In this work we investigate multipartition models, the subset of log-linear models for which one can perform the classical iterative proportional scaling (IPS) algorithm to numerically compute the maximum likelihood estimate (MLE). Multipartition models include families of models such as hierarchical models and balanced, stratified staged trees. We define a sufficient condition, called the Generalized Running Intersection Property (GRIP), on the matrix representation of a multipartition model under which the classical IPS algorithm produces the exact MLE in one cycle. In this case, the MLE is a rational function of the data. Additionally we connect the GRIP to the toric fiber product and to previous results for hierarchical models and balanced, stratified staged trees. This leads to a characterization of balanced, stratified staged trees in terms of the GRIP.

We apply numerical algebraic geometry to the invariant-theoretic problem of detecting symmetries between two plane algebraic curves. We describe an efficient equality test which determines, with "probability-one", whether or not two rational maps have the same image up to Zariski closure. The application to invariant theory is based on the construction of suitable signature maps associated to a group acting linearly on the respective curves. We consider two versions of this construction: differential and joint signature maps. In our examples and computational experiments, we focus on the complex Euclidean group, and introduce an algebraic joint signature that we prove determines equivalence of curves under this action. We demonstrate that the test is efficient and use it to empirically compare the sensitivity of differential and joint signatures to noise.