Standard basis of a polynomial ideal over commutative Artinian chain ring
-
E.V. Gorbatov
We construct a standard basis of an ideal of the polynomial ring R[X] = R[x1, . . . , xk] 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 χ of a polynomial ideal over the field 
= R/ rad(R) can be lifted to a standard basis of the same cardinality over R with respect to the natural epimorphism ν : R[X] → [X] if and only if there is an ideal I
R[X] such that I is a free R-module and Ī = (χ).
Copyright 2004, Walter de Gruyter
Articles in the same Issue
- Andrei Andreevich Markov (to the centenary of the birth)
- Some classes of random mappings of finite sets and non-homogeneous branching processes
- The method of boundary functionals for non-regular structures
- A formula for the radius of stability of a vector l∞-extremal trajectory problem
- Solving systems of polynomial equations over Galois–Eisenstein rings with the use of the canonical generating systems of polynomial ideals
- Standard basis of a polynomial ideal over commutative Artinian chain ring
- On some systems of generators of symmetric and alternating groups which allow a simple software implementation
Articles in the same Issue
- Andrei Andreevich Markov (to the centenary of the birth)
- Some classes of random mappings of finite sets and non-homogeneous branching processes
- The method of boundary functionals for non-regular structures
- A formula for the radius of stability of a vector l∞-extremal trajectory problem
- Solving systems of polynomial equations over Galois–Eisenstein rings with the use of the canonical generating systems of polynomial ideals
- Standard basis of a polynomial ideal over commutative Artinian chain ring
- On some systems of generators of symmetric and alternating groups which allow a simple software implementation
