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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 [ x  1 , . . . ,  x   k   ] 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.

Solving systems of polynomial equations over Galois–Eisenstein rings with the use of the canonical generating systems of polynomial ideals