- Usage:
`isGood = isGoodDegree(polynomialList,nu)`

- Inputs:
`polynomialList`, a list, with polynomials {f0,f1,f2,f3}`nu`, an integer`P`, a convex polyhedron, the polytope to be considered for the toric compactification

- Outputs:
`Bool`, a Boolean value

The integer nu needs to be a good degree for the first parameter {f0,f1,f2,f3} that can be verified by doing isGoodDegree(polinomialList,nu,P).

`isGoodDegree` verifies if the approximation complex Z associated to the homogeneous polynomials (wrt P) given is acyclic in degree nu.

Given a list of polynomials {f0,f1,f2,f3}, the method homoneizes the polynomials wrt P. Now, the approximation complex of cycles Z is bigraded. This funtion verifies if the strand of degree 'nu' of Z is acyclic.

Precisely, it computes the Euler characteristic of the nu-strand of the Z-complex, by computing the alternate sum of (-1)^i * hilbertFunction(nu+i*d,Z_i).

Note: the polynomials must be the inhomogeneous polynomials defining the affine parametrization.

i1 : needsPackage "Polyhedra" o1 = Polyhedra o1 : Package |

i2 : S = QQ[s,t]; |

i3 : f0 = s^2+s^3*t; |

i4 : f1 = s^3*t^6+1; |

i5 : f2 = s*t^2+2*s^3*t^5; |

i6 : f3 = s^2+s^3*t^6; |

i7 : l = {f0,f1,f2,f3}; |

i8 : P = convexHull(matrix{{0,0,1},{0,1,0}}); |

i9 : isGoodDegree (l,2,P) o9 = false |

- Polyhedra -- for computations with convex polyhedra, cones, and fans

- isGoodDegree(List,ZZ) -- verifies if the Euler Chrasteristric of the Z-complex is zero in the given degree
- isGoodDegree(Matrix,ZZ) -- verifies if the Euler Chrasteristric of the Z-complex is zero in the given degree