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2.4. GRÖBNER BASIS THEORY 17
but which contains (non-leading) terms that can also be reduced with respect to F .
Therefore we introduce the concept of the complete reduction of a polynomial:
Definition 2.13. (Complete polynomial reduction). Fix a monomial ordering ≻ and
let F be a finite generating system of polynomials {f 1 ,...,f m } in K[X]. A polynomial
g ∈ K[X] is completely reduced with respect to (or modulo) F if no monomial of g is
divisible by LM(f i ) for all i =1,...,m.
Example 2.9. Let g = x + y + z 2 and F = {xy − 1,xz+ y} with monomial ordering
≻ lex . Then g is completely reduced with respect to F because x, y and z 2 are not
divisible by LM(f 1 )=xy or LM(f 2 )=xz.
Example 2.10. Let g = x + y 2 + y and F = {f 1 } = {y − 1} with monomial ordering
≻ lex . Then g can be reduced by subtracting y times f 1 which gives x +2y. This is
still not completely reduced because 2y is still divisible by LM(f 1 )=y. Subtracting
again 2 times f 1 from this result, gives x+2 which is completely reduced with respect
to F .
This reduction process is finite and the completely reduced polynomial h is called
a Normal form of g with respect to F . This is denoted by: g −→ ∗ F h.
This reduction process is combined into an algorithm which is called the Generalized
Division Algorithm in K[X] formulated in the next Theorem (based on Theorem
3 of §2.3 of [28]:
Definition 2.14. (Generalized Division algorithm). Fix a monomial ordering ≻ and
let F be a finite generating system of polynomials {f 1 ,...,f m } in K[X]. Then every
f ∈ K[X] can be written as:
f = a 1 f 1 + a 2 f 2 + ...+ a m f m + r
where a i ,r ∈ K[X] and either the remainder of division r =0or r is completely
reduced with respect to F .
Note that the remainder r is generally not uniquely determined for division with
respect to F .
Suppose some polynomial f is reduced by F = {f 1 ,...,f m } and the normal form
of f turns out to be zero, then: f = a 1 f 1 +...+a m f m and therefore f ∈〈f 1 ,...,f m 〉.
But r = 0 is a sufficient condition for f being a member of 〈I〉 and not a necessary
condition. One can question what might be a good basis {g 1 ,...,g t } of the ideal
〈f 1 ,...,f m 〉, for which the remainder r on division by the generators g i is uniquely
determined and the condition r = 0 is equivalent to membership in the ideal. It will
turn out that a Gröbner basis will be such a basis.
Definition 2.15. Let I = 〈f 1 ,...,f m 〉 be an ideal in K[X], then we define for a given
ordering LT (I) ={LT (f) :f ∈ I}. The ideal generated by the elements of LT (I) is
denoted by 〈LT (I)〉.
Note that if I = 〈f 1 ,...,f m 〉 then 〈LT (f 1 ),...,LT(f m )〉 and 〈LT (I)〉 may be
different ideals but 〈LT (f 1 ),...,LT(f m )〉⊂〈LT (I)〉 is true.