The function call SparsePseudoRemainder(p, rc, R, out_h) returns a polynomial $r$ such that $hp$ equals $r$ modulo the ideal generate by rc where $h$ is a product of the initials of rc. Moreover, the returned polynomial $r$ is reduced with respect to rc.

If out_h is provided then it is assigned $h$.

It is assumed that pseudo-division of p by the successive polynomials of rc sorted by decreasing order of main variable.

This command is part of the RegularChains package, so it can be used in the form SparsePseudoRemainder(..) only after executing the command with(RegularChains).  However, it can always be accessed through the long form of the command by using RegularChains[SparsePseudoRemainder](..).

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

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

$R≔\mathrm{polynomial\_ring}$

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

$\mathrm{rc}≔\mathrm{regular\_chain}$

$p≔z^3$

$p≔z^3$

$\mathrm{SparsePseudoRemainder}\left(p\,\mathrm{rc}\,R\right)$

$-z$

$\mathrm{rc}≔\mathrm{Construct}\left(z{y}^2+1\,\mathrm{rc}\,R\right)\left[1\right]$

$\mathrm{rc}≔\mathrm{regular\_chain}$

$\mathrm{SparsePseudoRemainder}\left(y^2+1\,\mathrm{rc}\,R\right)$

$z-1$