IsRegular - Maple Help

RegularChains

 IsRegular
 check if a polynomial is regular modulo a regular chain

 Calling Sequence IsRegular(p, rc, R)

Parameters

 p - polynomial of R rc - regular chain of R R - polynomial ring

Description

 • The command IsRegular(in_p, in_rc, R) returns true if and only if p is regular modulo rc, that is if and only if p is not a zero-divisor modulo the saturated ideal of rc.
 • This command is part of the RegularChains package, so it can be used in the form IsRegular(..) only after executing the command with(RegularChains).  However, it can always be accessed through the long form of the command by using RegularChains[IsRegular](..).

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)
 > $T≔\mathrm{Empty}\left(R\right)$
 ${T}{≔}{\mathrm{regular_chain}}$ (2)
 > $T≔\mathrm{Chain}\left(\left[\left(z+1\right)\left(z+2\right),{y}^{2}+z,\left(x-z\right)\left(x-y\right)\right],T,R\right)$
 ${T}{≔}{\mathrm{regular_chain}}$ (3)
 > $\mathrm{Equations}\left(T,R\right)$
 $\left[{{x}}^{{2}}{+}\left({-}{y}{-}{z}\right){}{x}{+}{z}{}{y}{,}{{y}}^{{2}}{+}{z}{,}{{z}}^{{2}}{+}{3}{}{z}{+}{2}\right]$ (4)
 > $p≔\left(z+1\right)\left({x}^{3}+5\right)$
 ${p}{≔}\left({z}{+}{1}\right){}\left({{x}}^{{3}}{+}{5}\right)$ (5)
 > $\mathrm{IsRegular}\left(p,T,R\right)$
 ${\mathrm{false}}$ (6)
 > $\mathrm{regl},\mathrm{zdl}≔\mathrm{op}\left(\mathrm{Regularize}\left(p,T,R\right)\right)$
 ${\mathrm{regl}}{,}{\mathrm{zdl}}{≔}\left[{\mathrm{regular_chain}}\right]{,}\left[{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}\right]$ (7)
 > $\mathrm{map}\left(\mathrm{Equations},\mathrm{zdl},R\right)$
 $\left[\left[{x}{-}{1}{,}{y}{-}{1}{,}{z}{+}{1}\right]{,}\left[{x}{+}{1}{,}{y}{-}{1}{,}{z}{+}{1}\right]{,}\left[{x}{+}{1}{,}{y}{+}{1}{,}{z}{+}{1}\right]\right]$ (8)

The fact that the list zdl is not empty means that there are cases, modulo which, p is zero. This is clear from the definition of p and rc.

 > $q≔x+y+z$
 ${q}{≔}{x}{+}{y}{+}{z}$ (9)
 > $\mathrm{IsRegular}\left(q,T,R\right)$
 ${\mathrm{true}}$ (10)
 > $\mathrm{Regularize}\left(q,T,R\right)$
 $\left[\left[{T}\right]{,}\left[\right]\right]$ (11)
 > $\mathrm{Inverse}\left(q,T,R\right)$
 $\left[\left[\left[{-}{235}{}{x}{}{y}{}{{z}}^{{2}}{+}{94}{}{x}{}{{z}}^{{3}}{-}{515}{}{x}{}{y}{}{z}{+}{112}{}{x}{}{{z}}^{{2}}{-}{206}{}{z}{}{x}{,}{504}{,}{\mathrm{regular_chain}}\right]{,}\left[{-}{235}{}{x}{}{y}{}{{z}}^{{2}}{+}{94}{}{x}{}{{z}}^{{3}}{-}{515}{}{x}{}{y}{}{z}{+}{112}{}{x}{}{{z}}^{{2}}{-}{206}{}{z}{}{x}{,}{504}{,}{\mathrm{regular_chain}}\right]{,}\left[{-}{235}{}{x}{}{y}{}{{z}}^{{2}}{+}{94}{}{x}{}{{z}}^{{3}}{+}{470}{}{{y}}^{{2}}{}{{z}}^{{2}}{+}{282}{}{y}{}{{z}}^{{3}}{-}{188}{}{{z}}^{{4}}{-}{515}{}{x}{}{y}{}{z}{+}{112}{}{x}{}{{z}}^{{2}}{+}{1030}{}{{y}}^{{2}}{}{z}{+}{806}{}{y}{}{{z}}^{{2}}{-}{224}{}{{z}}^{{3}}{-}{206}{}{z}{}{x}{+}{412}{}{z}{}{y}{+}{412}{}{{z}}^{{2}}{,}{504}{,}{\mathrm{regular_chain}}\right]{,}\left[{-}{235}{}{x}{}{y}{}{{z}}^{{2}}{+}{94}{}{x}{}{{z}}^{{3}}{+}{470}{}{{y}}^{{2}}{}{{z}}^{{2}}{+}{282}{}{y}{}{{z}}^{{3}}{-}{188}{}{{z}}^{{4}}{-}{515}{}{x}{}{y}{}{z}{+}{112}{}{x}{}{{z}}^{{2}}{+}{1030}{}{{y}}^{{2}}{}{z}{+}{806}{}{y}{}{{z}}^{{2}}{-}{224}{}{{z}}^{{3}}{-}{206}{}{z}{}{x}{+}{412}{}{z}{}{y}{+}{412}{}{{z}}^{{2}}{,}{504}{,}{\mathrm{regular_chain}}\right]\right]{,}\left[\right]\right]$ (12)

Since q is regular with respect to T and since every variable q is algebraic with respect to T, we can compute the inverse of q modulo T.

 > $r≔z-x$
 ${r}{≔}{-}{x}{+}{z}$ (13)
 > $\mathrm{IsRegular}\left(r,T,R\right)$
 ${\mathrm{false}}$ (14)
 > $\mathrm{Regularize}\left(r,T,R\right)$
 $\left[\left[{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}\right]{,}\left[{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}\right]\right]$ (15)

For each case where r is regular modulo rc, we can compute its inverse.

 > $\mathrm{Inverse}\left(r,T,R\right)$
 $\left[\left[\left[{-1}{,}{2}{,}{\mathrm{regular_chain}}\right]{,}\left[{z}{+}{y}{,}{2}{,}{\mathrm{regular_chain}}\right]\right]{,}\left[{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}\right]\right]$ (16)