extended GCD of two polynomials with respect to a regular chain
ExtendedRegularGcd(p1, p2, v, rc, R)
ExtendedRegularGcd(p1, p2, v, rc, R, 'normalized'='yes')
ExtendedRegularGcd(p1, p2, v, rc, R, 'normalized'='strongly')
regular chain of R
polynomial of R
variable of R
boolean flag (optional)
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 p1⁢ai+bi⁢p2=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](..).
R ≔ PolynomialRing⁡x,y,z
rc ≔ Empty⁡R
rc ≔ Chain⁡z2−z−1,rc,R
p1 ≔ y−z3
p2 ≔ y3−z3
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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