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](..).

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

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

$R≔\mathrm{polynomial\_ring}$

$F≔\left[x\,x-y^2-z^2\,y-z^3\right]$

$F≔\left[x\,-y^2-z^2+x\,-z^3+y\right]$

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

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

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

$\left[\left\{\begin{array}{cc}x=0& \\ y-z^3=0& \\ z^4+1=0& \end{array}\right.\,\left\{\begin{array}{cc}x=0& \\ y=0& \\ z=0& \end{array}\right.\right]$

$\mathrm{IsTransverse}\left({\mathrm{dec}}_1\,F_3\,F_{1..2}\,R\right)$

$\mathrm{true}$

$\mathrm{IsTransverse}\left({\mathrm{dec}}_2\,F_3\,F_{1..2}\,R\right)$

$\mathrm{true}$

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.