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Ideal and subalgebra coefficients

Authors: Lorenzo Robbiano and Moss Sweedler
Journal: Proc. Amer. Math. Soc. 126 (1998), 2213-2219
MSC (1991): Primary 13P10; Secondary 12Y05
MathSciNet review: 1443407
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Abstract: For an ideal or $K$-subalgebra $E$ of $K[X_1,\dots,X_n]$, consider subfields $k\subset K$, where $E$ is generated - as ideal or $K$-subalgebra - by polynomials in $k[X_1,\dots,X_n]$. It is a standard result for ideals that there is a smallest such $k$. We give an algorithm to find it. We also prove that there is a smallest such $k$ for $K$-subalgebras. The ideal results use reduced Gröbner bases. For the subalgebra results we develop and then use subduced SAGBI (bases), the analog to reduced Gröbner bases.

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  • [AL] W. W. Adams, P. Loustaunau, An Introduction to Gröbner Bases, Graduate Studies in Mathematics, American Mathematical Society, Providence RI, 1994. MR 95g:13025
  • [CHV] A. Conca, J. Herzog, G. Valla, Sagbi Bases with Applications to Blow-up Algebras, J. Reine Angew. Math. 474(1996) 113-138. MR 97h:13023
  • [E] D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, Springer-Verlag, New York NY, 1995. MR 97a:13001
  • [KM] D. Kapur, K. Madlener, A completion procedure for computing a canonical basis for a $k$-subalgebra, in Computers and Mathematics (Cambridge, MA, 1989), (E. Kaltofen and S. M. Watt, Eds.) pp. 1-11, Springer-Verlag, New York NY, 1989. MR 90g:13001
  • [L58] S. Lang, Introduction to Algebraic Geometry, Tracts in Pure and Applied Mathematics, Interscience, New York NY, 1958. MR 20:7021
  • [L93] S. Lang, Algebra, Addison-Wesley, Reading MA, 1984. MR 86j:00003
  • [O] F. Ollivier, Canonical bases: relations with standard bases, finiteness conditions and application to tame automorphisms, in Effective Methods in Algebraic Geometry (Castiglioncello, 1990), (T. Mora and C. Traverso, Eds.), pp. 379-400, Progress in Mathematics 94, Birkhäuser Boston, Boston, MA, 1991. MR 92c:13026
  • [RS] L. Robbiano, M. Sweedler, Subalgebra bases, in Commutative Algebra (Salvador, 1988), (W. Bruns and A. Simis, Eds.), pp. 61-87, Lecture Notes in Mathematics 1430, Springer-Verlag, 1990. MR 91f:13027
  • [S] B. Sturmfels, Gröbner bases and convex polytopes, University Lecture Series 8, American Mathematical Society, Providence RI, 1996. MR 97b:13034
  • [T] C. Traverso, Metodi costruttivi e calcolo automatico in algebra commutativa, Boll. Un. Mat. Ital. 2-A(1988) 145-167. MR 89k:13001

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Additional Information

Lorenzo Robbiano
Affiliation: Department of Mathematics, University of Genoa, Italy

Moss Sweedler
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853

Keywords: Ideal, subalgebra, field of definition, reduced Gr\"obner basis, subduced SAGBI (basis)
Received by editor(s): August 29, 1996
Received by editor(s) in revised form: January 16, 1997
Additional Notes: The first author was partially supported by the Consiglio Nazionale delle Ricerche (CNR)
The second author was partially supported by the United States Army Research Office through the Army Center of Excellence for Symbolic Methods in Algorithmic Mathematics (ACSyAM), Mathematical Sciences Institute of Cornell University, Contract DAAL03-91-C-0027, and by the NSA
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 1998 American Mathematical Society

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