# isGoodDegree -- verifies if the Euler Chrasteristric of the Z-complex is zero in the given degree

## Synopsis

• Usage:
isGood = isGoodDegree(polynomialList,nu)
• Inputs:
• polynomialList, a list, with polynomials {f0,f1,f2,f3}
• nu, an integer
• P, , the polytope to be considered for the toric compactification
• Outputs:
• Bool,

## Description

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```