Cylindrify - Maple Help

RegularChains[AlgebraicGeometryTools]

 Cylindrify
 Simplify a polynomial system in the local ring of a point

 Calling Sequence Cylindrify(rc,F, R)

Parameters

 R - polynomial ring rc - regular chain of R F - list of polynomials of R

Description

 • 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:
 1 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
 2 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](..).

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$$\mathrm{with}\left(\mathrm{ChainTools}\right):$$\mathrm{with}\left(\mathrm{AlgebraicGeometryTools}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[z,y,x\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)
 > $F≔\left[{x}^{2}+y+z-1,x+{y}^{2}+z-1,z+y+{z}^{2}-1\right]$
 ${F}{≔}\left[{{x}}^{{2}}{+}{y}{+}{z}{-}{1}{,}{{y}}^{{2}}{+}{x}{+}{z}{-}{1}{,}{{z}}^{{2}}{+}{y}{+}{z}{-}{1}\right]$ (2)
 > $\mathrm{dec}≔\mathrm{Triangularize}\left(F,R\right)$
 ${\mathrm{dec}}{≔}\left[{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}\right]$ (3)
 > $\mathrm{Display}\left(\mathrm{dec},R\right)$
 $\left[\left\{\begin{array}{cc}{z}{-}{x}{=}{0}& {}\\ {y}{-}{x}{=}{0}& {}\\ {{x}}^{{2}}{+}{2}{}{x}{-}{1}{=}{0}& {}\end{array}\right\{,}\left\{\begin{array}{cc}{z}{+}{2}{=}{0}& {}\\ {y}{+}{1}{=}{0}& {}\\ {x}{-}{2}{=}{0}& {}\end{array}\right\{,}\left\{\begin{array}{cc}{z}{=}{0}& {}\\ {y}{-}{1}{=}{0}& {}\\ {x}{=}{0}& {}\end{array}\right\{,}\left\{\begin{array}{cc}{z}{-}{1}{=}{0}& {}\\ {y}{+}{1}{=}{0}& {}\\ {x}{+}{1}{=}{0}& {}\end{array}\right\{,}\left\{\begin{array}{cc}{z}{+}{1}{=}{0}& {}\\ {y}{-}{1}{=}{0}& {}\\ {x}{-}{1}{=}{0}& {}\end{array}\right\\right]$ (4)
 > $\mathrm{seq}\left(\mathrm{IsTransverse}\left(\mathrm{dec}\left[i\right],F\left[3\right],F\left[1..2\right],R\right),i=1..\mathrm{nops}\left(\mathrm{dec}\right)\right)$
 ${\mathrm{true}}{,}{\mathrm{true}}{,}{\mathrm{false}}{,}{\mathrm{true}}{,}{\mathrm{true}}$ (5)
 > $\mathrm{Cylindrify}\left(\mathrm{dec}\left[3\right],F,R\right)$
 $\left[{{x}}^{{2}}{+}{z}{+}{y}{-}{1}{,}{-}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{x}{-}{y}{,}{{x}}^{{4}}{+}{2}{}{{x}}^{{2}}{}{y}{-}{3}{}{{x}}^{{2}}{+}{{y}}^{{2}}{-}{2}{}{y}{+}{1}\right]$ (6)

References

 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.

Compatibility

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