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RegularChains

 Inverse
 inverse of a polynomial with respect to a regular chain

 Calling Sequence Inverse(p, rc, R) Inverse(p, rc, R, 'normalized'='yes')

Parameters

 R - polynomial ring rc - regular chain of R p - polynomial of R 'normalized'='yes' - boolean flag (optional)

Description

 • The function call Inverse(p, rc, R) returns a list $\left[\mathrm{inv},\mathrm{zdiv}\right]$. The list $\mathrm{inv}$  consists of pairs $\left[{q}_{i},{h}_{i},{\mathrm{rc}}_{i}\right]$ such that ${q}_{i}p$ equals ${h}_{i}$ modulo the saturated ideal of ${\mathrm{rc}}_{i}$, where ${h}_{i}$ is regular with respect to ${\mathrm{rc}}_{i}$. The list $\mathrm{zdiv}$ is a list of regular chains ${\mathrm{rc}}_{j}$ such that p is a zero-divisor modulo ${\mathrm{rc}}_{j}$. In addition, the set of all regular chains occurring in $\mathrm{inv}$ and $\mathrm{zdiv}$ is a triangular decomposition of rc. To be precise, they form a decomposition of rc in the sense of Kalkbrener.
 • If $'\mathrm{normalized}'='\mathrm{yes}'$ is passed, then the regular chain rc must be normalized. In addition, all the returned regular chains will be normalized.
 • If the regular chain rc is normalized but $'\mathrm{normalized}'='\mathrm{yes}'$ is not passed, then there is no guarantee that the returned regular chains will be normalized.
 • For zero-dimensional regular chains in prime characteristic, the commands RegularizeDim0 and NormalizePolynomialDim0 can be combined to obtain the same specification as the command Inverse  while gaining the advantages of  modular techniques and asymptotically fast polynomial arithmetic.
 • This command is part of the RegularChains package, so it can be used in the form Inverse(..) only after executing the command with(RegularChains). However, it can always be accessed through the long form of the command by using RegularChains[Inverse](..).

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$$\mathrm{with}\left(\mathrm{ChainTools}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[x,y,z\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)
 > $\mathrm{rc}≔\mathrm{Chain}\left(\left[{z}^{2}+1,{y}^{2}+z\right],\mathrm{Empty}\left(R\right),R\right)$
 ${\mathrm{rc}}{≔}{\mathrm{regular_chain}}$ (2)
 > $\mathrm{Inverse}\left(z,\mathrm{rc},R\right)$
 $\left[\left[\left[{-}{z}{,}{1}{,}{\mathrm{regular_chain}}\right]\right]{,}\left[\right]\right]$ (3)
 > $\mathrm{Inverse}\left(y,\mathrm{rc},R\right)$
 $\left[\left[\left[{z}{}{y}{,}{1}{,}{\mathrm{regular_chain}}\right]\right]{,}\left[\right]\right]$ (4)
 > $\mathrm{Inverse}\left(x,\mathrm{rc},R\right)$
 $\left[\left[\left[{1}{,}{x}{,}{\mathrm{regular_chain}}\right]\right]{,}\left[\right]\right]$ (5)
 > $p≔y-z$
 ${p}{≔}{y}{-}{z}$ (6)
 > $\mathrm{rc}≔\mathrm{Chain}\left(\left[{z}^{2}+1,{y}^{2}+1\right],\mathrm{Empty}\left(R\right),R\right)$
 ${\mathrm{rc}}{≔}{\mathrm{regular_chain}}$ (7)
 > $\mathrm{inv},\mathrm{zdiv}≔\mathrm{op}\left(\mathrm{Inverse}\left(p,\mathrm{rc},R\right)\right):$
 > $\mathrm{inv}$
 $\left[\left[{z}{,}{2}{,}{\mathrm{regular_chain}}\right]\right]$ (8)
 > $\mathrm{zdiv}$
 $\left[{\mathrm{regular_chain}}\right]$ (9)
 > $\mathrm{q1}≔\mathrm{inv}\left[1\right]\left[1\right];$$\mathrm{h1}≔\mathrm{inv}\left[1\right]\left[2\right];$$\mathrm{rc1}≔\mathrm{inv}\left[1\right]\left[3\right];$$\mathrm{Equations}\left(\mathrm{rc1},R\right)$
 ${\mathrm{q1}}{≔}{z}$
 ${\mathrm{h1}}{≔}{2}$
 ${\mathrm{rc1}}{≔}{\mathrm{regular_chain}}$
 $\left[{y}{+}{z}{,}{{z}}^{{2}}{+}{1}\right]$ (10)
 > $\mathrm{NormalForm}\left(\mathrm{q1}p-\mathrm{h1},\mathrm{rc1},R\right)$
 ${0}$ (11)
 > $\mathrm{rc2}≔\mathrm{zdiv}\left[1\right];$$\mathrm{Equations}\left(\mathrm{rc2},R\right)$
 ${\mathrm{rc2}}{≔}{\mathrm{regular_chain}}$
 $\left[{y}{-}{z}{,}{{z}}^{{2}}{+}{1}\right]$ (12)