The Homogenize(f, v) command homogenizes the polynomial $f$, by multiplying each term of $f$ by an appropriate power of $v$. The result is a polynomial $g$ in the same variables as $f$ plus one more variable $v$, such that all terms of $g$ have the same total degree, which equals the total degree of $f$.

The homogenization variable $v$ must be a new variable that does not appear in $f$.

The IsHomogeneous(f) command checks if the polynomial $f$ is homogeneous, i.e., all terms have the same total degree. If so, it returns $\mathrm{true}$, and $\mathrm{false}$ otherwise.

The Homogenize(f, v, X) command homogenizes the polynomial $f$ only w.r.t. the subset of the variables given by $X$. The resulting polynomial will be homogeneous in the variables $X\cup \left\{v\right\}$. The two-argument command Homogenize(f, v) is equivalent to Homogenize(f, v, indets(f,name)).

The IsHomogeneous(f, X) command checks if the polynomial $f$ is homogeneous w.r.t. the subset of the variables given by $X$. The one-argument command IsHomogeneous(f) is equivalent to IsHomogeneous(f, indets(f,name)).

The Homogenize(f, v, X, W) calling sequence performs a weighted homogenization, with weight $W_i$ given to variable $X_i$. If $v=\left[y\,e\right]$, then the homogenization variable $y$ is given weight $e$. Note that in this case the result may contain fractional powers of $y$.

The IsHomogeneous(f, X, W) command checks if the polynomial is weighted-homogeneous, with weight $W_i$ given to variable $X_i$.

If $f$ is a set or list of polynomials, then each element of $f$ will be homogenized / checked for homogeneity.

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

$f≔x^4+x^{2}y+yz+2z$

$f≔x^4+x^{2}y+yz+2z$

$\mathrm{IsHomogeneous}\left(f\right)$

$\mathrm{false}$

$g≔\mathrm{Homogenize}\left(f\,v\right)$

$g≔2v^{3}z+v^{2}yz+v{x}^{2}y+x^4$

$\mathrm{IsHomogeneous}\left(g\right)$

$\mathrm{true}$

$\mathrm{IsHomogeneous}\left(g\,\left[x\,y\,z\right]\right)$

$\mathrm{false}$

$\mathrm{Homogenize}\left(\left[f\,xy+z^3\right]\,v\right)$

$\left[2v^{3}z+v^{2}yz+v{x}^{2}y+x^4\,vxy+z^3\right]$

$\mathrm{IsHomogeneous}\left(\left[g\,ab-c^2\right]\right)$

$\mathrm{true}$

$\mathrm{Homogenize}\left(f\,v\,\left[x\,y\right]\right)$

$2v^{4}z+v^{3}yz+v{x}^{2}y+x^4$

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

$2v^{4}z+v^{2}yz+x^4+x^{2}y$

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

$x^4+2v^{2}z+vyz+x^{2}y$

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

$2v^{2}z+v^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}yz+\sqrt{v}{x}^{2}y+x^4$

$h≔x^6+x^{3}y+y^2$

$h≔x^6+x^{3}y+y^2$

$\mathrm{IsHomogeneous}\left(h\right)$

$\mathrm{false}$

$\mathrm{IsHomogeneous}\left(h\,\left[x\,y\right]\,\left[1\,3\right]\right)$

$\mathrm{true}$

$\mathrm{Homogenize}\left(h\,v\,\left[x\,y\right]\,\left[1\,3\right]\right)$

$x^6+x^{3}y+y^2$

The PolynomialTools[Homogenize] and PolynomialTools[IsHomogeneous] commands were introduced in Maple 2018.

For more information on Maple 2018 changes, see Updates in Maple 2018.