The function call NormalForm(p, rc, R) returns the normal form of p with respect to rc, that is, a rational polynomial $\frac{r}{h}$ such that $hp$ equals $r$ modulo the ideal generate by rc and such that $r$ is reduced with respect to rc.

For this call, the regular chain rc must be strongly normalized.

The algorithm is based on that of  SparsePseudoRemainder.

Please, refer to the paper of Boulier and Lemaire in Proc. ISSAC 2000 for detail about strongly normalized regular chains and normal forms.

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

The commands NormalFormDim0 and ReduceCoefficientsDim0 implement asymptotically fast algorithms for computing the normal form of a polynomial with respect to a zero-dimensional regular chain.

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

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

$R≔\mathrm{polynomial\_ring}$

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

$\mathrm{rc}≔\mathrm{regular\_chain}$

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

$z+1$

$\mathrm{rc}≔\mathrm{ListConstruct}\left(\left[z^2+1\,3y^2-z\right]\,\mathrm{Empty}\left(R\right)\,R\right)\left[1\right]$

$\mathrm{rc}≔\mathrm{regular\_chain}$

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

$\frac{z}{3}+1$

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

$z+3$