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Groebner

 InitialForm
 compute the initial form of a polynomial
 WeightedDegree
 compute the weighted degree of a polynomial

 Calling Sequence InitialForm(f, T) InitialForm(f, W, V) WeightedDegree(f, W, V)

Parameters

 f - polynomial or list or set of polynomials T - ShortMonomialOrder W - list or vector of rational weights V - list of variable names

Description

 • The WeightedDegree command computes the weighted degree of a polynomial f with respect to a vector of rational weights W.  The output is a rational number, the weighted degree of f, which is similar to the ordinary degree except that each power of a variable ${V}_{i}$ contributes ${W}_{i}$ to the degree of each term, instead of 1.
 • The InitialForm command extracts the terms of f with maximal weighted degree. The output is a polynomial.  One can also specify a monomial order T instead of a vector of weights, and the first row in a matrix representation for T will be used automatically.  See the MatrixOrder help page for more details.

Examples

 > $\mathrm{with}\left(\mathrm{Groebner}\right):$
 > $f≔{x}^{3}+{x}^{2}y+{y}^{2}$
 ${f}{≔}{{x}}^{{3}}{+}{{x}}^{{2}}{}{y}{+}{{y}}^{{2}}$ (1)
 > $\mathrm{WeightedDegree}\left(f,\left[1,2\right],\left[x,y\right]\right)$
 ${4}$ (2)
 > $\mathrm{InitialForm}\left(f,\left[1,2\right],\left[x,y\right]\right)$
 ${{x}}^{{2}}{}{y}{+}{{y}}^{{2}}$ (3)
 > $M≔\mathrm{MatrixOrder}\left(\mathrm{tdeg}\left(x,y\right),\left[x,y\right]\right)$
 ${M}{≔}\left[\left[{1}{,}{1}\right]{,}\left[{0}{,}{-1}\right]\right]$ (4)
 > $\mathrm{InitialForm}\left(f,\mathrm{tdeg}\left(x,y\right)\right)$
 ${{x}}^{{3}}{+}{{x}}^{{2}}{}{y}$ (5)