PolynomialTools - Maple Programming Help

Online Help

All Products    Maple    MapleSim


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

PolynomialTools

  

Homogenize

  

homogenize a multivariate polynomial

  

IsHomogeneous

  

check if a multivariate polynomial is homogeneous

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

Homogenize(f, v)

Homogenize(f, v, X)

Homogenize(f, v, X, W)

IsHomogeneous(f)

IsHomogeneous(f, X)

IsHomogeneous(f, X, W)

Parameters

f

-

multivariate polynomial, or list or set of multivariate polynomials

v

-

name or list of the form [name,posint]; the homogenization variable

X

-

(optional) list or set of names; variables w.r.t. which f is homogenized

W

-

(optional) list of nonnegative integers; weights

Description

• 

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 true, and 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 Xv. 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 Wi given to variable Xi. If v=y,e, 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 Wi given to variable Xi.

• 

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

Examples

withPolynomialTools:

fx4+x2y+yz+2z

fx4+x2y+yz+2z

(1)

IsHomogeneousf

false

(2)

gHomogenizef,v

g2v3z+v2yz+vx2y+x4

(3)

IsHomogeneousg

true

(4)

IsHomogeneousg,x,y,z

false

(5)

Homogenizef,xy+z3,v

2v3z+v2yz+vx2y+x4,vxy+z3

(6)

IsHomogeneousg,abc2

true

(7)

Homogenizef,v,x,y

2v4z+v3yz+vx2y+x4

(8)

Homogenizef,v,x,y,1,2

2v4z+v2yz+x4+x2y

(9)

Homogenizef,v,2,x,y,1,2

x4+2v2z+vyz+x2y

(10)

Homogenizef,v,2,x,y,1,1

2v2z+v32yz+vx2y+x4

(11)

hx6+x3y+y2

hx6+x3y+y2

(12)

IsHomogeneoush

false

(13)

IsHomogeneoush,x,y,1,3

true

(14)

Homogenizeh,v,x,y,1,3

x6+x3y+y2

(15)

Compatibility

• 

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

• 

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

See Also

PolynomialTools