Implicitization of surfaces via geometric tropicalization

In this paper we further develop the theory of geometric tropicalization due to Hacking, Keel and Tevelev and we describe tropical methods for implicitization of surfaces. More precisely, we enrich this theory with a combinatorial formula for tropical multiplicities of regular points in arbitrary dimension and we prove a conjecture of Sturmfels and Tevelev regarding sufficient combinatorial conditions to compute tropical varieties via geometric tropicalization. Using these two results, we extend previous work of Sturmfels, Tevelev and Yu for tropical implicitization of generic surfaces, and we provide methods for approaching the non-generic cases.Comment: 20 pages, 6 figures. Mayor reorganization and exposition improved. The results on geometric tropicalization have been extended to any dimension. In particular, Conjecture 2.8 is now Theorem 2.

Tropical secant graphs of monomial curves

The first secant variety of a projective monomial curve is a threefold with an action by a one-dimensional torus. Its tropicalization is a three-dimensional fan with a one-dimensional lineality space, so the tropical threefold is represented by a balanced graph. Our main result is an explicit construction of that graph. As a consequence, we obtain algorithms to effectively compute the multidegree and Chow polytope of an arbitrary projective monomial curve. This generalizes an earlier degree formula due to Ranestad. The combinatorics underlying our construction is rather delicate, and it is based on a refinement of the theory of geometric tropicalization due to Hacking, Keel and Tevelev.Comment: 30 pages, 8 figures. Major revision of the exposition. In particular, old Sections 4 and 5 are merged into a single section. Also, added Figure 3 and discussed Chow polytopes of rational normal curves in Section

An Implicitization Challenge for Binary Factor Analysis

We use tropical geometry to compute the multidegree and Newton polytope of the hypersurface of a statistical model with two hidden and four observed binary random variables, solving an open question stated by Drton, Sturmfels and Sullivant in "Lectures on Algebraic Statistics" (Problem 7.7). The model is obtained from the undirected graphical model of the complete bipartite graph $K_{2,4}$ by marginalizing two of the six binary random variables. We present algorithms for computing the Newton polytope of its defining equation by parallel walks along the polytope and its normal fan. In this way we compute vertices of the polytope. Finally, we also compute and certify its facets by studying tangent cones of the polytope at the symmetry classes vertices. The Newton polytope has 17214912 vertices in 44938 symmetry classes and 70646 facets in 246 symmetry classes.Comment: 25 pages, 5 figures, presented at Mega 09 (Barcelona, Spain

Test power properties of within-firm estimators of ownership and board-related explanatory variables with low time variation

Corporate governance research is often limited in its ability to employ within-firm estimators, which address time-invariant endogeneity, when the variables of interest exhibit low time variation (for example, ownership and board independence). The problem is further exacerbated if data for multiple points in time needs to be hand-collected. We offer simulation-based methodological guidance to improve the statistical power of within-firm estimators in the presence of low time variation. We illustrate the usefulness of our simulation results by replicating two influential studies on ownership and board independence and extending them with a within-firm estimator. Based on widely used databases as well as a novel granular database, we document the different degrees and nature of time variation of ownership and board independence across jurisdictions and subgroups by listed status, family control and complexity of ownership structure. Researchers can use our findings to optimize the hand-collection and pre-processing of governance data and thereby increase statistical power and/or to distinguish whether lack of significance is due to low time variation as opposed to absence of a true relationship between their governance variable of interest and the respective outcome