-
Let
be a finite subset of . The algebraic variety in is the set of all common zeros of of the polynomials in . -
Let
be an ideal in a Commutative Ring with unity. A subset is a basis for if -
(Fraleigh 28.4) Let
. The set of common zeros in of the polynomials for is the same as the set of common zeros in of all the polynomials in the entire ideal -
(Fraleigh 28.5) Hilbert Basis Theorem Every Ideal in
has a finite basis. -
(Fraleigh 28.6) Let
such that . If and are two members of a basis for an ideal of then replacement of by in the basis still yields a basis for . -
An ordering for the set of power products of
can be defined by having -
A set
of nonzero polynomials in with term ordering is a Groebner basis for the ideal if and only if for each nonzero , there exists some where such that divides ,. -
(Fraleigh 28.12) Denote by
the polynomial obtained as follows . First, multiply them by . Then add or subtract with suitable coefficients from so that cancellation results. A basis
is a Groebner basis for the ideal if and only if for all , the polynomial can be reduced to zero repeatedly dividing remainders by elements of as in the division algorithm.