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BigradedImplicit :: BigradedImplicit

BigradedImplicit -- A package for computing implicit equations of bigraded rational surfaces by means of approximation complexes

Description

BigradedImplicit is a package for computing implicit equations of bigraded rational surfaces by means of approximation complexes.

We provide a methd to compute a matrix representation (see representationMatrix) and the implicit equation (see implicitEq ) by means of the method developed in [Bot11]. As it is probably the most interesting case from a practical point of view, we restrict our computations to parametrizations of bigraded surfaces. This implementation allows to compute small examples for the better understanding of the theory developed in [Bot11].

[Bot11] "The implicit equation of a multigraded hypersurface. arXiv:1007.3437v3 . Journal of Algebra. Vol 348, Issue 1 (2011), 381-401"

Note: We require all the polynomial ring to be multigraded, i.e. the ring should be of the form QQ[s,u,t,v,Degrees=>1,1,0,1,1,0,1,0,1,1,0,1], where a+b,a,b stands for the multidegree, where the first coordinate is the sum of the degrees, and the seecond and third coordinate are the degreen in each group of variables (s,u) and (t,v) respectively.

Author

Version

This documentation describes version 1.4 of BigradedImplicit.

Source code

The source code from which this documentation is derived is in the file BigradedImplicit.m2.

Exports

  • Functions and commands
    • degreeImplicitEq -- computes the degree of det((Z)_nu)
    • implicitEq -- computes the gcd of the right-most map of the Z-complex in degree nu)
    • isGoodDegree -- verifies if the Z-complex is acyclic in the given degree
    • maxMinor -- Returns a maximal minor of the matrix of full rank.
    • representationMatrix -- computes the right-most map of the Z-complex in degree nu)
  • Symbols
    • Exact -- Strategy for functions that uses rank computation.
    • Numeric -- Strategy for functions that uses rank computation.