東華大學圖書館 |

語系: 繁體中文

說明(常見問題)

回圖書館首頁

手機版館藏查詢

登入

回首頁

切換: 標籤 | MARC模式 | ISBD

FindBook

Google Book

Amazon

博客來

Determinantal Representations and the Image of the Principal Minor Map.

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Determinantal Representations and the Image of the Principal Minor Map./
作者:	Al Ahmadieh, Abeer.
出版者:	Ann Arbor : ProQuest Dissertations & Theses, : 2022,
面頁冊數:	123 p.
附註:	Source: Dissertations Abstracts International, Volume: 84-01, Section: B.
Contained By:	Dissertations Abstracts International84-01B.
標題:	Mathematics. -
電子資源:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=29065489
ISBN:	9798837528910

Determinantal Representations and the Image of the Principal Minor Map.
Al Ahmadieh, Abeer.

Determinantal Representations and the Image of the Principal Minor Map. - Ann Arbor : ProQuest Dissertations & Theses, 2022 - 123 p.

Source: Dissertations Abstracts International, Volume: 84-01, Section: B.

Thesis (Ph.D.)--University of Washington, 2022.

This item must not be sold to any third party vendors.

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 principal minor map is the map that assigns to each n x n matrix the 2n-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 (SL2(R))n ⋊ Sn. 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 (SL2(ℝ))n ⋊ Sn. 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 \uD835\uDD3D, there is no finite set of equations whose orbit under (SL2(\uD835\uDD3D))n ⋊ Sn cuts out the image of n x 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 Ωh which maps birationally onto the graph of the gradient map ∇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 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.

ISBN: 9798837528910Subjects--Topical Terms:

515831
Mathematics.
Subjects--Index Terms:

Determinantal representations

Determinantal Representations and the Image of the Principal Minor Map.
LDR:04336nmm a2200337 4500 001 2352554
005 20221128104017.5
008 241004s2022 ||||||||||||||||| ||eng d
020 $a 9798837528910
035 $a (MiAaPQ)AAI29065489
035 $a AAI29065489
040 $a MiAaPQ $c MiAaPQ
100 1 $a Al Ahmadieh, Abeer. $3 3692190
245 1 0 $a Determinantal Representations and the Image of the Principal Minor Map.
260 1 $a Ann Arbor : $b ProQuest Dissertations & Theses, $c 2022
300 $a 123 p.
500 $a Source: Dissertations Abstracts International, Volume: 84-01, Section: B.
500 $a Advisor: Vinzant, Cynthia.
502 $a Thesis (Ph.D.)--University of Washington, 2022.
506 $a This item must not be sold to any third party vendors.
520 $a 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 principal minor map is the map that assigns to each n x n matrix the 2n-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 (SL2(R))n ⋊ Sn. 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 (SL2(ℝ))n ⋊ Sn. 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 \uD835\uDD3D, there is no finite set of equations whose orbit under (SL2(\uD835\uDD3D))n ⋊ Sn cuts out the image of n x 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 Ωh which maps birationally onto the graph of the gradient map ∇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 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.
590 $a School code: 0250.
650 4 $a Mathematics. $3 515831
650 4 $a Alternative dispute resolution. $3 3168470
653 $a Determinantal representations
653 $a Principal minor map
653 $a Algebraic geometry
690 $a 0405
690 $a 0649
710 2 $a University of Washington. $b Mathematics. $3 3173597
773 0 $t Dissertations Abstracts International $g 84-01B.
790 $a 0250
791 $a Ph.D.
792 $a 2022
793 $a English
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=29065489