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.

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

$\mathrm{with}\left(\mathrm{FastArithmeticTools}\right)\:$

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

$\mathrm{variables}≔\left[x\,y\,z\right]\:$$p≔957349889\:$

$\mathrm{sys}≔\left\{5y^4-3\,-20x+y-z\,-x^5+y^5-3y-1\right\}\:$

$R≔\mathrm{PolynomialRing}\left(\mathrm{variables}\,p\right)$

$R≔\mathrm{polynomial\_ring}$

$\mathrm{lrc}≔\mathrm{Triangularize}\left(\mathrm{sys}\,R\right)$

$\mathrm{lrc}≔\left[\mathrm{regular\_chain}\right]$

$\mathrm{rc}≔\mathrm{lrc}\left[1\right]$

$\mathrm{rc}≔\mathrm{regular\_chain}$

$\mathrm{Equations}\left(\mathrm{rc}\,R\right)$

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

$\mathrm{map}\left(\mathrm{Initial}\,\mathrm{Equations}\left(\mathrm{rc}\,R\right)\,R\right)$

$\left[z^{12}+94127136z^8+691135635z^7+676458799z^4+195425386z^3+326553470z^2+574327669\,z^{12}+94127136z^8+691135635z^7+676458799z^4+195425386z^3+326553470z^2+574327669\,1\right]$

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

$\mathrm{nrc}≔\mathrm{regular\_chain}$

$\mathrm{Equations}\left(\mathrm{nrc}\,R\right)\:$$\mathrm{map}\left(\mathrm{Initial}\,\mathrm{Equations}\left(\mathrm{nrc}\,R\right)\,R\right)$

$\left[1\,1\,1\right]$

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

$\mathrm{true}$