Entrywise Logarithmic Matrix Algebra and Dichotomy of Planar Graph Homomorphisms (Part I)

We prove a complexity classification of counting planar graph homomorphisms with non-negative weights. For a real symmetric matrix $M$ with non-negative entries, the problem $\text{PlGH}(M)$ is either (1) P-time computable over all graphs, or (2) #P-hard in general but P-time computable over planar graphs, or (3) #P-hard over planar graphs. Furthermore, $\text{PlGH}(M)$ in (2) consists of precisely those that involve the P-time FKT algorithm to count planar perfect matchings with a holographic transformation. The dichotomy is achieved by forming a (centered) logarithmic matrix algebra (a vector space with bilinear multiplication) by taking entrywise logarithms of all realizable matrices from $M$ using planar edge gadgets and polynomial interpolation. The current version is part I, which contains the proof for the dichotomy of entrywise positive and positive definite matrices, which is at the core of the dichotomy for non-negative matrices. Part II contains the extension from entrywise positive and positive definite matrices to non-negative matrices.