The command Cylindrify(rc,F, R) returns a list of polynomials G such that F and G have the same intersection multiplicity at every point defined by the zero-dimensional regular chain rc. Moreover, either G is F itself or there exists a variable v of R and a polynomial g of G such that:

the polynomial g has degree one in v and its leading coefficient in v is invertible in the local ring at p for every point p defined by the zero-dimensional regular chain rc; and

each other polynomial in G is independent of v.

In that latter case, the polynomial set G facilitates the study of the local properties of the zero set of F around every point solving rc.

It is assumed that F generates a zero-dimensional ideal and F consists of n polynomials where n is the number of variables in R.

It is assumed that rc is a zero-dimensional regular chain, the zero set of which is contained in that of F.

This is not a complete algorithm: in some rare cases, the command will signal an error and fail.

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

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

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

$R≔\mathrm{polynomial\_ring}$

$F≔\left[x^2+y+z-1\,x+y^2+z-1\,z+y+z^2-1\right]$

$F≔\left[x^2+y+z-1\,y^2+x+z-1\,z^2+y+z-1\right]$

$\mathrm{dec}≔\mathrm{Triangularize}\left(F\,R\right)$

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

$\mathrm{Display}\left(\mathrm{dec}\,R\right)$

$\left[\left\{\begin{array}{cc}z-x=0& \\ y-x=0& \\ x^2+2x-1=0& \end{array}\right.\,\left\{\begin{array}{cc}z+2=0& \\ y+1=0& \\ x-2=0& \end{array}\right.\,\left\{\begin{array}{cc}z=0& \\ y-1=0& \\ x=0& \end{array}\right.\,\left\{\begin{array}{cc}z-1=0& \\ y+1=0& \\ x+1=0& \end{array}\right.\,\left\{\begin{array}{cc}z+1=0& \\ y-1=0& \\ x-1=0& \end{array}\right.\right]$

$\mathrm{seq}\left(\mathrm{IsTransverse}\left(\mathrm{dec}\left[i\right]\,F\left[3\right]\,F\left[1..2\right]\,R\right)\,i=1..\mathrm{nops}\left(\mathrm{dec}\right)\right)$

$\mathrm{true},\mathrm{true},\mathrm{false},\mathrm{true},\mathrm{true}$

$\mathrm{Cylindrify}\left(\mathrm{dec}\left[3\right]\,F\,R\right)$

$\left[x^2+z+y-1\,-x^2+y^2+x-y\,x^4+2x^{2}y-3x^2+y^2-2y+1\right]$

Steffen Marcus, Marc Moreno Maza, Paul Vrbik "On Fulton's Algorithm for Computing Intersection Multiplicities." Computer Algebra in Scientific Computing (CASC), Lecture Notes in Computer Science - 7442, (2012): 198-211.

Parisa Alvandi, Marc Moreno Maza, Eric Schost, Paul Vrbik "A Standard Basis Free Algorithm for Computing the Tangent Cones of a Space Curve." Computer Algebra in Scientific Computing (CASC), Lecture Notes in Computer Science - 9301, (2015): 45-60.

The RegularChains[AlgebraicGeometryTools][Cylindrify] command was introduced in Maple 2020.

For more information on Maple 2020 changes, see Updates in Maple 2020.