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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 $\left[{g}_{i},{a}_{i},{b}_{i},{\mathrm{rc}}_{i}\right]$ where ${a}_{i}$, ${b}_{i}$, ${g}_{i}$ are polynomials of R and ${\mathrm{rc}}_{i}$ is a regular chain of R.
 • For each pair, the polynomial ${g}_{i}$ is a GCD of p1 and p2 modulo the saturated ideal of ${\mathrm{rc}}_{i}$.
 • For each pair, the polynomials ${a}_{i}$, ${b}_{i}$, ${g}_{i}$ satisfy $\mathrm{p1}{a}_{i}+{b}_{i}{p}_{2}={g}_{i}$ modulo the saturated ideal of ${\mathrm{rc}}_{i}$.
 • For each pair, the leading coefficient of the polynomial ${g}_{i}$ with respect to v is regular modulo the saturated ideal of ${\mathrm{rc}}_{i}$.
 • The returned regular chains ${\mathrm{rc}}_{i}$ 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

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$$\mathrm{with}\left(\mathrm{ChainTools}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[x,y,z\right]\right)$
 ${R}{:=}{\mathrm{polynomial_ring}}$ (1)
 > $\mathrm{rc}≔\mathrm{Empty}\left(R\right)$
 ${\mathrm{rc}}{:=}{\mathrm{regular_chain}}$ (2)
 > $\mathrm{rc}≔\mathrm{Chain}\left(\left[{z}^{2}-z-1\right],\mathrm{rc},R\right)$
 ${\mathrm{rc}}{:=}{\mathrm{regular_chain}}$ (3)
 > $\mathrm{p1}≔{\left(y-z\right)}^{3}$
 ${\mathrm{p1}}{:=}{\left({y}{-}{z}\right)}^{{3}}$ (4)
 > $\mathrm{p2}≔{y}^{3}-{z}^{3}$
 ${\mathrm{p2}}{:=}{{y}}^{{3}}{-}{{z}}^{{3}}$ (5)
 > $\mathrm{ExtendedRegularGcd}\left(\mathrm{p1},\mathrm{p2},y,\mathrm{rc},R\right)$
 $\left[\left[{9}{}{y}{}{{z}}^{{4}}{-}{9}{}{{z}}^{{5}}{,}{3}{}{y}{}{z}{+}{3}{}{{z}}^{{2}}{,}{-}{3}{}{y}{}{z}{+}{6}{}{{z}}^{{2}}{,}{\mathrm{regular_chain}}\right]\right]$ (6)
 > $\mathrm{ExtendedRegularGcd}\left(\mathrm{p1},\mathrm{p2},y,\mathrm{rc},R,\mathrm{normalized}=\mathrm{strongly}\right)$
 $\left[\left[{9}{}{y}{}{{z}}^{{4}}{-}{9}{}{{z}}^{{5}}{,}{3}{}{y}{}{z}{+}{3}{}{{z}}^{{2}}{,}{-}{3}{}{y}{}{z}{+}{6}{}{{z}}^{{2}}{,}{\mathrm{regular_chain}}\right]\right]$ (7)

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.