Groebner - Maple Programming Help

Home : Support : Online Help : Mathematics : Algebra : Polynomials : Groebner : Groebner/LeadingTerm

Groebner

 Compute the leading term of a polynomial
 Compute the leading monomial of a polynomial
 Compute the leading coefficient of a polynomial
 TrailingTerm
 Compute the trailing term of a polynomial

Parameters

 f - polynomial or list or set of polynomials T - MonomialOrder or ShortMonomialOrder J - PolynomialIdeal tord - ShortMonomialOrder

Description

 • The LeadingTerm command computes the largest (or leading) term of a polynomial f with respect to the monomial order T and returns the sequence (leading coefficient, leading monomial).  If T is a ShortMonomialOrder then f must be a polynomial in the ring implied by T.  If T is a MonomialOrder created with the Groebner[MonomialOrder] command, then f must be a member of the algebra used to define T.  The LeadingTerm command automatically maps onto lists and sets.
 • The LeadingMonomial and LeadingCoefficient commands behave identically to LeadingTerm, but return only leading monomials or coefficients, respectively.  The LeadingMonomial command has an additional syntax, LeadingMonomial(J, tord) computes the ideal of leading monomials for a PolynomialIdeal J. This typically requires the computation of a Groebner basis.
 • The TrailingTerm command is identical to LeadingTerm, except the smallest (or trailing) term of a polynomial f is computed.  It returns the sequence (trailing coefficient, trailing monomial).
 • To compare or sort monomials with respect to a monomial order, use the TestOrder command. For a description of the monomial orders that are available in Maple, see the Monomial Orders help page.
 • Note that the leadcoeff, leadterm, and leadmon commands have been superseded by LeadingCoefficient, LeadingMonomial, and LeadingTerm, respectively.  (Warning: the notions of monomials and terms were interchanged; see Groebner[terminology] for details).  The lowercase commands may not be supported in a future Maple release.

Examples

 > $\mathrm{with}\left(\mathrm{Groebner}\right):$
 > $p≔-18x{y}^{5}z-96x{y}^{4}{z}^{2}+9x{y}^{4}-592x{y}^{3}z+45{y}^{5}+240{y}^{4}z+320x{y}^{2}+1600{y}^{3}$
 ${p}{≔}{-}{18}{}{x}{}{{y}}^{{5}}{}{z}{-}{96}{}{x}{}{{y}}^{{4}}{}{{z}}^{{2}}{+}{9}{}{x}{}{{y}}^{{4}}{-}{592}{}{x}{}{{y}}^{{3}}{}{z}{+}{45}{}{{y}}^{{5}}{+}{240}{}{{y}}^{{4}}{}{z}{+}{320}{}{x}{}{{y}}^{{2}}{+}{1600}{}{{y}}^{{3}}$ (1)
 > $\mathrm{LeadingTerm}\left(p,\mathrm{plex}\left(x,y\right)\right)$
 ${-}{18}{}{z}{,}{x}{}{{y}}^{{5}}$ (2)
 > $\mathrm{LeadingTerm}\left(p,\mathrm{plex}\left(x,y,z\right)\right)$
 ${-18}{,}{x}{}{{y}}^{{5}}{}{z}$ (3)
 > $\mathrm{TrailingTerm}\left(p,\mathrm{plex}\left(x,y,z\right)\right)$
 ${1600}{,}{{y}}^{{3}}$ (4)
 > $\mathrm{LeadingTerm}\left(p,\mathrm{plex}\left(z,y,x\right)\right)$
 ${-96}{,}{x}{}{{y}}^{{4}}{}{{z}}^{{2}}$ (5)
 > $\mathrm{LeadingCoefficient}\left(p,\mathrm{plex}\left(z,y,x\right)\right)$
 ${-96}$ (6)
 > $\mathrm{LeadingMonomial}\left(p,\mathrm{plex}\left(z,y,x\right)\right)$
 ${x}{}{{y}}^{{4}}{}{{z}}^{{2}}$ (7)
 > $P≔5{x}^{2}+y+{z}^{2}:$
 > $Q≔3xy-1:$
 > $\mathrm{LeadingMonomial}\left(\left[P,Q\right],\mathrm{tdeg}\left(x,y,z\right)\right)$
 $\left[{{x}}^{{2}}{,}{x}{}{y}\right]$ (8)
 > $\mathrm{with}\left(\mathrm{PolynomialIdeals}\right):$
 > $\mathrm{LeadingMonomial}\left(⟨P,Q⟩,\mathrm{tdeg}\left(x,y,z\right)\right)$
 $⟨{{x}}^{{2}}{,}{x}{}{y}{,}{y}{}{{z}}^{{2}}⟩$ (9)