check whether an hypersurface and a space curve meet transversely
IsTransverse(rc,f, F, R)
regular chain of R
a polynomial of R
list of polynomials of R
The command IsTransverse(rc,f, F, R) returns true if and only if the hypersurface defined by f and the space curve defined by F meet transversely at every point defined by the zero-dimensional regular chain rc.
In other words, this command returns true if and only if the hypersurface defined by f and the tangent cone of the space curve defined by F at p intersect at p, and only at p, in a neighbourhood of p, for every point p defined by the regular chain rc.
It is assumed that rc is a zero-dimensional regular chain.
It is assumed that F generates a one-dimensional ideal and F consists of n-1 polynomials where n is the number of variables in R.
It is assumed that the hypersurface defined by f is non-singular at every point defined by rc.
This command is part of the RegularChains[AlgebraicGeometryTools] package, so it can be used in the form IsTransverse(..) 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][IsTransverse](..).
R ≔ PolynomialRing⁡x,y,z
F ≔ x,x−y2−z2,y−z3
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][IsTransverse] command was introduced in Maple 2020.
For more information on Maple 2020 changes, see Updates in Maple 2020.
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