Abstract. We formulate the Secant Conjecture, which is a generalization of the Shapiro Conjecture for Grassmannians. It asserts that an intersection of Schubert varieties in a Grassmannian is transverse with all points real if the flags defining the Schubert varieties are secant along disjoint intervals of a rational normal curve. We present theoretical evidence for this conjecture as well as computational evidence obtained in over one terahertz-year of computing, and we discuss some of the phenomena we observed in our data.
Abstract. In many areas of applied mathematics and statistics, it is a fundamental problem to find the best representative of a model by optimizing an objective function. This can be done by determining critical points of the objective function restricted to the model.We compile ideas arising from numerical algebraic geometry to compute the critical points of an objective function. Our method consists of using numerical homotopy continuation and a monodromy action on the total critical space to compute all of the complex critical points of an objective function. To illustrate the relevance of our method, we apply it to the Euclidean distance function to compute ED-degrees and the likelihood function to compute maximum likelihood degrees.
We study chains of lattice ideals that are invariant under a symmetric group action. In our setting, the ambient rings for these ideals are polynomial rings which are increasing in (Krull) dimension. Thus, these chains will fail to stabilize in the traditional commutative algebra sense. However, we prove a theorem which says that "up to the action of the group", these chains locally stabilize. We also give an algorithm, which we have implemented in software, for explicitly constructing these stabilization generators for a family of Laurent toric ideals involved in applications to algebraic statistics. We close with several open problems and conjectures arising from our theoretical and computational investigations.
We study the three state toric homogeneous Markov chain model and three special cases of it, namely: (i) when the initial state parameters are constant, (ii) without self-loops, and (iii) when both cases are satisfied at the same time. Using as a key tool a directed multigraph associated to the model, the state-graph, we give a bound on the number of vertices of the polytope associated to the model which does not depend on the time. Based on our computations, we also conjecture the stabilization of the f-vector of the polytope, analyze the normality of the semigroup, give conjectural bounds on the degree of the Markov bases.
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