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The standard basis of a polynomial ideal over a commutative Artinian chain ring
E. V. Gorbatov
Abstract:
We construct a standard basis of an ideal of the polynomial ring
$R[X]=R[x_1,\ldots,x_k]$
over commutative Artinian chain ring
$R$, which generalises a
Gröbner
base of a polynomial ideal over fields. We adopt the notion of the leading
term of a polynomial suggested by D. A. Mikhailov and A. A. Nechaev, but
using the simplification schemes introduced by V. N. Latyshev. We prove that any canonical
generating system
constructed by D. A. Mikhailov and A. A. Nechaev is a standard basis of the special
form. We give an
algorithm (based on the notion of
$S$-polynomial) which constructs standard bases
and canonical generating systems of an ideal.
We define minimal
and reduced standard bases and give their characterisations. We prove that a
Gröbner
base
$\chi$ of a polynomial ideal over the field
$\bar R=R/\operatorname{rad}(R)$ can be lifted
to a
standard basis of the same cardinality over
$R$ with respect to the natural
epimorphism
$\nu\colon R[X]\to \bar R[X]$ if and only if there is an
ideal
$I\triangleleft R[X]$ such that
$I$ is a free
$R$-module and
$\bar{I}=(\chi)$.
The research was supported by the Russian Foundation for Basic Research, grant
02-01-00218, and by the President of the Russian Federation program of support of
leading scientific schools,
grant 1910.2003.1.
UDC:
512.8
Received: 10.11.2003
DOI:
10.4213/dm142
Fulltext:
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