homogenize a multivariate polynomial
check if a multivariate polynomial is homogeneous
Homogenize(f, v, X)
Homogenize(f, v, X, W)
IsHomogeneous(f, X, W)
multivariate polynomial, or list or set of multivariate polynomials
name or list of the form [name,posint]; the homogenization variable
(optional) list or set of names; variables w.r.t. which f is homogenized
(optional) list of nonnegative integers; weights
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 X∪v. 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.
f ≔ x4+x2⁢y+y⁢z+2⁢z
g ≔ Homogenize⁡f,v
h ≔ x6+x3⁢y+y2
The PolynomialTools[Homogenize] and PolynomialTools[IsHomogeneous] commands were introduced in Maple 2018.
For more information on Maple 2018 changes, see Updates in Maple 2018.
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