Computing Hermitian Determinantal Representations of Plane Curves Cynthia Vinzant University of Michigan 0.2 2 0.0 1 0 0.2 1 0.4 2 0.6 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2 1 0 1 2 joint with Daniel Plaumann, Rainer Sinn, and David Speyer. Cynthia Vinzant Computing Determinantal Representations
Determinantal Representations A determinantal representation of f ∈ R [ x 1 , . . . , x n ] d is � n � � f = det x i M i where M i are d × d matrices. i =1 Cynthia Vinzant Computing Determinantal Representations
Determinantal Representations A determinantal representation of f ∈ R [ x 1 , . . . , x n ] d is � n � � f = det x i M i where M i are d × d matrices. i =1 � x − y � z Ex: x 2 − y 2 − z 2 = det z x + y � � 1 � � − 1 � � 0 �� 0 0 1 = det x + y + z 0 1 0 1 1 0 Cynthia Vinzant Computing Determinantal Representations
Determinantal Representations A determinantal representation of f ∈ R [ x 1 , . . . , x n ] d is � n � � f = det x i M i where M i are d × d matrices. i =1 � x − y � z Ex: x 2 − y 2 − z 2 = det z x + y � � 1 � � − 1 � � 0 �� 0 0 1 = det x + y + z 0 1 0 1 1 0 A representation � i x i M i is definite if the M i are real symmetric or Hermitian and there is a positive definite matrix in their span, � e ∈ R n . M ( e ) = e i M i ≻ 0 for some i Cynthia Vinzant Computing Determinantal Representations
Hyperbolic Polynomials f = det( � i x i M i ) with � i e i M i ≻ 0 ⇒ f is hyperbolic with respect to e . (roots of f ( e + tx ) are real for every x ∈ R n ) 3 2 2 2 1 1 1 e e 0 0 0 1 1 1 2 3 2 2 2 1 0 1 2 3 4 2 1 0 1 2 2 1 0 1 2 a hyperbolic cubic a hyperbolic quartic a not-hyperbolic quartic Cynthia Vinzant Computing Determinantal Representations
Hyperbolic Polynomials f = det( � i x i M i ) with � i e i M i ≻ 0 ⇒ f is hyperbolic with respect to e . (roots of f ( e + tx ) are real for every x ∈ R n ) 3 2 2 2 1 1 1 e e 0 0 0 1 1 1 2 3 2 2 2 1 0 1 2 3 4 2 1 0 1 2 2 1 0 1 2 a hyperbolic cubic a hyperbolic quartic a not-hyperbolic quartic Hyperbolic plane curves consist of degree/2 nested ovals in P 2 ( R ). Cynthia Vinzant Computing Determinantal Representations
Definite Determinantal Representations Question: What hyperbolic polynomials have definite determinantal representations f = det( � i x i M i )? Cynthia Vinzant Computing Determinantal Representations
Definite Determinantal Representations Question: What hyperbolic polynomials have definite determinantal representations f = det( � i x i M i )? Related Question: What convex semialgebraic sets can be written as a slice of the cone of positive semidefinite matrices? Cynthia Vinzant Computing Determinantal Representations
Definite Determinantal Representations Question: What hyperbolic polynomials have definite determinantal representations f = det( � i x i M i )? Related Question: What convex semialgebraic sets can be written as a slice of the cone of positive semidefinite matrices? Theorem (Helton-Vinnikov 2007) If a polynomial f ∈ R [ x , y , z ] d is hyperbolic with respect to e ∈ R 3 then there exist real symmetric matrices A , B , C ∈ R d × d sym with f = det( xA + yB + zC ) and e 1 A + e 2 B + e 3 C ≻ 0 . Cynthia Vinzant Computing Determinantal Representations
Constructions Computing real symmetric determinantal representations is hard . One can use . . . o theta functions (` a la Helton and Vinnikov) o homotopy continuation (Leykin and Plaumann) Cynthia Vinzant Computing Determinantal Representations
Constructions Computing real symmetric determinantal representations is hard . One can use . . . o theta functions (` a la Helton and Vinnikov) o homotopy continuation (Leykin and Plaumann) These slow down around degree ≈ 6 , 7. Cynthia Vinzant Computing Determinantal Representations
Constructions Computing real symmetric determinantal representations is hard . One can use . . . o theta functions (` a la Helton and Vinnikov) o homotopy continuation (Leykin and Plaumann) These slow down around degree ≈ 6 , 7. Computing Hermitian determinantal representations is easier . Cynthia Vinzant Computing Determinantal Representations
Interlacing and Distinguishing Definiteness Theorem (Plaumann-V. 2013) For a Hermitian matrix of linear forms M ( x ) = � i x i M i , the matrix M ( e ) is (positive or negative) definite if and only if the top left ( d − 1) × ( d − 1) minor of M interlaces det( M ) with respect to e. Cynthia Vinzant Computing Determinantal Representations
Interlacing and Distinguishing Definiteness Theorem (Plaumann-V. 2013) For a Hermitian matrix of linear forms M ( x ) = � i x i M i , the matrix M ( e ) is (positive or negative) definite if and only if the top left ( d − 1) × ( d − 1) minor of M interlaces det( M ) with respect to e. 2 2 1 1 e e 0 0 1 1 2 2 2 1 0 1 2 2 1 0 1 2 interlacer non-interlacer Cynthia Vinzant Computing Determinantal Representations
Interlacing and Distinguishing Definiteness Theorem (Plaumann-V. 2013) For a Hermitian matrix of linear forms M ( x ) = � i x i M i , the matrix M ( e ) is (positive or negative) definite if and only if the top left ( d − 1) × ( d − 1) minor of M interlaces det( M ) with respect to e. 2 2 Example of an interlacer: 1 the directional derivative 1 e e 0 0 n ∂ f � e i 1 1 ∂ x i i =1 2 2 2 1 0 1 2 2 1 0 1 2 interlacer non-interlacer Cynthia Vinzant Computing Determinantal Representations
Interlude: adjugate matrices For a matrix M , let M adj denote its adjugate (or classical adjoint). Some observations about the matrix M adj . . . Cynthia Vinzant Computing Determinantal Representations
Interlude: adjugate matrices For a matrix M , let M adj denote its adjugate (or classical adjoint). Some observations about the matrix M adj . . . i x i M i , the entries of M adj have degree d − 1. o If M = � Cynthia Vinzant Computing Determinantal Representations
Interlude: adjugate matrices For a matrix M , let M adj denote its adjugate (or classical adjoint). Some observations about the matrix M adj . . . i x i M i , the entries of M adj have degree d − 1. o If M = � M adj · M o M · M adj = = det( M ) I . Cynthia Vinzant Computing Determinantal Representations
Interlude: adjugate matrices For a matrix M , let M adj denote its adjugate (or classical adjoint). Some observations about the matrix M adj . . . i x i M i , the entries of M adj have degree d − 1. o If M = � M adj · M o M · M adj = = det( M ) I . o M adj ( p ) has rank ≤ 1 for every point p in V (det( M )). Cynthia Vinzant Computing Determinantal Representations
Interlude: adjugate matrices For a matrix M , let M adj denote its adjugate (or classical adjoint). Some observations about the matrix M adj . . . i x i M i , the entries of M adj have degree d − 1. o If M = � M adj · M o M · M adj = = det( M ) I . o M adj ( p ) has rank ≤ 1 for every point p in V (det( M )). Idea (Dixon 1902) : Construct a d × d matrix of forms of degree d − 1 whose 2 × 2 minors lie in the ideal � f � . Cynthia Vinzant Computing Determinantal Representations
Interlude: adjugate matrices For a matrix M , let M adj denote its adjugate (or classical adjoint). Some observations about the matrix M adj . . . i x i M i , the entries of M adj have degree d − 1. o If M = � M adj · M o M · M adj = = det( M ) I . o M adj ( p ) has rank ≤ 1 for every point p in V (det( M )). Idea (Dixon 1902) : Construct a d × d matrix of forms of degree d − 1 whose 2 × 2 minors lie in the ideal � f � . o ( M adj ) 11 interlaces det( M ) ⇒ M ( e ) ≻ 0. Cynthia Vinzant Computing Determinantal Representations
Interlacers → Definite Determinantal Representations Theorem (Plaumann-V. 2013) Suppose g 1 ∈ R [ x , y , z ] d − 1 interlaces f with respect to e ∈ R 3 and split the points V ( f , g 1 ) into disjoint sets S ∪ S. 2 1 e 0 1 2 2 1 0 1 2 Cynthia Vinzant Computing Determinantal Representations
Interlacers → Definite Determinantal Representations Theorem (Plaumann-V. 2013) Suppose g 1 ∈ R [ x , y , z ] d − 1 interlaces f with respect to e ∈ R 3 and split the points V ( f , g 1 ) into disjoint sets S ∪ S. If g = ( g 1 , . . . , g d ) is a basis C [ x , y , z ] d − 1 ∩ I ( S ) , 2 1 e 0 1 2 2 1 0 1 2 Cynthia Vinzant Computing Determinantal Representations
Interlacers → Definite Determinantal Representations Theorem (Plaumann-V. 2013) Suppose g 1 ∈ R [ x , y , z ] d − 1 interlaces f with respect to e ∈ R 3 and split the points V ( f , g 1 ) into disjoint sets S ∪ S. If g = ( g 1 , . . . , g d ) is a basis C [ x , y , z ] d − 1 ∩ I ( S ) , then there is a Hermitian matrix M = xA + yB + zC with ( M adj ) (1 , · ) = g , 2 1 M ( e ) ≻ 0 , and e 0 f = det( M ) . 1 2 2 1 0 1 2 Cynthia Vinzant Computing Determinantal Representations
Algorithm (PSSV) Input: f ∈ R [ x , y , z ] d and e ∈ R 3 with f hyperbolic w.resp. to e . 2 1 e 0 1 2 2 1 0 1 2 Cynthia Vinzant Computing Determinantal Representations
Algorithm (PSSV) Input: f ∈ R [ x , y , z ] d and e ∈ R 3 with f hyperbolic w.resp. to e . 2 o Let g 1 = e 1 ∂ f ∂ x + e 2 ∂ f ∂ y + e 3 ∂ f ∂ z . 1 e 0 1 2 2 1 0 1 2 Cynthia Vinzant Computing Determinantal Representations
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