Robert Krone: The tropical Cayley-Menger variety
Abstract
Abstract: The Cayley-Menger variety is the Zariski closure of the set
of vectors specifying the pairwise squared distances between n points in R^d.
For a graph on n vertices, a coordinate projection of the Cayley-Menger variety
gives the possible edge lengths of the embeddings of the graph into R^d.
Tropicalization converts an algebraic set into a polyhedral complex, the
"combinatorial shadow" of the original variety. When d=2, the tropical
Cayley-Menger variety is the set of sums of two ultrametrics on n leaves.
We can describe its polyhedral structure in terms of pairs of rooted trees.
This description leads to a new, tropical, proof of Laman's theorem, which is a
characterization of the minimal generically rigid graphs in R^2.
This is joint work with Daniel Bernstein.