Research

Tropicalizing Principal Minors of Positive Definite Matrices

Abeer Al Ahmadieh, Felipe Rincón, Cynthia Vinzant, and Josephine Yu.

We study the tropicalization of the image of the cone of positive definite matrices under the principal minors map. It is a polyhedral subset of the set of M-concave functions on the discrete n-dimensional cube. We show it coincides with the intersection of the affine tropical flag variety with the submodular cone. In particular, any cell in the regular subdivision of the cube induced by a point in this tropicalization can be subdivided into base polytopes of realizable matroids. We also use this tropicalization as a guide to discover new algebraic inequalities among the principal minors of positive definite matrices of a fixed size.

Real and Positive Tropicalizations of Symmetric Determinantal Varieties

Abeer Al Ahmadieh, May Cai and Josephine Yu

We study real and positive tropicalizations of the varieties of low rank symmetric matrices over real or complex Puiseux series. We show that real tropicalization coincides with complex tropicalization for rank two and corank one cases. We also show that the two notions of positive tropicalization introduced by Speyer and Williams coincide for symmetric rank two matrices, but they differ for symmetric corank one matrices.

The Fiber of the Principal Minor Map

Abeer Al Ahmadieh

submitted

This paper explores the fibers of the principal minor map over a general field. The principal minor map is the map that assigns to each n×n matrix the 2ⁿ-vector of its principal minors. In 1984, Hartfiel and Loewy proposed a condition that was sufficient to ensure that the fiber of the principal minor map is a single point up to diagonal equivalence. Loewy later improved upon this condition in 1986. In this paper, we provide a necessary and sufficient condition for the fiber to be a point up to diagonal equivalence. Additionally, we establish a connection between the reducibility of a matrix and the reducibility of its determinantal representation. Using this connection, we fully characterize the fiber of symmetric and Hermitian matrices in the space of n×n matrices over any field F. We also use these techniques to answer a question of Borcea, Brändén, and Liggett concerning real stable matrices.

Determinantal Representation and the image of the principal minor map

Abeer Al Ahmadieh and Cynthia Vinzant

International Mathematics Research Notices Volume 2024, Issue 10, May 2024, Pages 8930–8958.

In this paper we explore determinantal representations of multiaffine polynomials and consequences for the image of various spaces of matrices under the principal minor map. We show that a real multiaffine polynomial has a definite Hermitian determinantal representation if and only if all of its so-called Rayleigh differences factor as Hermitian squares and use this characterization to conclude that the image of the space of Hermitian matrices under the principal minor map is cut out by the orbit of finitely many equations and inequalities under the action of (SL₂(R))ⁿxS_n. We also study such representations over more general fields with quadratic extensions. Factorizations of Rayleigh differences prove an effective tool for capturing subtle behavior of the principal minor map. In contrast to the Hermitian case, we give examples to show for any field F, there is no finite set of equations whose orbit under (SL₂(F))ⁿxS_n cuts out the image of n×n matrices over F under the principal minor map for every n.

Characterizing principal minors of symmetric matrices via determinantal multiaffine polynomials

Abeer Al Ahmadieh and Cynthia Vinzant

Journal of Algebra 638, (2024) pp 255-278.

Here we consider the image of the principal minor map of symmetric matrices over an arbitrary unique factorization domain R. By exploiting a connection with symmetric determinantal representations, we characterize the image of the principal minor map through the condition that certain polynomials coming from so-called Rayleigh differences are squares in the polynomial ring over R. In almost all cases, one can characterize the image of the principal minor map using the orbit of Cayley's hyperdeterminant under the action of (SL₂(R))ⁿxS_n. Over the complex numbers, this recovers a characterization of Oeding from 2011, and over the reals, the orbit of a single additional quadratic inequality suffices to cut out the image. Applications to other symmetric determinantal representations are also discussed.

A generalization of the space of complete quadrics

Abeer Al Ahmadieh , Mario Kummer , and Miruna-Stefana Sorea

Matematiche (Catania) 76 (2021), no. 2,pp 431–446.

To any homogeneous polynomial h we naturally associate a variety Ωh which maps birationally onto the graph Γh of the gradient map ∇h and which agrees with the space of complete quadrics when h is the determinant of the generic symmetric matrix. We give a sufficient criterion for Ωh being smooth which applies for example when h is an elementary symmetric polynomial. In this case Ωh is a smooth toric variety associated to a certain generalized permutohedron. We also give examples when Ωh is not smooth.