11] Krick, T.; Pardo, L. M.; Sombra, M., Sharp estimates for the arithmetic Nullstellensatz, Duke Math. J., 109, 3, 521-598 (2001) · Zbl 1010.11035
[12] Müller-Quade, J.; Beth, T., Calculating generators for invariant fields of linear algebraic groups, (Applied Algebra, Algebraic Algorithms and Error-correcting Codes (Honolulu, HI, 1999). Applied Algebra, Algebraic Algorithms and Error-correcting Codes (Honolulu, HI, 1999), Lecture Notes in Computer Science, vol. 1719 (1999), Springer-Verlag: Springer-Verlag Berlin), 392-403 · Zbl 0959.14029
[13] Müller-Quade, J.; Steinwandt, R., Basic algorithms for rational function fields, J. Symbolic Comput., 27, 2, 143-170 (1999) · Zbl 0935.11046
[14] Rosenlicht, M., Some basic theorems on algebraic groups, Amer. J. Math., 78, 401-443 (1956) · Zbl 0073.37601
[15] Shafarevich, I. R., Basic Algebraic Geometry. 1 (1994), Springer-Verlag: Springer-Verlag Berlin, (Varieties in Projective Space, Translated from the 1988 Russian edition and with notes by Miles Reid) · Zbl 0675.14001
[16] van der Waerden, Modern Algebra (1971), Springer-Verlag: Springer-Verlag New York · Zbl 0221.12001
[17] Vinberg, E. B.; Popov, V. L., Invariant theory, (Parshin, A.; Shafarevich, I. R., Algebraic Geometry IV. Algebraic Geometry IV, Encyclopaedia of Mathematical Sciences, vol. 55 (1994), Springer-Verlag: Springer-Verlag Berlin), 122-278 · Zbl 0783.14028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.