Algebraic Matroids in Action

Abstract

In recent years, various notions of algebraic independence have emerged as a central and unifying theme in a number of areas of applied mathematics, including algebraic statistics and the rigidity theory of bar-and-joint frameworks. In each of these settings, the fundamental problem is to determine the extent to which certain unknowns depend algebraically on given data. This has, in turn, led to a resurgence of interest in algebraic matroids, which are the combinatorial formalism for algebraic (in)dependence. We give a self-contained introduction to algebraic matroids together with examples highlighting their potential application.

MSC:

• Primary 05B35

Notes

ACKNOWLEDGMENTS

The first and third authors wish to thank Franz Király for many helpful conversations during previous projects which have influenced their understanding of algebraic matroids. We also wish to thank Bernd Sturmfels and David Cox for their encouragement, Will Traves for helpful conversations, and Dustin Cartwright for comments on the Lindström valuation.

Notes

1 Jan Peter Schäfermeyer brought Pollaczek-Geiringer’s work to the attention of the framework rigidity community in 2017.

Additional information

Notes on contributors

Zvi Rosen

ZVI ROSEN is an assistant professor at Florida Atlantic University. He enjoys studying polynomial systems, particularly those arising in biology and statistics. He received a B.A. and M.A. at the University of Pennsylvania, and completed a Ph.D. under Bernd Sturmfels at the University of California, Berkeley.

Jessica Sidman

JESSICA SIDMAN loves to work on problems at the intersection of computational algebra, algebraic geometry, and combinatorics. Her recent work in rigidity theory combines aspects of these three fields, and all got started when an undergraduate doing a thesis on protein folding asked her a question about projective space. She got her Ph.D. from the University of Michigan and a B.A. in mathematics from Scripps College.

Louis Theran

LOUIS THERAN works on problems around rigidity and flexibility of frameworks, and the surrounding geometric, combinatorial and algebraic objects. He holds a B.S., M.S., and Ph.D. from the University of Massachusetts, Amherst.