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PolynomialTools

  

GcdFreeBasis

  

compute a gcd free basis of a set or list of polynomials

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

GcdFreeBasis(S)

Parameters

S

-

set or list of polynomials

Description

• 

The GcdFreeBasis command uses repeated gcd computations to compute a gcd free basis B of the polynomials in S. B satisfies the following properties.

– 

Each polynomial in S can be written as a constant times a product of polynomials from B.

– 

The polynomials in B are pairwise coprime, that is, their gcd is constant.

– 

With respect to cardinality, B is minimal with these properties. (This is equivalent to saying that B can be computed using gcds and divisions only, but not factorization.)

• 

The gcd free basis is unique up to ordering and multiplication of the basis elements by constants.

• 

GcdFreeBasis can handle the same types of coefficients as the Maple function gcd.

• 

If S is a set, then the output is a set as well. If S is a list, then the output is also a list. The ordering of the elements in the result is not determined in either case.

• 

Zero polynomials and constants are ignored. In particular, for a constant const, GcdFreeBasis([const]) returns [ ]. The empty set or list is a valid input and is returned unchanged.

• 

A polynomial f is considered constant by GcdFreeBasis if degreef returns 0.

Examples

withPolynomialTools:

GcdFreeBasisabcd,abce,abde

c,d,e,ab

(1)

GcdFreeBasisx61,x101,x151

x1,x+1,x2x+1,x2+x+1,x4x3+x2x+1,x4+x3+x2+x+1,x8x7+x5x4+x3x+1

(2)

References

  

Bach, Eric.; Driscoll, James.; and Shallit, Jeffrey. "Factor Refinement." Journal of Algorithms Vol. 15(2), (1993): 199-222.

See Also

factors

gcd

PolynomialTools

sqrfree

 


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