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.

$\mathrm{with}\left(\mathrm{Groebner}\right)\:$

$f≔x^3+x^{2}y+y^2$

$f≔x^3+x^{2}y+y^2$

$\mathrm{WeightedDegree}\left(f\,\left[1\,2\right]\,\left[x\,y\right]\right)$

$4$

$\mathrm{InitialForm}\left(f\,\left[1\,2\right]\,\left[x\,y\right]\right)$

$x^{2}y+y^2$

$M≔\mathrm{MatrixOrder}\left(\mathrm{tdeg}\left(x\,y\right)\,\left[x\,y\right]\right)$

$M≔\left[\left[1\,1\right]\,\left[0\,\mathrm{-1}\right]\right]$

$\mathrm{InitialForm}\left(f\,\mathrm{tdeg}\left(x\,y\right)\right)$

$x^3+x^{2}y$