18.3 Multigraded lex ordering : A variety of elimination orders
There are other useful monomial orders which one can use other than graded lex and lex. Another
type of order is what we call multigraded lex and is a mixture of graded lex and lex order. This
multigraded order is set using SetMonomialOrder, SetKnowns and SetUnknowns which
are described in Section 18.4. As an example, suppose that we execute the following
commands:
SetMonomialOrder[{A,B,C},{a,b,c},{d,e,f}];
|
We use the notation
to
denote this order.
For an intuitive idea of why multigraded lex is helpful, we think of A, B and C as corresponding to
variables in some engineering problem which represent quantities which are known and a, b, c, d, e and f
to be unknown.
The fact that d, e and f are in the top level indicates that we are very interested in solving for d, e
and f in terms of A, B, C, a, b and c, but are not willing to solve for b in terms of expressions
involving either d, e or f.
For example,
-
- (1) d > a **a **A **b
-
- (2) d **a **A **b > a
-
- (3) e **d > d **e
-
- (4) b **a > a **b
-
- (5) a **b **b > b **a
-
- (6) a > A **B **A **B **A **B
This order induces an order on monomials in the following way. One does the following steps
in determining whether a monomial m is greater in the order than a monomial n or
not.
-
- (1) First, compute the total degree of m with respect to only the variables d, e and f.
-
- (2) Second, compute the total degree of n with respect to only the variables d, e and f.
-
- (3) If the number from item (2) is smaller than the number from item (1), then m is smaller
than n. If the number from item (2) is bigger than the number from item (1), then m
is bigger than n. If the numbers from items (1) and (2) are equal, then proceed to the
next item.
-
- (4) First, compute the total degree of m with respect to only the variables a, b and c.
-
- (5) Second, compute the total degree of n with respect to only the variables a, b and c.
-
- (6) If the number from item (5) is smaller than the number from item (4), then m is smaller
than n. If the number from item (5) is bigger than the number from item (4), then m
is bigger than n. If the numbers from items (4) and (5) are equal, then proceed to the
next item.
-
- (7) First, compute the total degree of m with respect to only the variables A, B and C.
-
- (8) Second, compute the total degree of n with respect to only the variables A, B and C.
-
- (9) If the number from item (8) is smaller than the number from item (7), then m is smaller
than n. If the number from item (8) is bigger than the number from item (7), then m
is bigger than n. If the numbers from items (7) and (8) are equal, then proceed to the
next item.
-
- (10) At this point, say that m is smaller than n if and only if m is smaller than n with
respect to the graded lex order A < B < C < a < b < c < d < e < f
For more information on multigraded lex orders, consult [HSStrat].