extended normalized GCD of two polynomials with respect to a regular chain - Maple Help

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RegularChains[ChainTools][ExtendedNormalizedGcd] - extended normalized GCD of two polynomials with respect to a regular chain

Calling Sequence

ExtendedNormalizedGcd(p1, p2, v, rc, R)

Parameters

p1

-

polynomial of R

p2

-

polynomial of R

v

-

variable of R

rc

-

regular chain of R

R

-

polynomial ring

Description

• 

The command ExtendedNormalizedGcd(p1, p2, v, rc, R) returns a list of pairs gi,ai,bi,rci where ai, bi, gi are polynomials of R and rci is a regular chain of R.

• 

For each pair, the polynomial gi is a normalized GCD of p1 and p2 modulo the saturated ideal of rci.

• 

For each pair, the polynomials ai, bi, gi satisfy p1ai+p2bi=gi modulo the saturated ideal of rci.

• 

For each pair, the leading coefficient of the polynomial gi with respect to v is normalized (and thus regular) modulo the saturated ideal of rci.

• 

The returned regular chains rci form a triangular decomposition of rc (in the sense of Kalkbrener).

• 

The returned regular chains are strongly normalized.

• 

Comparing to ExtendedRegularGcd, the output of ExtendedNormalizedGcd will look simpler in general when rc is zero-dimensional.

• 

However, the output of ExtendedNormalizedGcd may be much larger and much more expensive to get than the one of ExtendedRegularGcd, when rc is not zero-dimensional.

• 

rc must be strongly normalized.

• 

v must be the common main variable of p1 and p2.

• 

The initials of p1 and p2 must be regular with respect to rc.

• 

This command is part of the RegularChains[ChainTools] package, so it can be used in the form ExtendedNormalizedGcd(..) only after executing the command with(RegularChains[ChainTools]).  However, it can always be accessed through the long form of the command by using RegularChains[ChainTools][ExtendedNormalizedGcd](..).

Examples

withRegularChains:withChainTools:

R:=PolynomialRingx,y,z

R:=polynomial_ring

(1)

rc:=EmptyR

rc:=regular_chain

(2)

rc:=Chainz2z1,rc,R

rc:=regular_chain

(3)

p1:=yz3

p1:=yz3

(4)

p2:=y3z3

p2:=y3z3

(5)

ExtendedNormalizedGcdp1,p2,y,rc,R

9y9z,6yz9y3z+6,6yz+9y6z+12,regular_chain

(6)

ExtendedRegularGcdp1,p2,y,rc,R

9yz49z5,3yz+3z2,3yz+6z2,regular_chain

(7)

See Also

Chain, Empty, ExtendedRegularGcd, PolynomialRing, RegularChains, RegularGcd, Regularize, RegularizeInitial

References

  

Moreno Maza, M. "On triangular decompositions of algebraic varieties" Technical Report 4/99, NAG, UK, Presented at the MEGA-2000 Conference, Bath, UK. Available at http://www.csd.uwo.ca/~moreno.


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