construct a MonomialOrder
MonomialOrder(A, tord, M)
a polynomial algebra created with the Ore_algebra package
ShortMonomialOrder or a user-defined monomial order
(optional) module placeholder variables
The MonomialOrder command constructs a term order on a polynomial algebra A. This object (called a MonomialOrder) may be used with other commands in the Groebner package to do computations with modules and (left) ideals of non-commutative skew polynomials.
For commutative problems you typically do not need to construct a MonomialOrder, you can use a short monomial order description (such as plex(x,y,z)) instead. For a list of these short monomial orders, see the Monomial Orders help page.
The first argument is an algebra A defined by the Ore_algebra package. It can be one of the following:
a commutative polynomial algebra defined by Ore_algebra[poly_algebra]
a Weyl algebra defined by Ore_algebra[diff_algebra]
a difference algebra defined by Ore_algebra[shift_algebra]
a q-difference algebra defined by Ore_algebra[qshift_algebra]
a general Ore algebra defined by Ore_algebra[skew_algebra]
The second argument should be a ShortMonomialOrder, which describes the order imposed on variables of the algebra. For example, plex(x,y,z) describes lexicographic order with x > y > z. All of the variables of the algebra A must be ordered. For a list of short monomial orders, see the Monomial Orders help page.
The second argument can also describe a user-defined monomial order, which can have the form user(P, L) or user(P, Q, L). In both cases P is a procedure taking two monomials as arguments and returning true if and only if they are in increasing order, L is the list of indeterminates with respect to which the monomial order is defined, and (optionally) Q is a procedure for computing the leading term of a polynomial (see LeadingTerm). The procedure Q should take a polynomial as input and return the sequence (leading coefficient, leading monomial).
The optional third argument M is a list or set of module placeholder variables. These variables must be included in the algebra A and ordered by the monomial order tord. Multiplication by elements of M is not allowed, so distinct monomials in the variables of M indicate different module components. See the Groebner[Basis_details] help page for more information.
Due to technical limitations in the Ore_algebra package, you can not use algebras with pairs of variables that depend on each other, such as d[x] and x, to construct a MonomialOrder. In general, depends(u,v) must return false for each pair of variables u≠v, otherwise MonomialOrder will return an error. Use variable names such as dx and x or d and x instead. This limitation does not apply to ordinary commutative computations using ShortMonomialOrders.
Note: the termorder command is deprecated. It may not be supported in a future Maple release.
First we construct a monomial order on the polynomial ring Q[x,y,z]. The monomials are ordered by lexicographic order with x > y > z.
A ≔ poly_algebra⁡x,y,z
T ≔ MonomialOrder⁡A,plex⁡x,y,z
We can now do Groebner basis computations using T, although in this case an equivalent short syntax can be used.
f ≔ −4⁢y⁢x−3⁢z2+y⁢z2−5⁢x4+x3⁢z2
For more complicated domains, such as modules or skew polynomial rings, it is necessary to define a MonomialOrder. Below we define a Weyl algebra with D[i]*x[i] = x[i]*D[i] + 1 for i=1..2. The D[i] are variables and the x[i] are parameters. The ring is ordered using graded-reverse lexicographic order with D > D.
A ≔ diff_algebra⁡D1,x1,D2,x2
T ≔ MonomialOrder⁡A,tdeg⁡D1,D2
A user-defined (lexicographic) order
A ≔ poly_algebra⁡x,y,z:
P := proc(t1, t2) global x,y,z; (degree(t1,x) < degree(t2,x)) or (degree(t1,x) = degree(t2,x) and degree(t1,y) < degree(t2,y)) or (degree(t1,x) = degree(t2,x) and degree(t1,y) = degree(t2,y) and degree(t1,z) <= degree(t2,z)) end proc:
T ≔ MonomialOrder⁡A,'user'⁡P,x,y,z
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