Gröbner Bases and Normal Forms in a Subring of the Power Series Ring on Countably Many Variables
✍ Scribed by J. Snellman
- Publisher
- Elsevier Science
- Year
- 1998
- Tongue
- English
- Weight
- 511 KB
- Volume
- 25
- Category
- Article
- ISSN
- 0747-7171
No coin nor oath required. For personal study only.
✦ Synopsis
If K is a field, let the ring R consist of finite sums of homogeneous elements in
Then, R contains M, the free semi-group on the countable set of variables {x 1 , x 2 , x 3 , . . .}. In this paper, we generalize the notion of admissible order from finitely generated sub-monoids of M to M itself; assume that > is such an admissible order on M. We show that we can define leading power products, with respect to >, of elements in R , and thus the initial ideal gr(I) of an arbitrary ideal I ⊂ R . If I is what we call a locally finitely generated ideal, then we show that gr(I) is also locally finitely generated; this implies that I has a finite truncated Gröbner basis up to any total degree. We give an example of a finitely generated homogeneous ideal which has a non-finitely generated initial ideal with respect to the lexicographic initial order > lex on M.