Solving systems of polynomial equations over Galois–Eisenstein rings with the use of the canonical generating systems of polynomial ideals
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D.A. Mikhailov
A Galois–Eisenstein ring or a GE-ring is a finite commutative chain ring. We consider
two methods of enumeration of all solutions of some system of polynomial equations over a GE-ring
R. The first method is the general method of coordinate-wise linearisation. This method reduces to
solving the initial polynomial system over the quotient field 
= R/ Rad R and then to solving a series of linear equations systems over the same field. For an arbitrary ideal of the ring R[x1, . . . , xk ] a standard base called the canonical generating system (CGS) is constructed. The second method consists of finding a CGS of the ideal generated by the polynomials forming the left-hand side of the initial system of equations and solving instead of the initial system the system with polynomials of the CGS in the left-hand side. For systems of such type a modification of the coordinate-wise linearisation method is presented.
Copyright 2004, Walter de Gruyter
Artikel in diesem Heft
- 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
Artikel in diesem Heft
- 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
