Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



SAGBI bases for rings of invariant Laurent polynomials

Authors: Alexander Duncan and Zinovy Reichstein
Journal: Proc. Amer. Math. Soc. 137 (2009), 835-844
MSC (2000): Primary 13A50, 13P99
Published electronically: September 15, 2008
MathSciNet review: 2457421
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ k$ be a field, let $ L_n = k[x_1^{\pm 1}, \dots, x_n^{\pm 1}]$ be the Laurent polynomial ring in $ n$ variables and let $ G$ be a finite group of $ k$-algebra automorphisms of $ L_n$. We give a necessary and sufficient condition for the ring of invariants $ L_n^G$ to have a SAGBI basis. We show that if this condition is satisfied, then $ L_n^G$ has a SAGBI basis relative to any choice of coordinates in $ L_n$ and any term order.

References [Enhancements On Off] (What's this?)

  • 1. Winfried Bruns and Joseph Gubeladze, Polytopes, rings and k-theory,
  • 2. David Cox, John Little, and Donal O'Shea, Ideals, varieties, and algorithms, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992, An introduction to computational algebraic geometry and commutative algebra. MR 1189133 (93j:13031)
  • 3. Leonard Eugene Dickson, Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors, American Journal of Mathematics 35 (1913), no. 4, 413-422. MR 1506194
  • 4. Manfred Göbel, The optimal lower bound for generators of invariant rings without finite SAGBI bases with respect to any admissible order, Discrete Math. Theor. Comput. Sci. 3 (1999), no. 2, 65-70 (electronic). MR 1695195 (2001b:13008)
  • 5. Manfred Göbel, Finite SAGBI bases for polynomial invariants of conjugates of alternating groups, Math. Comp. 71 (2002), no. 238, 761-765 (electronic). MR 1885626 (2002m:13009)
  • 6. Shigeru Kuroda, The infiniteness of the SAGBI bases for certain invariant rings, Osaka J. Math. 39 (2002), no. 3, 665-680. MR 1932287 (2003k:13033)
  • 7. Martin Lorenz, Multiplicative invariant theory, Encyclopaedia of Mathematical Sciences, vol. 135, Springer-Verlag, Berlin, 2005, Invariant Theory and Algebraic Transformation Groups, VI. MR 2131760 (2005m:13012)
  • 8. Zinovy Reichstein, SAGBI bases in rings of multiplicative invariants, Comment. Math. Helv. 78 (2003), no. 1, 185-202. MR 1966757 (2004c:13005)
  • 9. Lorenzo Robbiano and Moss Sweedler, Subalgebra bases, Commutative algebra (Salvador, 1988), Lecture Notes in Math., vol. 1430, Springer, Berlin, 1990, pp. 61-87. MR 1068324 (91f:13027)
  • 10. Bernd Sturmfels, Gröbner bases and convex polytopes, University Lecture Series, vol. 8, American Mathematical Society, Providence, RI, 1996. MR 1363949 (97b:13034)
  • 11. Mohammed Tesemma, On multiplicative invariants of finite reflection groups, Comm. Algebra 35 (2007), no. 7, 2258-2274. MR 2331844
  • 12. N. M. Thiéry and S. Thomassé, Convex cones and SAGBI bases of permutation invariants, Invariant theory in all characteristics, CRM Proc. Lecture Notes, vol. 35, Amer. Math. Soc., Providence, RI, 2004, pp. 259-263. MR 2066473 (2005e:13006)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 13A50, 13P99

Retrieve articles in all journals with MSC (2000): 13A50, 13P99

Additional Information

Alexander Duncan
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada

Zinovy Reichstein
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada

Keywords: SAGBI basis, subduction algorithm, G\"obel's conjecture, group action, algebra of invariants, reflection group, abelian semigroup
Received by editor(s): February 6, 2008
Received by editor(s) in revised form: February 28, 2008
Published electronically: September 15, 2008
Additional Notes: The first author was partially supported by an NSERC Canada Graduate Scholarship.
The second author was partially supported by NSERC Discovery and Accelerator Supplement grants
Communicated by: Martin Lorenz
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society