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Groebner

  

Homogenize

  

homogenize polynomials and ideals

 

Calling Sequence

Parameters

Description

Examples

References

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

withGroebner:

fx5+xy2+y4+1

f:=x5+y4+xy2+1

(1)

Homogenizef,h

h5+h2xy2+hy4+x5

(2)

Homogenizef,h,x

h5y4+h4xy2+h5+x5

(3)

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

withPolynomialIdeals&colon;

Fx21&comma;xy1

F:=x21&comma;xy1

(4)

IdealMembershipxy&comma;F

true

(5)

FhHomogenizeF&comma;h

Fh:=h2&plus;x2&comma;h2&plus;xy

(6)

IdealMembershipxy&comma;Fh

false

(7)

Groebner&lsqb;Basis&rsqb;Fh&comma;tdegx&comma;y&comma;h

h2&plus;xy&comma;h2&plus;x2&comma;h2xh2y&comma;h4&plus;h2y2

(8)

IdealMembershipxy&comma;HomogenizeF&comma;h

true

(9)

HomogenizeGroebner&lsqb;Basis&rsqb;F&comma;tdegx&comma;y&comma;&apos;h&apos;

xy&comma;h2&plus;y2

(10)

References

  

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

See Also

degree

Groebner[Basis]

PolynomialIdeals

 


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