Simplify a polynomial system in the local ring of a point
regular chain of R
list of polynomials of R
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](..).
R ≔ PolynomialRing⁡z,y,x
F ≔ x2+y+z−1,x+y2+z−1,z+y+z2−1
dec ≔ Triangularize⁡F,R
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.
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