An algebraic approach to circle packing

We show that for certain triangulations of surfaces, circle packings realising the triangulation can be found by solving a system of polynomial equations. We also present a similar system of equations for unbranched circle packings. The variables in these equations are associated to corners of triangles in the complex, with equations for interior vertices, edges, faces, and generators of first homology. The vertex equations are generalisations of the Descartes circle theorem, of higher degree but more symmetric than those previously found by the authors. We also provide some connections between the spinorial approach of previous work of the authors, and classical Euclidean geometry.