The command RegularizeInitial(f, rc, R) returns a list of pairs $\left[f_i\,{\mathrm{rc}}_i\right]$ where each ${\mathrm{rc}}_i$ is a regular chain of R and each $f_i$ is a polynomial of R.

The set of all the regular chains ${\mathrm{rc}}_i$ form a decomposition of in_rc in the sense of Kalkbrener.

Each polynomial $f_i$ is either constant or its initial is regular modulo ${\mathrm{rc}}_i$.

Each polynomial $f_i$ is equal to f modulo the saturated ideal of ${\mathrm{rc}}_i$.

The function is based on Regularize.

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

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

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

$R≔\mathrm{polynomial\_ring}$

$T≔\mathrm{Empty}\left(R\right)$

$T≔\mathrm{regular\_chain}$

$T≔\mathrm{Chain}\left(\left[\left(z+1\right)\left(z+2\right)\,y^2+z\,\left(x-z\right)\left(x-y\right)\right]\,T\,R\right)$

$T≔\mathrm{regular\_chain}$

$\mathrm{rtl}≔\mathrm{RegularizeInitial}\left(\left(z+1\right)\left(x^3+5\right)\,T\,R\right)$

$\mathrm{rtl}≔\left[\left[x^{3}z+x^3+5z+5\,\mathrm{regular\_chain}\right]\,\left[5z+5\,\mathrm{regular\_chain}\right]\right]$

$$\begin{gathered} \mathbf{for}i\mathbf{to}\mathrm{nops}\left(\mathrm{rtl}\right)\mathbf{do} \\ \mathrm{Equations}\left(\mathrm{rtl}\left[i\right]\left[2\right]\,R\right); \\ \mathrm{IsRegular}\left(\mathrm{rtl}\left[i\right]\left[1\right]\,\mathrm{rtl}\left[i\right]\left[2\right]\,R\right) \\ \mathbf{end}\mathbf{do} \end{gathered}$$

$\mathrm{false}$