compute a shiftless decomposition of a univariate polynomial - Maple Help

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PolynomialTools[ShiftlessDecomposition] - compute a shiftless decomposition  of a univariate polynomial

Calling Sequence

ShiftlessDecomposition(f,x)

Parameters

f

-

polynomial in x

x

-

indeterminate

Description

• 

The ShiftlessDecomposition command computes a shiftless decomposition [c,[[g1,[[h11,e11],[h12,e12],...]],[g2,[[h21,e21],[h22,e22],...]],...]] of f w.r.t. x.

  

It satisfies the following properties.

  

 f=cg1x+h11e11g1x+h12e12...g2x+h21e21g2x+h22e22... 

  

 g1,... are squarefree and pairwise shift coprime, that is, for 1i,j and all integers h, we have gcdgix,gjx+h1 if and only if i=j and h=0

  

c is constant w.r.t. x, and g1,... are nonconstant primitive polynomials w.r.t. x.

  

The hij and eij are non-negative integers with 0=hi1<hi2<... and 0<eij for all i&comma;j.

• 

The shiftless decomposition is unique up to reordering and multiplication by units. The gi are ordered by ascending degree in x, but the ordering within the same degree is not determined.

• 

If f is constant w.r.t. x, then the return value is f&comma;.

• 

Partial factorizations of the input are not taken into account.

Examples

withPolynomialTools&colon;

ShiftlessDecompositionexpandpochhammerx&comma;3pochhammerx&comma;5&comma;x

1&comma;x&comma;0&comma;2&comma;1&comma;2&comma;2&comma;2&comma;3&comma;1&comma;4&comma;1

(1)

ShiftlessDecompositionx61x101x151&comma;x

1&comma;x1&comma;0&comma;3&comma;2&comma;2&comma;x2x&plus;1&comma;0&comma;1&comma;1&comma;2&comma;x4&plus;x3&plus;x2&plus;x&plus;1&comma;0&comma;2&comma;x122x11&plus;2x10x9&plus;2x73x6&plus;2x5x3&plus;2x22x&plus;1&comma;0&comma;1

(2)

See Also

gcd, PolynomialTools, PolynomialTools[GreatestFactorialFactorization], PolynomialTools[ShiftEquivalent], PolynomialTools[Translate], sqrfree

References

  

Gerhard, J.; Giesbrecht, M.; Storjohann, A.; and Zima, E.V. "Shiftless decomposition and polynomial-time rational summation." Proceedings International Symposium on Symbolic and Algebraic Computation, pp. 119-126. ed. J.R. Sendra. 2003.


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