Abstract. We prove that the number of tropical critical points of an affine matroid [Formula: see text] is equal to the beta invariant of [Formula: see text]. Motivated by the computation of maximum likelihood degrees, this number is defined to be the degree of the intersection of the Bergman fan of [Formula: see text] and the inverted Bergman fan of [Formula: see text], where [Formula: see text] is an element of [Formula: see text] that is neither a loop nor a coloop. Equivalently, for a generic weight vector [Formula: see text] on [Formula: see text], this is the number of ways to find weights [Formula: see text] on [Formula: see text] and [Formula: see text] on [Formula: see text] with [Formula: see text] such that, on each circuit of [Formula: see text] (resp., [Formula: see text]), the minimum [Formula: see text]-weight (resp., [Formula: see text]-weight) occurs at least twice. This answers a question of Sturmfels.

We study the motion of a robotic arm inside a rectangular tunnel. We prove that the configuration space of all possible positions of the robot is a ${CAT}(0)$ cubical complex. This allows us to use techniques from geometric group theory to find the optimal way of moving the arm from one position to another. We also compute the diameter of the configuration space, that is, the longest distance between two positions of the robot.

Given a reconfigurable system $X$, such as a robot moving on a grid or a set of particles traversing a graph without colliding, the possible positions of $X$ naturally form a cubical complex ${\mathcal{S}}(X)$. When ${\mathcal{S}}(X)$ is a CAT(0) space, we can explicitly construct the shortest path between any two points for any of the four most natural metrics: distance, time, number of moves, and number of steps of simultaneous moves. Using Ardila, Owen, and Sullivant's result that CAT(0) cubical complexes are in correspondence with posets with inconsistent pairs (PIPs), we can prove that a state complex ${\mathcal{S}}(X)$ is CAT(0) by identifying the corresponding PIP. We illustrate this very general strategy with one known and one new example: Abrams and Ghrist's positive robotic arm on a square grid, and the robotic arm in a strip. We then use the PIP as a combinatorial “remote control" to move these robots efficiently from one position to another.

The convex hull of the roots of a classical root lattice is called a root polytope. We determine explicit unimodular triangulations of the boundaries of the root polytopes associated to the root lattices $A_n$, $C_n$, and $D_n$, and we compute their f- and h-vectors. This leads us to recover formulae for the growth series of these root lattices, which were first conjectured by Conway, Mallows, and Sloane and Baake and Grimm and were proved by Conway and Sloane and Bacher, de la Harpe, and Venkov. We also prove the formula for the growth series of the root lattice $B_n$, which requires a modification of our technique.