 Definitions for leading monomials, coefficients, and terms - Maple Programming Help

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Definitions for leading monomials, coefficients, and terms

Description

 • The current terminology used in Groebner is that of Ideals, Varieties and Algorithms by David Cox, John Little, and Donal O'Shea, Springer-Verlag, (1992).  In releases of Maple prior to Maple 10, Groebner used a different, conflicting convention. In particular, what is now called a leading monomial used to be called a leading term, and vice versa.
 • Note: the old commands Groebner[leadmon] and Groebner[leadterm] still respect the old convention, however they are now deprecated.  You should replace them with Groebner[LeadingTerm] or Groebner[LeadingMonomial] respectively.
 • The current convention is as follows:
 – A monomial is a product of indeterminates from a fixed set X, possibly with repetitions. The coefficient of a monomial is always one.
 – A term of a polynomial (with respect to X) is the product of a monomial in X and a coefficient whose degree in X is zero. This coefficient may include other indeterminates not in X.  For example, the coefficients may be rational functions in other variables.
 – The "leading term" of a polynomial with respect to a monomial order is the term whose monomial is greatest with respect to the order and whose coefficient is non-zero.  The coefficient and monomial of this term are called the "leading coefficient" and "leading monomial" of the polynomial, respectively.
 • Note that the LeadingTerm command does not actually output terms, but rather the sequence (leading coefficient, leading monomial). This may be changed in a future release of Maple.

Examples

 > $\mathrm{with}\left(\mathrm{Groebner}\right):$
 > $f≔5{x}^{3}y+{x}^{2}{w}^{2}t+5{x}^{3}yzt-2xz{w}^{3}t+3{y}^{2}{w}^{3}t$
 ${f}{≔}{-}{2}{}{x}{}{z}{}{{w}}^{{3}}{}{t}{+}{3}{}{{y}}^{{2}}{}{{w}}^{{3}}{}{t}{+}{5}{}{{x}}^{{3}}{}{y}{}{z}{}{t}{+}{{x}}^{{2}}{}{{w}}^{{2}}{}{t}{+}{5}{}{{x}}^{{3}}{}{y}$ (1)

With respect to the definitions above, we will compute the leading coefficient, leading monomial, and leading term of the polynomial f with respect to lexicographic order with x > y > z > w > t.

 > $\mathrm{LeadingCoefficient}\left(f,\mathrm{plex}\left(x,y,z,w,t\right)\right)$
 ${5}$ (2)
 > $\mathrm{LeadingMonomial}\left(f,\mathrm{plex}\left(x,y,z,w,t\right)\right)$
 ${{x}}^{{3}}{}{y}{}{z}{}{t}$ (3)
 > $\mathrm{LeadingTerm}\left(f,\mathrm{plex}\left(x,y,z,w,t\right)\right)$
 ${5}{}{{x}}^{{3}}{}{y}{}{z}{}{t}$ (4)

In releases of Maple prior to Maple 10, Groebner[leadmon] computed what is now returned by LeadingTerm and Groebner[leadterm] computed what is now returned by LeadingMonomial.

 > $\mathrm{leadcoeff}\left(f,\mathrm{plex}\left(x,y,z,w,t\right)\right)$
 ${5}$ (5)
 > $\mathrm{leadterm}\left(f,\mathrm{plex}\left(x,y,z,w,t\right)\right)$
 ${{x}}^{{3}}{}{y}{}{z}{}{t}$ (6)
 > $\mathrm{leadmon}\left(f,\mathrm{plex}\left(x,y,z,w,t\right)\right)$
 ${5}{}{{x}}^{{3}}{}{y}{}{z}{}{t}$ (7)