# degreeImplicitEq -- computes the degree of det((Z)_nu)

## Synopsis

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

## 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)

degreeImplicitEq computes the degree of the gcd of the maximal minors of the matrix representation of {f0,f1,f2,f3} in degree 'nu' after homogeneization wrt P, equivalently, the degree of determinant of the Z-complex associated to {f0,f1,f2,f3} in degree 'nu'. This is:

degreeImplicitEq({f0,f1,f2,f3}, nu,P) = degree(representationMatrix(teToricRationalMap({f0,f1,f2,f3},P),nu) that computes 'deg(det((Z(f0,f1,f2,f3))_nu))'

Given a list of polynomials {f0,f1,f2,f3}, it computes the degree of the gcd of the maximal minors of the right-most map of the strand of degree 'nu'.

Precisely, 'degreeImplicitEq' returns an integer which is the degree of H^{deg(f)}.G, where H is the implicit equation, deg(f) is the degree of f={f0,f1,f2,f3}, and G is an extra factor that may appear.

 ```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 : degreeImplicitEq (l,2,P) o9 = 0```
 `i10 : S = QQ[s,t]; ` `i11 : f0 = s^2+s^3*t; ` `i12 : f1 = s^3*t^6+1; ` `i13 : f2 = s*t^2+2*s^3*t^5; ` `i14 : f3 = s^2+s^3*t^6; ` ```i15 : degreeImplicitEq ( {f0,f1,f2,f3} ,2, polynomialsToPolytope {f0,f1,f2,f3}) o15 = 17```