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Groebner

 MonomialOrder
 construct a MonomialOrder

 Calling Sequence MonomialOrder(A, tord, M)

Parameters

 A - a polynomial algebra created with the Ore_algebra package tord - ShortMonomialOrder or a user-defined monomial order M - (optional) module placeholder variables

Description

 • 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 $u\ne v$, otherwise MonomialOrder will return an error. Use variable names such as dx and x or d[1] and x[1] 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.

Examples

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.

 > $\mathrm{with}\left(\mathrm{Ore_algebra}\right):$
 > $\mathrm{with}\left(\mathrm{Groebner}\right):$
 > $A≔\mathrm{poly_algebra}\left(x,y,z\right)$
 ${A}{≔}{\mathrm{Ore_algebra}}$ (1)
 > $T≔\mathrm{MonomialOrder}\left(A,\mathrm{plex}\left(x,y,z\right)\right)$
 ${T}{≔}{\mathrm{monomial_order}}$ (2)

We can now do Groebner basis computations using T, although in this case an equivalent short syntax can be used.

 > $f≔-4yx-3{z}^{2}+y{z}^{2}-5{x}^{4}+{x}^{3}{z}^{2}$
 ${f}{≔}{{x}}^{{3}}{}{{z}}^{{2}}{-}{5}{}{{x}}^{{4}}{+}{y}{}{{z}}^{{2}}{-}{4}{}{y}{}{x}{-}{3}{}{{z}}^{{2}}$ (3)
 > $\mathrm{LeadingTerm}\left(f,T\right)$
 ${-5}{,}{{x}}^{{4}}$ (4)
 > $\mathrm{LeadingTerm}\left(f,\mathrm{plex}\left(x,y,z\right)\right)$
 ${-5}{,}{{x}}^{{4}}$ (5)

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[1] > D[2].

 > $A≔\mathrm{diff_algebra}\left(\left[\mathrm{D}\left[1\right],x\left[1\right]\right],\left[\mathrm{D}\left[2\right],x\left[2\right]\right]\right)$
 ${A}{≔}{\mathrm{Ore_algebra}}$ (6)
 > $A\left["polynomial_indets"\right]$
 $\left\{{{\mathrm{D}}}_{{1}}{,}{{\mathrm{D}}}_{{2}}\right\}$ (7)
 > $A\left["rational_indets"\right]$
 $\left\{{{x}}_{{1}}{,}{{x}}_{{2}}\right\}$ (8)
 > $T≔\mathrm{MonomialOrder}\left(A,\mathrm{tdeg}\left(\mathrm{D}\left[1\right],\mathrm{D}\left[2\right]\right)\right)$
 ${T}{≔}{\mathrm{monomial_order}}$ (9)
 > $\mathrm{SPolynomial}\left(\mathrm{D}\left[1\right]x\left[2\right]-x\left[2\right],\mathrm{D}\left[2\right]x\left[1\right]-x\left[1\right],T\right)$
 $\left({{x}}_{{2}}{}{{x}}_{{1}}{+}{{x}}_{{1}}\right){}{{\mathrm{D}}}_{{1}}{+}\left({-}{{x}}_{{2}}{}{{x}}_{{1}}{-}{{x}}_{{2}}\right){}{{\mathrm{D}}}_{{2}}{-}{{x}}_{{1}}{+}{{x}}_{{2}}$ (10)

A user-defined (lexicographic) order

 > $A≔\mathrm{poly_algebra}\left(x,y,z\right):$
 > 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:
 > $T≔\mathrm{MonomialOrder}\left(A,'\mathrm{user}'\left(P,\left[x,y,z\right]\right)\right)$
 ${T}{≔}{\mathrm{monomial_order}}$ (11)
 > $\mathrm{LeadingTerm}\left(f,T\right)$
 ${-5}{,}{{x}}^{{4}}$ (12)