Paramotopy

Paramotopy: Software for parameter homotopies. It has long been known in the numerical algebraic geometry community that parameter homotopies provide a powerful tool for solving parametrized polynomial systems at a large number of parameter values. Paramotopy is a new software package dedicated to the efficient, parallel processing of a large number of parameter homotopies. Paramotopy makes use of Bertini’s implementation of user-defined homotopies, paired with Boost and other libraries for the efficient handling of large numbers of parameter values and solutions. It was initially developed to solve problems from redundant kinematics - see joint work with D. Brake (North Carolina State University), A. Maciejewski (Colorado State University), and V. Putkaradze (University of Alberta). This talk will include a brief introduction to parameter homotopies, a short tutorial on Paramotopy, and details about an application from biochemistry. In particular, the number of equilibria of a biochemical reaction network depends on the choice of a number of parameter values. With D. Brake, J. Gunawardena (Harvard Medical School), B. Gyori (National University of Singapore), and K-M Nam (Swarthmore College), we have devised a number of methods for better understanding the geography of the parameter space for such problems, with Paramotopy as a key ingredient.