 NormalizeRegularChainDim0 - Maple Help

RegularChains[FastArithmeticTools]

 NormalizeRegularChainDim0
 normalize a zero-dimensional regular chain Calling Sequence NormalizeRegularChainDim0(rc, R) Parameters

 R - polynomial ring rc - a regular chain of R Description

 • Returns a normalized regular chain generating the same ideal as rc.
 • rc is a zero-dimensional non-empty regular chain.
 • Moreover R must have a prime characteristic $p$ such that FFT-based polynomial arithmetic can be used for this actual computation. The higher the degrees of f and rc are, the larger must be $e$ such that ${2}^{e}$ divides $p-1$.  If the degree of  f or rc is too large, then an error is raised. Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$
 > $\mathrm{with}\left(\mathrm{FastArithmeticTools}\right):$
 > $\mathrm{with}\left(\mathrm{ChainTools}\right):$
 > $\mathrm{variables}≔\left[x,y,z\right]:$$p≔957349889:$
 > $\mathrm{sys}≔\left\{5{y}^{4}-3,-20x+y-z,-{x}^{5}+{y}^{5}-3y-1\right\}:$
 > $R≔\mathrm{PolynomialRing}\left(\mathrm{variables},p\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)

We solve a system in 3 variables and 3 unknowns

 > $\mathrm{lrc}≔\mathrm{Triangularize}\left(\mathrm{sys},R\right)$
 ${\mathrm{lrc}}{≔}\left[{\mathrm{regular_chain}}\right]$ (2)

Its triangular decomposition consists of only one regular chain

 > $\mathrm{rc}≔\mathrm{lrc}\left[1\right]$
 ${\mathrm{rc}}{≔}{\mathrm{regular_chain}}$ (3)
 > $\mathrm{Equations}\left(\mathrm{rc},R\right)$
 $\left[\left({{z}}^{{12}}{+}{94127136}{}{{z}}^{{8}}{+}{691135635}{}{{z}}^{{7}}{+}{676458799}{}{{z}}^{{4}}{+}{195425386}{}{{z}}^{{3}}{+}{326553470}{}{{z}}^{{2}}{+}{574327669}\right){}{x}{+}{27352854}{}{{z}}^{{13}}{+}{673373922}{}{{z}}^{{9}}{+}{410681381}{}{{z}}^{{8}}{+}{817312291}{}{{z}}^{{5}}{+}{308837227}{}{{z}}^{{4}}{+}{32655347}{}{{z}}^{{3}}{+}{116876413}{}{z}{+}{880926729}{,}\left({{z}}^{{12}}{+}{94127136}{}{{z}}^{{8}}{+}{691135635}{}{{z}}^{{7}}{+}{676458799}{}{{z}}^{{4}}{+}{195425386}{}{{z}}^{{3}}{+}{326553470}{}{{z}}^{{2}}{+}{574327669}\right){}{y}{+}{547057079}{}{{z}}^{{13}}{+}{927802747}{}{{z}}^{{9}}{+}{821042762}{}{{z}}^{{8}}{+}{352188797}{}{{z}}^{{5}}{+}{237219820}{}{{z}}^{{4}}{+}{326553470}{}{{z}}^{{3}}{+}{805850702}{}{z}{+}{386236578}{,}{{z}}^{{20}}{+}{957349886}{}{{z}}^{{16}}{+}{944549889}{}{{z}}^{{15}}{+}{886639826}{}{{z}}^{{12}}{+}{458149889}{}{{z}}^{{11}}{+}{156173647}{}{{z}}^{{10}}{+}{568152312}{}{{z}}^{{8}}{+}{120112423}{}{{z}}^{{7}}{+}{434195336}{}{{z}}^{{6}}{+}{398220483}{}{{z}}^{{5}}{+}{536874419}{}{{z}}^{{4}}{+}{604689895}{}{{z}}^{{3}}{+}{446611758}{}{{z}}^{{2}}{+}{237311560}{}{z}{+}{665813406}\right]$ (4)

Each initial is not equal to 1, hence this regular chain is not normalized

 > $\mathrm{map}\left(\mathrm{Initial},\mathrm{Equations}\left(\mathrm{rc},R\right),R\right)$
 $\left[{{z}}^{{12}}{+}{94127136}{}{{z}}^{{8}}{+}{691135635}{}{{z}}^{{7}}{+}{676458799}{}{{z}}^{{4}}{+}{195425386}{}{{z}}^{{3}}{+}{326553470}{}{{z}}^{{2}}{+}{574327669}{,}{{z}}^{{12}}{+}{94127136}{}{{z}}^{{8}}{+}{691135635}{}{{z}}^{{7}}{+}{676458799}{}{{z}}^{{4}}{+}{195425386}{}{{z}}^{{3}}{+}{326553470}{}{{z}}^{{2}}{+}{574327669}{,}{1}\right]$ (5)

We compute here a regular chain which is normalized and which describes the same solution as the previous one

 > $\mathrm{nrc}≔\mathrm{NormalizeRegularChainDim0}\left(\mathrm{rc},R\right)$
 ${\mathrm{nrc}}{≔}{\mathrm{regular_chain}}$ (6)

We check that it is normalized

 > $\mathrm{Equations}\left(\mathrm{nrc},R\right):$$\mathrm{map}\left(\mathrm{Initial},\mathrm{Equations}\left(\mathrm{nrc},R\right),R\right)$
 $\left[{1}{,}{1}{,}{1}\right]$ (7)

We check that the two regular chains describe the set of solutions

 > $\mathrm{EqualSaturatedIdeals}\left(\mathrm{rc},\mathrm{nrc},R\right)$
 ${\mathrm{true}}$ (8)