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PolynomialTools

  

ShiftlessDecomposition

  

compute a shiftless decomposition  of a univariate polynomial

 

Calling Sequence

Parameters

Description

Examples

References

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)

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.

See Also

gcd

PolynomialTools

PolynomialTools[GreatestFactorialFactorization]

PolynomialTools[ShiftEquivalent]

PolynomialTools[Translate]

sqrfree

 


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