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

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RegularChains[ExtendedRegularGcd] - extended GCD of two polynomials with respect to a regular chain

Calling Sequence

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

ExtendedRegularGcd(p1, p2, v, rc, R, 'normalized'='yes')

ExtendedRegularGcd(p1, p2, v, rc, R, 'normalized'='strongly')

Parameters

R

-

polynomial ring

rc

-

regular chain of R

p1

-

polynomial of R

p2

-

polynomial of R

v

-

variable of R

'normalized'='yes'

-

boolean flag (optional)

'normalized'='strongly'

-

boolean flag (optional)

Description

• 

The function call ExtendedRegularGcd(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 GCD of p1 and p2 modulo the saturated ideal of rci.

• 

For each pair, the polynomials ai, bi, gi satisfy p1ai+bip2=gi modulo the saturated ideal of rci.

• 

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

• 

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

• 

If 'normalized'='yes' is present, the returned regular chains are normalized.

• 

If 'normalized'='strongly' is present, the returned regular chains are strongly normalized.

• 

If 'normalized'='yes' is present, rc must be normalized.

• 

If 'normalized'='strongly' is present, 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 package, so it can be used in the form ExtendedRegularGcd(..) only after executing the command with(RegularChains). However, it can always be accessed through the long form of the command by using RegularChains[ExtendedRegularGcd](..).

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)

ExtendedRegularGcdp1,p2,y,rc,R

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

(6)

ExtendedRegularGcdp1,p2,y,rc,R,normalized=strongly

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

(7)

See Also

Chain, ChainTools, Empty, ExtendedNormalizedGcd, IsRegular, 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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