The Chow form of a reciprocal linear space Cynthia Vinzant North - PowerPoint PPT Presentation

The Chow form of a reciprocal linear space Cynthia Vinzant North Carolina State University joint work with Mario Kummer, Universit¨ at Konstanz Cynthia Vinzant The Chow form of a reciprocal linear space

Hyperbolicity and determinantal representations A polynomial f ∈ R [ x 1 , . . . , x n ] d is hyperbolic with respect to a point v if every real line through v meets V ( f ) in only real points. Cynthia Vinzant The Chow form of a reciprocal linear space

Hyperbolicity and determinantal representations A polynomial f ∈ R [ x 1 , . . . , x n ] d is hyperbolic with respect to a point v if every real line through v meets V ( f ) in only real points. Example: f = x 2 − y 2 − z 2 , v = (1 , 0 , 0) Cynthia Vinzant The Chow form of a reciprocal linear space

Hyperbolicity and determinantal representations A polynomial f ∈ R [ x 1 , . . . , x n ] d is hyperbolic with respect to a point v if every real line through v meets V ( f ) in only real points. Example: f = x 2 − y 2 − z 2 , v = (1 , 0 , 0) Example: f = det( � i x i A i ) where A 1 , . . . , A n ∈ R d × d sym and the matrix � i v i A i is positive definite Cynthia Vinzant The Chow form of a reciprocal linear space

Hyperbolicity and determinantal representations A polynomial f ∈ R [ x 1 , . . . , x n ] d is hyperbolic with respect to a point v if every real line through v meets V ( f ) in only real points. Example: f = x 2 − y 2 − z 2 , v = (1 , 0 , 0) Example: f = det( � i x i A i ) where A 1 , . . . , A n ∈ R d × d sym and the matrix � i v i A i is positive definite � � x + y z e.g. x 2 − y 2 − z 2 = det z x − y Cynthia Vinzant The Chow form of a reciprocal linear space

Hyperbolicity and determinantal representations A polynomial f ∈ R [ x 1 , . . . , x n ] d is hyperbolic with respect to a point v if every real line through v meets V ( f ) in only real points. Cynthia Vinzant The Chow form of a reciprocal linear space

Hyperbolicity and determinantal representations A polynomial f ∈ R [ x 1 , . . . , x n ] d is hyperbolic with respect to a point v if every real line through v meets V ( f ) in only real points. Theorem (Helton-Vinnikov 2007). A polynomial f ∈ R [ x 1 , x 2 , x 3 ] d is hyperbolic if and only if there exist A 1 , A 2 , A 3 ∈ R d × d sym with �� � � f = det and v i A i ≻ 0 . x i A i i i Cynthia Vinzant The Chow form of a reciprocal linear space

Chow forms: making varieties into hypersurfaces since 1937 Let X ⊂ P n − 1 be an irreducible variety of dimension d − 1. Then { L : L ⊥ intersects X } is a hypersurface in G ( d − 1 , n − 1) Cynthia Vinzant The Chow form of a reciprocal linear space

Chow forms: making varieties into hypersurfaces since 1937 Let X ⊂ P n − 1 be an irreducible variety of dimension d − 1. Then { L : L ⊥ intersects X } is a hypersurface in G ( d − 1 , n − 1) defined by a polynomial in the Pl¨ ucker coordinates on G ( d − 1 , n − 1) called the Chow form of X . Cynthia Vinzant The Chow form of a reciprocal linear space

Chow forms: making varieties into hypersurfaces since 1937 Let X ⊂ P n − 1 be an irreducible variety of dimension d − 1. Then { L : L ⊥ intersects X } is a hypersurface in G ( d − 1 , n − 1) defined by a polynomial in the Pl¨ ucker coordinates on G ( d − 1 , n − 1) called the Chow form of X . Example: X = { [ s 3 : s 2 t : st 2 : t 3 ] : [ s : t ] ∈ P 1 } Cynthia Vinzant The Chow form of a reciprocal linear space

Chow forms: making varieties into hypersurfaces since 1937 Let X ⊂ P n − 1 be an irreducible variety of dimension d − 1. Then { L : L ⊥ intersects X } is a hypersurface in G ( d − 1 , n − 1) defined by a polynomial in the Pl¨ ucker coordinates on G ( d − 1 , n − 1) called the Chow form of X . Example: X = { [ s 3 : s 2 t : st 2 : t 3 ] : [ s : t ] ∈ P 1 } L = span { a , b } ⊂ P 3 , L ⊥ ∩ X � = 0 ⇔ a 0 + a 1 t + a 2 t 2 + a 3 t 3 , b 0 + b 1 t + b 2 t 2 + b 3 t 3 have a common root Cynthia Vinzant The Chow form of a reciprocal linear space

Chow forms: making varieties into hypersurfaces since 1937 Let X ⊂ P n − 1 be an irreducible variety of dimension d − 1. Then { L : L ⊥ intersects X } is a hypersurface in G ( d − 1 , n − 1) defined by a polynomial in the Pl¨ ucker coordinates on G ( d − 1 , n − 1) called the Chow form of X . Example: X = { [ s 3 : s 2 t : st 2 : t 3 ] : [ s : t ] ∈ P 1 } L = span { a , b } ⊂ P 3 , L ⊥ ∩ X � = 0 ⇔ a 0 + a 1 t + a 2 t 2 + a 3 t 3 , b 0 + b 1 t + b 2 t 2 + b 3 t 3 have a common root The Chow form of X is the resultant of these polynomials. Cynthia Vinzant The Chow form of a reciprocal linear space

Hyperbolicity and Chow forms A real variety X ⊂ P n − 1 ( C ) of codim ( X ) = c is hyperbolic with respect to a linear space L of dim c − 1 if X ∩ L = ∅ and for all real linear spaces L ′ ⊃ L of dim( L ′ ) = c , all points X ∩ L ′ are real. Cynthia Vinzant The Chow form of a reciprocal linear space

Hyperbolicity and Chow forms A real variety X ⊂ P n − 1 ( C ) of codim ( X ) = c is hyperbolic with respect to a linear space L of dim c − 1 if X ∩ L = ∅ and for all real linear spaces L ′ ⊃ L of dim( L ′ ) = c , all points X ∩ L ′ are real. Theorem (Shamovich-Vinnikov 2015). If a curve X ⊂ P n − 1 is hyperbolic with respect to L , then its Chow form is a determinant �� � 2 ) p I ( L ⊥ ) A I ≻ 0 � det 2 ) p I ( M ) A I with I ∈ ( [ n ] I ∈ ( [ n ] for some matrices A I ∈ C D × D Herm with D = deg( X ). Cynthia Vinzant The Chow form of a reciprocal linear space

Reciprocal linear spaces Given a linear space L ∈ Gr ( d , n ), its reciprocal linear space is L − 1 = P ��� : x ∈ L ∩ ( C ∗ ) n �� x − 1 1 , . . . , x − 1 � . n Cynthia Vinzant The Chow form of a reciprocal linear space

Reciprocal linear spaces Given a linear space L ∈ Gr ( d , n ), its reciprocal linear space is L − 1 = P ��� : x ∈ L ∩ ( C ∗ ) n �� x − 1 1 , . . . , x − 1 � . n Varchenko (1995): L − 1 is hyperbolic with respect to L ⊥ . Cynthia Vinzant The Chow form of a reciprocal linear space

Reciprocal linear spaces Given a linear space L ∈ Gr ( d , n ), its reciprocal linear space is L − 1 = P ��� : x ∈ L ∩ ( C ∗ ) n �� x − 1 1 , . . . , x − 1 � . n Varchenko (1995): L − 1 is hyperbolic with respect to L ⊥ . Proudfoot-Speyer (2006): deg( L − 1 ) a matroid invariant of L � n − 1 � generically = d − 1 Cynthia Vinzant The Chow form of a reciprocal linear space

Reciprocal linear spaces Given a linear space L ∈ Gr ( d , n ), its reciprocal linear space is L − 1 = P ��� : x ∈ L ∩ ( C ∗ ) n �� x − 1 1 , . . . , x − 1 � . n Varchenko (1995): L − 1 is hyperbolic with respect to L ⊥ . Proudfoot-Speyer (2006): deg( L − 1 ) a matroid invariant of L � n − 1 � generically = d − 1 De Loera-Sturmfels-V. (2012): L − 1 ∩ ( L ⊥ + v ) are analytic centers of the bounded regions in a hyperplane arrangement. Cynthia Vinzant The Chow form of a reciprocal linear space

Example: ( d , n ) = (2 , 4) Take ℓ 0 , ℓ 1 , ℓ 2 , ℓ 3 ∈ R [ s , t ]. Then L = { [ ℓ 0 : ℓ 1 : ℓ 2 : ℓ 3 ] : [ s : t ] ∈ P 1 } ∈ G (1 , 3). Cynthia Vinzant The Chow form of a reciprocal linear space

Example: ( d , n ) = (2 , 4) Take ℓ 0 , ℓ 1 , ℓ 2 , ℓ 3 ∈ R [ s , t ]. Then L = { [ ℓ 0 : ℓ 1 : ℓ 2 : ℓ 3 ] : [ s : t ] ∈ P 1 } ∈ G (1 , 3). L intersects the coordinate hyperplanes { x i = 0 } in 4 points. Remove them and take inverses to get L − 1 = { [ 1 ℓ 0 : 1 ℓ 1 : 1 ℓ 2 : 1 ℓ 3 ] } = { [ ℓ 1 ℓ 2 ℓ 3 : ℓ 0 ℓ 2 ℓ 3 : ℓ 0 ℓ 1 ℓ 3 : ℓ 0 ℓ 1 ℓ 2 ] } . Cynthia Vinzant The Chow form of a reciprocal linear space

Example: ( d , n ) = (2 , 4) Take ℓ 0 , ℓ 1 , ℓ 2 , ℓ 3 ∈ R [ s , t ]. Then L = { [ ℓ 0 : ℓ 1 : ℓ 2 : ℓ 3 ] : [ s : t ] ∈ P 1 } ∈ G (1 , 3). L intersects the coordinate hyperplanes { x i = 0 } in 4 points. Remove them and take inverses to get L − 1 = { [ 1 ℓ 0 : 1 ℓ 1 : 1 ℓ 2 : 1 ℓ 3 ] } = { [ ℓ 1 ℓ 2 ℓ 3 : ℓ 0 ℓ 2 ℓ 3 : ℓ 0 ℓ 1 ℓ 3 : ℓ 0 ℓ 1 ℓ 2 ] } . 1.5 L − 1 is a rational cubic curve. Any plane L ′ containing L ⊥ intersects L − 1 in 3 = deg( L − 1 ) real points. - - Cynthia Vinzant The Chow form of a reciprocal linear space

Determinantal representation for L − 1 Let L ∈ G ( d − 1 , n − 1) not contained in a hyperplane { x i = 0 } . Define p ( L ) ∈ P ( � d R n ) and B = { I ∈ � [ n ] � : p I ( L ) � = 0 } . d Cynthia Vinzant The Chow form of a reciprocal linear space

Determinantal representation for L − 1 Let L ∈ G ( d − 1 , n − 1) not contained in a hyperplane { x i = 0 } . Define p ( L ) ∈ P ( � d R n ) and B = { I ∈ � [ n ] � : p I ( L ) � = 0 } . d Theorem (Kummer-V. 2016). The Chow form of L − 1 can be written as a determinant � � � p I ( M ) det p I ( L ) A I I ∈B for some rank-one, p.s.d. matrices A I = v I v T of size deg( L − 1 ). I Cynthia Vinzant The Chow form of a reciprocal linear space