An Invitation to Tropical Geometry Eva Maria Feichtner - PowerPoint PPT Presentation

An Invitation to Tropical Geometry Eva Maria Feichtner feichtne@igt.uni-stuttgart.de http://www.igt.uni-stuttgart.de/AbGeoTop/Feichtner/ DIAMANT/EIDMA Symposium 2007 Eva Maria Feichtner: An Invitation to Tropical Geometry ; DIAMANT/EIDMA Symposium, May 31/June 1, 2007. – p.1/30

Outline A -Discriminants ∆ A 1. 2. Tropical Geometry 3. Tropical A -Discriminants 4. The Newton Polytope of ∆ A This is joint work with Alicia Dickenstein and Bernd Sturmfels arXiv:math.AG/0510126 , J. Amer. Math. Soc., to appear. Eva Maria Feichtner: An Invitation to Tropical Geometry ; DIAMANT/EIDMA Symposium, May 31/June 1, 2007. – p.2/30

1. Discriminants: Classical Examples 1. Discriminant of a quadratic polynomial in 1 variable f ( t ) = x 2 t 2 + x 1 t + x 0 , x 2 � = 0 ∆ f = x 2 ⇐ ⇒ 1 − 4 x 2 x 0 = 0 f has a double root 2. Discriminant of a cubic polynomial in 1 variable f ( t ) = x 3 t 3 + x 2 t 2 + x 1 t + x 0 , x 3 � = 0 ⇐ ⇒ f has a double root ∆ f = 27 x 2 1 x 2 4 − 18 x 1 x 2 x 3 x 4 +4 x 1 x 3 3 +4 x 3 2 x 4 − x 2 2 x 2 3 = 0 Eva Maria Feichtner: An Invitation to Tropical Geometry ; DIAMANT/EIDMA Symposium, May 31/June 1, 2007. – p.3/30

A -Discriminants [Gelfand, Kapranov, Zelevinsky 1992] � � ∈ Z d × n , (1 , . . . , 1) ∈ row span A , a 1 , . . . , a n span Z d a 1 · · · a n A = A represents a family of hypersurfaces in ( C ∗ ) d defined by n n x j t a j = x j t a 1 j 1 t a 2 j . . . t a dj � � f A ( t ) = . 2 d j =1 j =1 A = cl { ( x 1 : . . . : x n ) ∈ CP n − 1 | f A ( t ) = 0 has a singular point in ( C ∗ ) d } X ∗ Generically, codim X ∗ A = 1 , and X ∗ A = V (∆ A ) , where ∆ A irreducible polynomial in Z [ x 1 , . . . , x n ] , the A -discriminant. Eva Maria Feichtner: An Invitation to Tropical Geometry ; DIAMANT/EIDMA Symposium, May 31/June 1, 2007. – p.4/30

A -Discriminants: Classical Examples 1. Discriminant of a quadratic polynomial in 1 variable � � f ( t ) = x 2 t 2 + x 1 t + x 0 , 1 1 1 x 2 � = 0 A = 0 1 2 ∆ A = x 2 f has a double root ⇐ ⇒ 1 − 4 x 2 x 0 = 0 2. Discriminant of a cubic polynomial in 1 variable � � f ( t ) = x 3 t 3 + x 2 t 2 + x 1 t + x 0 , x 3 � = 0 1 1 1 1 A = 0 1 2 3 f has a double root ⇐ ⇒ ∆ A = 27 x 2 1 x 2 4 − 18 x 1 x 2 x 3 x 3 +4 x 1 x 3 3 +4 x 3 2 x 4 − x 2 2 x 2 3 = 0 Eva Maria Feichtner: An Invitation to Tropical Geometry ; DIAMANT/EIDMA Symposium, May 31/June 1, 2007. – p.5/30

A -Discriminants: Classical Examples 3. Resultant of two polynomials in 1 variable n m � � x i t i , y i t i , f ( t ) = x n � = 0 , g ( t ) = y m � = 0 , i =0 i =0 ⇐ ⇒ f and g have a common root Res( f, g ) = 0 Res( f, g ) = ∆ A ∈ Z [ x 0 , . . . , x n , y 0 , . . . , y m ] for   1 1 . . . 1 0 0 . . . 0 A = 0 0 . . . 0 1 1 . . . 1     0 1 . . . n 0 1 . . . m Res( f, g ) = determinant of the Sylvester matrix Eva Maria Feichtner: An Invitation to Tropical Geometry ; DIAMANT/EIDMA Symposium, May 31/June 1, 2007. – p.6/30

A -Discriminants: More Examples 4. Discriminant of a deg 2 homogeneous polynomial in 3 variables   1 1 1 1 1 1 A = 0 1 2 0 1 0     0 0 0 1 1 2   2 x 1 x 2 x 4 ∆ A = 1 / 2 det x 2 2 x 3 x 5     x 4 x 5 2 x 6 5. Discriminant of a deg 3 homogeneous polynomial in 3 variables   1 1 1 1 1 1 1 1 1 1 A = 0 0 0 0 1 1 1 2 2 3     0 1 2 3 0 1 2 0 1 0 deg ∆ A = 12 , 2040 terms Eva Maria Feichtner: An Invitation to Tropical Geometry ; DIAMANT/EIDMA Symposium, May 31/June 1, 2007. – p.7/30

Newton Polytopes γ c x c = � � γ c x c 1 1 · · · x c n γ c ∈ C ∗ , C ⊂ Z n g = n , c ∈ C c ∈ C New( g ) = conv { c | c ∈ C } ⊆ R n Newton polytope Example: g = 27 x 2 1 x 2 4 − 18 x 1 x 2 x 3 x 4 + 4 x 1 x 3 3 + 4 x 3 2 x 4 − x 2 2 x 2 2002 3 1111 0301 Once we know New(∆ A ) , determining ∆ A is merely a linear algebra problem! 0220 1030 Eva Maria Feichtner: An Invitation to Tropical Geometry ; DIAMANT/EIDMA Symposium, May 31/June 1, 2007. – p.8/30

A -Discriminants: Our Goals Goal: Derive information on ∆ A , resp. X ∗ A , for instance deg ∆ A Newton polytope of ∆ A directly from the matrix, i.e., the point configuration A . Ansatz: Study the tropicalization of X ∗ A ! Eva Maria Feichtner: An Invitation to Tropical Geometry ; DIAMANT/EIDMA Symposium, May 31/June 1, 2007. – p.9/30

2. Tropical Geometry Tropical geometry is algebraic geometry over the tropical semiring ( R ∪ {∞} , ⊕ , ⊗ ) , x ⊕ y := min { x, y } , x ⊗ y := x + y . tropical varieties, τ algebraic varieties − → i.e. polyhedral fans Eva Maria Feichtner: An Invitation to Tropical Geometry ; DIAMANT/EIDMA Symposium, May 31/June 1, 2007. – p.10/30

Tropical Varieties – the Algebraic Approach Y ⊆ CP n − 1 irreducible variety, dim Y = r , I Y ⊆ C [ x 1 , . . . , x n ] defining prime ideal. For w ∈ R n and f = � c ∈ C γ c x c , γ c ∈ C , C ⊂ Z n , define � γ c x c in w f = initial term of f , w · c min in w ( I Y ) = � in w f | f ∈ I Y � initial ideal of I Y . τ ( Y ) = { w ∈ R n | in w ( I Y ) does not contain a monomial } tropicalization of Y τ ( Y ) is a pure r -dimensional polyhedral fan in R n , resp. TP n − 1 . Eva Maria Feichtner: An Invitation to Tropical Geometry ; DIAMANT/EIDMA Symposium, May 31/June 1, 2007. – p.11/30

Examples of Tropicalized Varieties 1. The discriminant of a cubic polynomial in 1 variable ∆ = 27 x 2 1 x 2 4 − 18 x 1 x 2 x 3 x 4 + 4 x 1 x 3 3 + 4 x 3 2 x 4 − x 2 2 x 2 3 Eva Maria Feichtner: An Invitation to Tropical Geometry ; DIAMANT/EIDMA Symposium, May 31/June 1, 2007. – p.12/30

Examples of Tropicalized Varieties 1. The discriminant of a cubic polynomial in 1 variable ∆ = 27 x 2 1 x 2 4 − 18 x 1 x 2 x 3 x 4 + 4 x 1 x 3 3 + 4 x 3 2 x 4 − x 2 2 x 2 3 in ( − 1 , − 1 , − 1 , 0) (∆) = Eva Maria Feichtner: An Invitation to Tropical Geometry ; DIAMANT/EIDMA Symposium, May 31/June 1, 2007. – p.12/30

Examples of Tropicalized Varieties 1. The discriminant of a cubic polynomial in 1 variable ∆ = 27 x 2 1 x 2 4 − 18 x 1 x 2 x 3 x 4 + 4 x 1 x 3 3 + 4 x 3 2 x 4 − x 2 2 x 2 3 in ( − 1 , − 1 , − 1 , 0) (∆) = 4 x 1 x 3 3 − x 2 2 x 2 3 Eva Maria Feichtner: An Invitation to Tropical Geometry ; DIAMANT/EIDMA Symposium, May 31/June 1, 2007. – p.12/30

Examples of Tropicalized Varieties 1. The discriminant of a cubic polynomial in 1 variable ∆ = 27 x 2 1 x 2 4 − 18 x 1 x 2 x 3 x 4 + 4 x 1 x 3 3 + 4 x 3 2 x 4 − x 2 2 x 2 3 in ( − 1 , − 1 , − 1 , 0) (∆) = 4 x 1 x 3 3 − x 2 2 x 2 ( − 1 , − 1 , − 1 , 0) ∈ τ ( X ∗ A ) 3 Eva Maria Feichtner: An Invitation to Tropical Geometry ; DIAMANT/EIDMA Symposium, May 31/June 1, 2007. – p.12/30

Examples of Tropicalized Varieties 1. The discriminant of a cubic polynomial in 1 variable ∆ = 27 x 2 1 x 2 4 − 18 x 1 x 2 x 3 x 4 + 4 x 1 x 3 3 + 4 x 3 2 x 4 − x 2 2 x 2 3 in ( − 1 , − 1 , − 1 , 0) (∆) = 4 x 1 x 3 3 − x 2 2 x 2 ( − 1 , − 1 , − 1 , 0) ∈ τ ( X ∗ A ) 3 in (1 , 0 , 1 , 0) (∆) = Eva Maria Feichtner: An Invitation to Tropical Geometry ; DIAMANT/EIDMA Symposium, May 31/June 1, 2007. – p.12/30

Examples of Tropicalized Varieties 1. The discriminant of a cubic polynomial in 1 variable ∆ = 27 x 2 1 x 2 4 − 18 x 1 x 2 x 3 x 4 + 4 x 1 x 3 3 + 4 x 3 2 x 4 − x 2 2 x 2 3 in ( − 1 , − 1 , − 1 , 0) (∆) = 4 x 1 x 3 3 − x 2 2 x 2 ( − 1 , − 1 , − 1 , 0) ∈ τ ( X ∗ A ) 3 in (1 , 0 , 1 , 0) (∆) = 4 x 3 2 x 4 Eva Maria Feichtner: An Invitation to Tropical Geometry ; DIAMANT/EIDMA Symposium, May 31/June 1, 2007. – p.12/30

Examples of Tropicalized Varieties 1. The discriminant of a cubic polynomial in 1 variable ∆ = 27 x 2 1 x 2 4 − 18 x 1 x 2 x 3 x 4 + 4 x 1 x 3 3 + 4 x 3 2 x 4 − x 2 2 x 2 3 in ( − 1 , − 1 , − 1 , 0) (∆) = 4 x 1 x 3 3 − x 2 2 x 2 ( − 1 , − 1 , − 1 , 0) ∈ τ ( X ∗ A ) 3 in (1 , 0 , 1 , 0) (∆) = 4 x 3 (1 , 0 , 1 , 0) �∈ τ ( X ∗ 2 x 4 A ) Eva Maria Feichtner: An Invitation to Tropical Geometry ; DIAMANT/EIDMA Symposium, May 31/June 1, 2007. – p.12/30

Examples of Tropicalized Varieties 1. The discriminant of a cubic polynomial in 1 variable ∆ = 27 x 2 1 x 2 4 − 18 x 1 x 2 x 3 x 4 + 4 x 1 x 3 3 + 4 x 3 2 x 4 − x 2 2 x 2 3 in ( − 1 , − 1 , − 1 , 0) (∆) = 4 x 1 x 3 3 − x 2 2 x 2 ( − 1 , − 1 , − 1 , 0) ∈ τ ( X ∗ A ) 3 in (1 , 0 , 1 , 0) (∆) = 4 x 3 (1 , 0 , 1 , 0) �∈ τ ( X ∗ 2 x 4 A ) 2002 0301 0220 1030 Eva Maria Feichtner: An Invitation to Tropical Geometry ; DIAMANT/EIDMA Symposium, May 31/June 1, 2007. – p.12/30

Examples of Tropicalized Varieties 2. Y hypersurface in CP n − 1 f ∈ C [ x 1 , . . . , x n ] irreducible polynomial defining Y New( f ) Newton polytope, N New( f ) its normal fan τ ( Y ) = codim 1 -skeleton of N New( f ) Proof: { w ∈ R n | in w ( f ) is not a monomial } τ ( Y ) = { w ∈ R n | dim � � > 0 } = New(in w ( f )) { w ∈ R n | dim � � > 0 } = w -minimal face of New( f ) � = σ σ ∈N New( f ) codim σ> 0 Eva Maria Feichtner: An Invitation to Tropical Geometry ; DIAMANT/EIDMA Symposium, May 31/June 1, 2007. – p.13/30