Let A be an n-by-n matrix and M(x,y,z)=zIn+xâ(A)+yâ(A), where â(A)=(A+Aâ)/2 and â(A)=(AâAâ)/(2i). The inverse numerical range problem seeks a unit vector x corresponding to a given point z of the numerical range of A satisfying z=xâAx. A kernel vector function Ο=Ο(x,y,z) of M(x,y,z) with point (x,y,z) on the curve FA(x,y,z)=detâĄ(M(x,y,z))=0 plays the role of the unit vector x for the inverse numerical range. The columns of the adjugate matrix L(x,y,z)=(Ljk(x,y,z)) of M(x,y,z) are kernel vector functions of M(x,y,z). We prove the Abel theorem on the intersections of the algebraic curves FA(x,y,z)=0 and Ljk(x,y,z)=0. A concrete numerical example is provided to verify the result using the Maple package algcurves.