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MatrixRepToric :: degreeImplicitEq

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



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

See also

Ways to use degreeImplicitEq :