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 $a_i\mathrm{p1}+b_i\mathrm{p2}=g_i$ 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](..).

$\mathrm{with}\left(\mathrm{RegularChains}\right)\:$$\mathrm{with}\left(\mathrm{ChainTools}\right)\:$

$R≔\mathrm{PolynomialRing}\left(\left[x\,y\,z\right]\right)$

$R≔\mathrm{polynomial\_ring}$

$\mathrm{rc}≔\mathrm{Chain}\left(\left[z^2-z-1\right]\,\mathrm{Empty}\left(R\right)\,R\right)$

$\mathrm{rc}≔\mathrm{regular\_chain}$

$\mathrm{p1}≔{\left(y-z\right)}^3$

$\mathrm{p1}≔{\left(y-z\right)}^3$

$\mathrm{p2}≔y^3-z^3$

$\mathrm{p2}≔y^3-z^3$

$\mathrm{ExtendedNormalizedGcd}\left(\mathrm{p1}\,\mathrm{p2}\,y\,\mathrm{rc}\,R\right)$

$\left[\left[9y-9z\,6yz-9y-3z+6\,-6yz+9y-6z+12\,\mathrm{regular\_chain}\right]\right]$

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.