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

Full-text PDF Free Access

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]**William W. Adams and Philippe Loustaunau,*An introduction to Gröbner bases*, Graduate Studies in Mathematics, vol. 3, American Mathematical Society, Providence, RI, 1994. MR**1287608****[CHV]**Aldo Conca, Jürgen Herzog, and Giuseppe Valla,*Sagbi bases with applications to blow-up algebras*, J. Reine Angew. Math.**474**(1996), 113–138. MR**1390693**, 10.1515/crll.1996.474.113**[E]**David Eisenbud,*Commutative algebra*, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR**1322960****[KM]**Deepak Kapur and Klaus Madlener,*A completion procedure for computing a canonical basis for a 𝑘-subalgebra*, Computers and mathematics (Cambridge, MA, 1989) Springer, New York, 1989, pp. 1–11. MR**1005954****[L58]**Serge Lang,*Introduction to algebraic geometry*, Interscience Publishers, Inc., New York-London, 1958. MR**0100591****[L93]**Serge Lang,*Algebra*, 2nd ed., Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1984. MR**783636****[O]**François Ollivier,*Canonical bases: relations with standard bases, finiteness conditions and application to tame automorphisms*, Effective methods in algebraic geometry (Castiglioncello, 1990) Progr. Math., vol. 94, Birkhäuser Boston, Boston, MA, 1991, pp. 379–400. MR**1106435****[RS]**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**, 10.1007/BFb0085537**[S]**Bernd Sturmfels,*Gröbner bases and convex polytopes*, University Lecture Series, vol. 8, American Mathematical Society, Providence, RI, 1996. MR**1363949****[T]**Carlo Traverso,*Constructive methods and automatic computation in commutative algebra*, Boll. Un. Mat. Ital. A (7)**2**(1988), no. 2, 145–167 (Italian). MR**948289**

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:
http://dx.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