IsTransverse - Maple Help

RegularChains[AlgebraicGeometryTools]

 IsTransverse
 check whether an hypersurface and a space curve meet transversely

 Calling Sequence IsTransverse(rc,f, F, R)

Parameters

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

Description

 • 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](..).

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$$\mathrm{with}\left(\mathrm{ChainTools}\right):$$\mathrm{with}\left(\mathrm{AlgebraicGeometryTools}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[x,y,z\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)
 > $F≔\left[x,x-{y}^{2}-{z}^{2},y-{z}^{3}\right]$
 ${F}{≔}\left[{x}{,}{-}{{y}}^{{2}}{-}{{z}}^{{2}}{+}{x}{,}{-}{{z}}^{{3}}{+}{y}\right]$ (2)
 > $\mathrm{dec}≔\mathrm{Triangularize}\left(F,R\right)$
 ${\mathrm{dec}}{≔}\left[{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}\right]$ (3)
 > $\mathrm{Display}\left(\mathrm{dec},R\right)$
 $\left[\left\{\begin{array}{cc}{x}{=}{0}& {}\\ {y}{-}{{z}}^{{3}}{=}{0}& {}\\ {{z}}^{{4}}{+}{1}{=}{0}& {}\end{array}\right\{,}\left\{\begin{array}{cc}{x}{=}{0}& {}\\ {y}{=}{0}& {}\\ {z}{=}{0}& {}\end{array}\right\\right]$ (4)
 > $\mathrm{IsTransverse}\left(\mathrm{dec}\left[1\right],F\left[3\right],F\left[1..2\right],R\right)$
 ${\mathrm{true}}$ (5)
 > $\mathrm{IsTransverse}\left(\mathrm{dec}\left[2\right],F\left[3\right],F\left[1..2\right],R\right)$
 ${\mathrm{true}}$ (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][IsTransverse] command was introduced in Maple 2020.