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

DOI:
https://doi.org/10.1090/S0002-9939-98-04306-8

MathSciNet review:
1443407

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For an ideal or -subalgebra of , consider subfields , where is generated - as ideal or -subalgebra - by polynomials in . It is a standard result for ideals that there is a smallest such . We give an algorithm to find it. We also prove that there is a smallest such for -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.

**[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 -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**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
13P10,
12Y05

Retrieve articles in all journals with MSC (1991): 13P10, 12Y05

Additional Information

**Lorenzo Robbiano**

Affiliation:
Department of Mathematics, University of Genoa, Italy

Email:
robbiano@dima.unige.it

**Moss Sweedler**

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

Email:
moss_sweedler@cornell.edu

DOI:
https://doi.org/10.1090/S0002-9939-98-04306-8

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