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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 $\left[{g}_{i},{a}_{i},{b}_{i},{\mathrm{rc}}_{i}\right]$ 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 normalized 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 $\mathrm{p1}{a}_{i}+\mathrm{p2}{b}_{i}={g}_{i}$ modulo the saturated ideal of ${\mathrm{rc}}_{i}$.
 • For each pair, the leading coefficient of the polynomial ${g}_{i}$ with respect to v is normalized (and thus 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).
 • 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

 > $\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)
 > $\mathrm{ExtendedNormalizedGcd}\left(\mathrm{p1},\mathrm{p2},y,\mathrm{rc},R\right)$
 $\left[\left[{9}{}{y}{-}{9}{}{z}{,}{6}{}{y}{}{z}{-}{9}{}{y}{-}{3}{}{z}{+}{6}{,}{-}{6}{}{y}{}{z}{+}{9}{}{y}{-}{6}{}{z}{+}{12}{,}{\mathrm{regular_chain}}\right]\right]$ (6)
 > $\mathrm{ExtendedRegularGcd}\left(\mathrm{p1},\mathrm{p2},y,\mathrm{rc},R\right)$
 $\left[\left[{9}{}{y}{}{{z}}^{{4}}{-}{9}{}{{z}}^{{5}}{,}{3}{}{y}{}{z}{+}{3}{}{{z}}^{{2}}{,}{-}{3}{}{y}{}{z}{+}{6}{}{{z}}^{{2}}{,}{\mathrm{regular_chain}}\right]\right]$ (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.