# isGoodDegree -- verifies if the Z-complex is acyclic in the given degree

## Synopsis

• Usage:
isGoodDeg = isGoodDegree(polynomialList, nu)
• Inputs:
• polynomialList, a list, polynomialList 'f={f0,...,fn}' defining the rational map
• nu, a list, the multidegree where ti take the homogeneous strand of the map f (i.e.: R_d)
• Outputs:
• bool, , A boolean 'True' or 'False'

## Description

The list 'nu' needs to be a 'good degree' for the first parameter '{f0,...,fn}' that can be verified by doing isGoodDegree(polinomialList,nu)

isGoodDegree({f0,...,fn}, nu) verifies if the approximation complex Z associated to the polynomials given is acyclic in degree 'nu'

Given a list of polynomials {f_0,...,f_n}, the approximation complex of cycles is multigraded. This funtion verifies if the strand of degree 'nu' 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)

 `i1 : A=QQ[s,u,t,v,Degrees=>{{1,1,0},{1,1,0},{1,0,1},{1,0,1}}];` `i2 : f0=s*t*u*v+t*u^2*v; f1=s*u*v^2; f2=u*t^2*s+u^2*t^2; f3=s^2*t*v+s*t*u*v;` ```i6 : isGoodDegree ({f0,f1,f2,f3},{2,1,1}) o6 = true```

## Ways to use isGoodDegree :

• isGoodDegree(List,List)