Abraham Martín del Campo

THE SECANT CONJECTURE IN THE REAL SCHUBERT CALCULUS

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.

Degree Bounds for a Minimal Markov Basis for the Three-state Toric Homogeneous Markov Chain Model

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.

The Monotone Secant Conjecture in the Real Schubert Calculus

The monotone secant conjecture posits a rich class of polynomial systems, all of whose solutions are real. These systems come from the Schubert calculus on flag manifolds, and the monotone secant conjecture is a compelling generalization of the Shapiro conjecture for Grassmannians (theorem of Mukhin, Tarasov, and Varchenko). We present some theoretical evidence for this conjecture, as well as computational evidence obtained by 1.9 terahertz-years of computing, and we discuss some of the phenomena we observed in our data.

Galois groups of Schubert problems of lines are at least alternating

We show that the Galois group of any Schubert problem involving lines in projective space contains the alternating group. This constitutes the largest family of enu- merative problems whose Galois groups have been largely determined. Using a criterion of Vakil and a special position argument due to Schubert, our result follows from a particular inequality among Kostka numbers of two-rowed tableaux. In most cases, a combinatorial injection proves the inequality. For the remaining cases, we use the Weyl integral formula to obtain an integral formula for these Kostka numbers. This rewrites the inequality as an integral, which we estimate to establish the inequality.

Corrigendum to Finiteness theorems and algorithms for permutation invariant chains of Laurent lattice ideals J. Symb. Comput. 50 (March 2013) 314-334

We correct errors in the proof of Hillar and del Campo (2013, Theorem 19) and in the description of Hillar and del Campo (2013, Table 1). We also fix some typos and notational issues. We thank Thomas Kahle for pointing out the mistake in the table and Robert Krone for finding the gap in our original proof. We begin with a comment to avoid confusion in the proof of Hillar and del Campo (2013, Proposition 13). Let dlex be the partial ordering of [P]k (as defined in Hillar and del Campo, 2013, Equation (6)) induced by the degree lexicographic order ≤dlex and the action of the group SP . To prove Proposition 13, we used Higman’s lemma (Hillar and del Campo, 2013, Lemma 14) to demonstrate that the order dlex is a well-partial-ordering of [P]k . In the argument, we considered the alphabet = {0, 1, . . . , k} yielding the set ∗ of finite sequences of elements of . For w = (w1, . . . , wk) ∈ [P]k , we used an additive notation to define w∗ ∈ ∗ by wi := ∑