Groebner
Homogenize
homogenize polynomials and ideals
Calling Sequence
Parameters
Description
Examples
References
Homogenize(f, h, vars)
f
-
polynomial or list or set of polynomials, or a PolynomialIdeal
h
variable
vars
(optional) list or set of variables
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.
withGroebner:
f≔x5+xy2+y4+1
f≔x5+y4+xy2+1
Homogenizef,h
h5+h2xy2+hy4+x5
Homogenizef,h,x
h5y4+h4xy2+h5+x5
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.
withPolynomialIdeals:
F≔x2−1,xy−1
IdealMembershipx−y,F
true
Fh≔HomogenizeF,h
Fh≔−h2+x2,−h2+xy
IdealMembershipx−y,Fh
false
GroebnerBasisFh,tdegx,y,h
−h2+xy,−h2+x2,h2x−h2y,−h4+h2y2
IdealMembershipx−y,HomogenizeF,h
HomogenizeGroebnerBasisF,tdegx,y,h
x−y,−h2+y2
Froberg, R. An Introduction to Grobner Bases. West Sussex: Wiley & Sons, 1997.
See Also
degree
Groebner[Basis]
PolynomialIdeals
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