# implicitEq -- computes the gcd of the right-most map of the Z-complex in degree nu)

## Synopsis

• Usage:
iEq = implicitEq(polynomialList,nu,P)
• Inputs:
• polynomialList, a list, with polynomials {f0,f1,f2,f3}
• nu, an integer
• Outputs:
• G,

## 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)

implicitEq({f0,f1,f2,f3},nu,P) computes the determinant of the maximal minors of representationMatrix({f0,f1,f2,f3},nu,P). Equivalently, it computes the determinant of the Z-complex in degree 'nu'. This is:

implicitEq({f0,f1,f2,f3}, nu,P)=det((Z(f0,f1,f2,f3))_nu)

We also have that: degree(implicitEq({f0,f1,f2,f3}, nu))=degreeImplicitEq({f0,f1,f2,f3}, nu,P)

 ```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 : implicitEq ({f0,f1,f2,f3},2,P) WARNING: representation matrix is not of full rank (rank target M != rank M). o9 = 1 o9 : QQ[X , X , X , X ] 0 1 2 3```

For the programmer: it computes the determinant of a maximal minor of 'representationMatrix (teToricRationalMap({f0,f1,f2,f3},P),2)' using the function 'maxMinor'