Groebner - Maple Programming Help

Home : Support : Online Help : Mathematics : Algebra : Polynomials : Groebner : Groebner/Homogenize

Groebner

 Homogenize
 homogenize polynomials and ideals

 Calling Sequence Homogenize(f, h, vars)

Parameters

 f - polynomial or list or set of polynomials, or a PolynomialIdeal h - variable vars - (optional) list or set of variables

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{Groebner}\right):$
 > $f≔{x}^{5}+x{y}^{2}+{y}^{4}+1$
 ${f}{:=}{{x}}^{{5}}{+}{{y}}^{{4}}{+}{x}{}{{y}}^{{2}}{+}{1}$ (1)
 > $\mathrm{Homogenize}\left(f,h\right)$
 ${{h}}^{{5}}{+}{{h}}^{{2}}{}{x}{}{{y}}^{{2}}{+}{h}{}{{y}}^{{4}}{+}{{x}}^{{5}}$ (2)
 > $\mathrm{Homogenize}\left(f,h,\left\{x\right\}\right)$
 ${{h}}^{{5}}{}{{y}}^{{4}}{+}{{h}}^{{4}}{}{x}{}{{y}}^{{2}}{+}{{h}}^{{5}}{+}{{x}}^{{5}}$ (3)

It does not suffice to simply homogenize the generators of an ideal. In the example below $x-y$ is in the ideal <F>, and since the polynomial is homogeneous it should be in the homogenization of <F> as well.

 > $\mathrm{with}\left(\mathrm{PolynomialIdeals}\right):$
 > $F≔\left[{x}^{2}-1,xy-1\right]$
 ${F}{:=}\left[{{x}}^{{2}}{-}{1}{,}{x}{}{y}{-}{1}\right]$ (4)
 > $\mathrm{IdealMembership}\left(x-y,⟨F⟩\right)$
 ${\mathrm{true}}$ (5)
 > $\mathrm{Fh}≔\mathrm{Homogenize}\left(F,h\right)$
 ${\mathrm{Fh}}{:=}\left[{-}{{h}}^{{2}}{+}{{x}}^{{2}}{,}{-}{{h}}^{{2}}{+}{x}{}{y}\right]$ (6)
 > $\mathrm{IdealMembership}\left(x-y,⟨\mathrm{Fh}⟩\right)$
 ${\mathrm{false}}$ (7)
 > $\mathrm{Groebner}[\mathrm{Basis}]\left(\mathrm{Fh},\mathrm{tdeg}\left(x,y,h\right)\right)$
 $\left[{-}{{h}}^{{2}}{+}{x}{}{y}{,}{-}{{h}}^{{2}}{+}{{x}}^{{2}}{,}{{h}}^{{2}}{}{x}{-}{{h}}^{{2}}{}{y}{,}{-}{{h}}^{{4}}{+}{{h}}^{{2}}{}{{y}}^{{2}}\right]$ (8)
 > $\mathrm{IdealMembership}\left(x-y,\mathrm{Homogenize}\left(⟨F⟩,h\right)\right)$
 ${\mathrm{true}}$ (9)
 > $\mathrm{Homogenize}\left(\mathrm{Groebner}[\mathrm{Basis}]\left(F,\mathrm{tdeg}\left(x,y\right)\right),'h'\right)$
 $\left[{x}{-}{y}{,}{-}{{h}}^{{2}}{+}{{y}}^{{2}}\right]$ (10)

References

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