Determinantal Rings Associated With Symmetric Matrices

Author : Abeer Al Ahmadieh
Publisher :
ISBN 13 :
Total Pages : 0 pages
Book Rating : 4.:/5 (137 download)

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Download or read book Determinantal Representations and the Image of the Principal Minor Map written by Abeer Al Ahmadieh and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Research in algebraic geometry has interfaces with other fields, such as matrix theory, combinatorics, and convex geometry. It is a branch of mathematics that studies solution to systems of polynomial equations and inequalities. This dissertation consists of three projects, all of which use techniques from matrix theory, convex geometry and symbolic computation to approach problems in algebraic geometry. In the first chapter we introduce some of the necessary background in classical, convex and real algebraic geometry. We also introduce the principal minor problem and its applications. In the second chapter we study the image of the principal minor map of symmetric matrices. The \textit{principal minor map} is the map that assigns to each $n\times n$ matrix the $2^n$-vector of its principal minors. By exploiting a connection with symmetric determinantal representations, we characterize the image of the subspace of symmetric matrices through the condition that certain polynomials coming from the so-called Rayleigh differences are squares in the polynomial ring over any unique factorization domain $R$. In almost all cases, one can characterize this image using the orbit of Cayley's hyperdeterminant under the action of $(\SL_2(R))^{n} \rtimes S_{n}$. Over $\C$, this recovers a characterization of Oeding from 2011, and over $\R$, the orbit of a single additional quadratic inequality suffices to cut out the image. In the third chapter 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 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 $({\rm SL}_2(\mathbb{R}))^{n} \rtimes S_{n}$. 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 over any field $\mathbb{F}$, there is no finite set of equations whose orbit under $({\rm SL}_2(\mathbb{F}))^{n} \rtimes S_{n}$ cuts out the image of $n\times n$ matrices under the principal minor map for every $n$. In the fourth chapter we study the variety of the space of complete quadrics. It is the space of nondegenerate quadrics, representing nonsingular quadrics, in addition to the so-called degenerate quadrics. We aim at generalizing the space of complete quadrics associated to any hyperbolic polynomial. To any homogeneous polynomial $h$ we naturally associate a variety $\Omega_h$ which maps birationally onto the graph of the gradient map $\nabla h$ and which agrees with the space of complete quadrics when $h$ is the determinant of a generic symmetric matrix. We give a sufficient criterion for $\Omega_h$ being smooth which applies, for example, when $h$ is an elementary symmetric polynomial. In this case $\Omega_h$ is a smooth toric variety associated to a certain generalized permutohedron. We also give examples when $\Omega_h$ is not smooth.