The Homogenize command homogenizes polynomials and polynomial ideals. If f is a polynomial, then a minimal power of h is added to each term so that all resulting terms have the same total degree.  The variables of f can be specified explicitly by an optional third argument vars. Homogenize also maps onto lists and sets of polynomials automatically.

If the first argument f is a PolynomialIdeal, then Homogenize constructs the ideal generated by all homogenizations of polynomials in f.  This is done by homogenizing a total degree Groebner basis for f.

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

$f≔x^5+x{y}^2+y^4+1$

$f≔x^5+y^4+x{y}^2+1$

$\mathrm{Homogenize}\left(f\,h\right)$

$h^5+h^{2}x{y}^2+h{y}^4+x^5$

$\mathrm{Homogenize}\left(f\,h\,\left\{x\right\}\right)$

$h^5{y}^4+h^{4}x{y}^2+h^5+x^5$

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

$F≔\left[x^2-1\,xy-1\right]$

$F≔\left[x^2-1\,xy-1\right]$

$\mathrm{IdealMembership}\left(x-y\,⟨F⟩\right)$

$\mathrm{true}$

$\mathrm{Fh}≔\mathrm{Homogenize}\left(F\,h\right)$

$\mathrm{Fh}≔\left[-h^2+x^2\,-h^2+xy\right]$

$\mathrm{IdealMembership}\left(x-y\,⟨\mathrm{Fh}⟩\right)$

$\mathrm{false}$

$\mathrm{Groebner}\left[\mathrm{Basis}\right]\left(\mathrm{Fh}\,\mathrm{tdeg}\left(x\,y\,h\right)\right)$

$\left[-h^2+xy\,-h^2+x^2\,h^{2}x-h^{2}y\,-h^4+h^2{y}^2\right]$

$\mathrm{IdealMembership}\left(x-y\,\mathrm{Homogenize}\left(⟨F⟩\,h\right)\right)$

$\mathrm{true}$

$\mathrm{Homogenize}\left(\mathrm{Groebner}\left[\mathrm{Basis}\right]\left(F\,\mathrm{tdeg}\left(x\,y\right)\right)\,'h'\right)$

$\left[x-y\,-h^2+y^2\right]$

Froberg, R. An Introduction to Grobner Bases. West Sussex: Wiley & Sons, 1997.