The RegularChains:-AlgebraicGeometryTools subpackage contains a collection of commands for manipulating algebraic curves, surfaces, and algebraic sets of higher dimension. The commands currently available mainly focus on computing the limit of a family of sets like limits of a family of secants in the case of tangent cone computation.

The command RegularChainBranches computes the Puiseux series solutions of an algebraic set (given by a regular chain of dimension one) around one of its points. A key feature of this command is the fact that it can separate the real Puiseux series solutions from those which are not real.

The computation of regular chain branches supports the computation of limits of real rational functions as well as the the computation of limit points of one-dimensional constructible sets (with the option of focusing on the real points, that is, those obtained from real Puiseux series solutions).

The command RationalFunctionLimit computes the limit (when it exists) of a real rational function (with an arbitrary number of variables) at any point which is an isolated pole of its denominator.

The command LimitPoints computes the limit points (for the Euclidean topology) of a constructible set given by the quasi-component of a one-dimensional regular chain.

The computation of limit points support the computation of the tangent cone of a space curve at one its points; this is performed by the command TangentCone.

The command TangentPlane computes the tangent plane of a hypersurface at one of its points.

Computing tangent cones and tangent planes is used to decide whether a hypersurface and a space curve meet transversely at one of their common points; this is performed by the command IsTransverse.

The command IntersectionMultiplicity returns the intersection multiplicity of a zero-dimensional algebraic set at a given point of that set.

The underlying algorithm coincides with the well-known Fulton algorithm in the case of an input polynomial system in two variables. With more than two variables, the algorithm attempts to reduce the intersection multiplicity computation to the case of two variables. This reduction is attempted by means of a procedure called cylindrification and implemented by the command Cylindrify.

The command TriangularizeWithMultiplicity returns a triangular decomposition of the algebraic set of its input system (assumed to be square and zero-dimensional) together with the multiplicity of every point of that set.

The following is a list of available commands.

Parisa Alvandi, Changbo Chen, Marc Moreno Maza "Computing the Limit Points of the Quasi-component of a Regular Chain in Dimension One." Computer Algebra in Scientific Computing (CASC), Lecture Notes in Computer Science - 8136, (2013): 30-45.

Parisa Alvandi, Masoud Ataei, Mahsa Kazemi, Marc Moreno Maza "On the Extended Hensel Construction and its application to the computation of real limit points." J. Symb. Comput. 98: 120-162 (2020)

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.