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compoly

determine a possible composition of a polynomial

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

compoly(r)

compoly(r, x)

compoly(r, {x, ... })

Parameters

r

-

polynomial

x

-

variable on which the composition will be made

Description

• 

The function compoly returns a pair px, x=qx such that subs(x=q(x), p(x)) is equal to r, the input polynomial. If such a pair cannot be found, it returns FAIL. px and qx are nonlinear polynomials and qx has a low-degree in x greater or equal to 1.

• 

When compoly is called without additional arguments, it is equivalent to being called with a second argument which is indets(r). When the second argument is a set with more than one variable, then a multivariate composition is attempted. In this case, the result has the same form, but the second polynomial is multivariate, that is, px, x=qx,....  For the multivariate case, the second polynomial, qx, may be of degree 1.

• 

Note that the composition may not be unique. In particular, if px, x=qx,... is a composition, then we can find another px such that replacing qx,... by  cqx,+b will also result in a valid composition. This non-determinacy is eliminated by selecting c and b such that the q polynomial has integer content 1 and its independent term is 0.

Examples

compolyx69x5+27x427x32x2y+6xy+1,x

x32xy+1,x=x23x

(1)

compolyx43x3x+5,x

FAIL

(2)

compolyx2+2xy7x+y27y+16

x27x+16,x=x+y

(3)

compolyx4+4x3y3+6x2y6+4xy9+y12+x+y31,x,y

x4+x1,x=y3+x

(4)

See Also

indets

polynomials

subs

 


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