Abstract: We present a new algorithmic framework which utilizes tropical geometry and homotopy continuation for solving systems of polynomial equations where some of the polynomials are generic elements in linear subspaces of the polynomial ring. This approach generalizes the polyhedral homotopies by Huber and Sturmfels.
The Meeting on Algebraic Geometry and its Applications was intended as a one-day meeting that attracts people in the South-East of the US. The first two incarnations — at Georgia Tech in 2015 and at Clemson in 2016 — gathered 25-30 participants each.
MAGA had a year off in 2017 due to a large international meeting scheduled in the same year at GT, but the meeting will run in 2018. There is also a plan to continue this as a regular annual event held in April in Atlanta.
However, the world has changed a bit in the last couple of years and a lot of people associate the acronym MAGA… with something completely different. Therefore a question: what could be a new name/abbreviation?
This week’s Macaulay2 event featured presentations, Q&A, and ample practice problems on a wide variety of topics. Many thanks to the speakers and organizers for making this event so successful. Stay tuned for next week’s conference!
People in algebraic geometry have used monodromy groups for a long time—in particular, use of monodromy group to solve polynomial systems has appeared elsewhere recently (see here, and here) and is certainly implicit in earlier algorithms. In fact, the possible variations on this idea seem limitless. Our implementation allows one to compare various approaches and unifies them in a general framework.
The package currently works best for the following problem: we are given a family of square polynomial systems in variables over the complex numbers, where the parameter space is a finite-dimensional affine space, such that, for a Zariski open subset of there exists such that However, the various methods are modular enough to incorporate them into solving more general systems—see here for a great example. Understanding what users want in more general settings will help us improve the package in the future, so try it out and let us know what you think!
Abstract: The toric h-numbers of a dual hypersimplex and the Chow Betti numbers of the normal fan of a hypersimplex are the ranks of intersection cohomology and Chow cohomology respectively of the torus orbit closure of a generic point in the Grassmannian. We give explicit formulas for these numbers. We also show that similar formulas hold for the coordinator numbers of type A^* lattices.