Equations governing physico-chemical processes are usually known at microscopic spatial scales, yet one suspects that there exist equations, e.g. in the form of Partial Differential (PDEs), can explain system evolution much coarser, meso- or macroscopic length scales. Discovering those coarse-grained effective PDEs lead to considerable savings computation-intensive tasks like prediction control...

We study the interplay between geometry and partial differential equations. show how fundamental ideas we use require ability to correctly calculate dimensions of spaces associated varieties zeros symbols those This brings center analysis several classical results from algebraic geometry, including Cayley-Bacharach theorem some its variants as Serret’s theorem, Brill-Noether Restsatz theorem.