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2.3. MONOMIAL ORDERINGS 15
Let X = {x 1 ,...,x n } be a set of variables. The set containing all power products
of x 1 ,...,x n is denoted by T X :
T X = {x α } = {x α1
1 ···xαn n :(α 1 ,...,α n ) ∈ N n 0 }. (2.5)
Since a polynomial is a linear combination of monomials, we would often like to
arrange the terms in a polynomial in an unambiguous way in descending (or ascending)
order. So we must be able to compare monomials to each other to establish their
proper relative positions by use of monomial orderings.
Every x α ∈ T X is called a monomial and the total degree of a monomial is defined
as |α| = α 1 + ...+ α n . The most significant term of a non-zero polynomial p with
respect to a monomial ordering ≻ is called the leading term LT ≻ (p) of that polynomial
with respect to the chosen ordering. A leading term LT ≻ (p) of some polynomial p
consists of a leading coefficient LC ≻ (p) and the corresponding leading monomial
LM ≻ (p) such that LT ≻ (p) =LC ≻ (p)LM ≻ (p).
Definition 2.8. (Monomial ordering). Let X be a non-empty set of variables. A
monomial ordering ≻ on T X is a total ordering with the additional property that
u ≻ 1 for all u ∈ T X and that if u ≻ v then tu ≻ tv for all u, v, t ∈ T X .
There are several monomial orderings of which we will explicitly review three: the
lexicographic ordering, the graded lex ordering and the graded reverse lex ordering.
Definition 2.9. (Lexicographic Ordering). Let X be a set of n variables x 1 ,...,x n
and let α =(α 1 ,...,α n ) and β =(β 1 ,...,β n ) ∈ N n 0 . Then α ≻ lex β is a monomial
ordering on T X if in the vector difference α−β the left-most non-zero entry is positive.
We write x α ≻ lex x β if α ≻ lex β. The lexicographic ordering is often abbreviated as
lex ordering.
Example 2.5. (3, 2, 4) ≻ lex (2, 5, 3) since the left-most entry in (3, 2, 4) − (2, 5, 3) =
(1, −3, 1) is positive. Hence, it holds that x 3 1x 2 2x 4 3 ≻ lex x 2 1x 5 2x 3 3.
Definition 2.10. (Graded Lex Ordering). Let X be a set of n variables x 1 ,...,x n
and let α =(α 1 ,...,α n ) and β =(β 1 ,...,β n ) ∈ N n 0 . Then α ≻ grlex β is a monomial
ordering on T X if |α| = ∑ n
i=1 α i > |β| = ∑ n
i=1 β i,or|α| = |β| and α ≻ lex β. The
graded lex ordering is often abbreviated as grlex ordering.
Example 2.6. (4, 3, 4) ≻ grlex (3, 5, 3) since |(4, 3, 4)| = |(3, 5, 3)| = 11 and (4, 3, 4)
≻ lex (3, 5, 3). This holds since the left-most entry in (4, 3, 4) − (3, 5, 3)=(1, −2, 1) is
positive. Hence, it holds that x 4 1x 3 2x 4 3 ≻ grlex x 3 1x 5 2x 3 3.
In the literature the graded lex ordering is also called the total degree ordering.
Definition 2.11. (Graded Reverse Lex Ordering). Let X be a set of n variables
x 1 ,...,x n and let α =(α 1 ,...,α n ) and β =(β 1 ,...,β n ) ∈ N n 0 . Then α ≻ grevlex β is
a monomial ordering on T X if |α| = ∑ n
i=1 α i > |β| = ∑ n
i=1 β i,or|α| = |β| and the
right-most non-zero entry in α − β ∈ Z n is negative. The graded reverse lex ordering
is often abbreviated as grevlex ordering.