1 Tropical ideals do not realise all Bergman fans Jan Draisma Universit¨ at Bern and Eindhoven University of Technology FPSAC, Ljubljana, Juli 2019 joint work with Felipe Rinc´ on
Tropicalisation 2 - 3 Setting: K an infinite field and v : K → R := R ∪ {∞} a non- Archimedean valuation: • v − 1 ( ∞ ) = { 0 } • v ( ab ) = v ( a ) + v ( b ) • v ( a + b ) ≥ min {v ( a ) , v ( b ) } R is a semifield with respect to ⊕ = min and ⊙ = + Tropicalising polynomials Trop : K [ x 1 , . . . , x n ] → R [ x 1 , . . . , x n ] f = � ≥ 0 c α x α �→ � α v ( c α ) ⊙ x ⊙ α = Trop( f ) α ∈ Z n Example: K = Q , v = 2 -adic, f = x 2 − 2 � Trop( f ) = (0 ⊙ x ⊙ 2 ) ⊕ 1 : x �→ min { 2 x, 1 } 1 2
Tropicalisation 3 - 4 V R ( x 1 ⊕ x 2 ⊕ 0) Tropical hypersurface h ∈ R [ x 1 , . . . , x n ] � V R ( h ) := { p ∈ R n | the min in h ( p ) is attained ≥ twice } Tropicalising an ideal I ⊆ K [ x 1 , . . . , x n ] � Trop( I ) := { Trop( f ) | f ∈ I } ⊆ R [ x 1 , . . . , x n ] [ Bieri-Groves, Einsiedler-Kapranov-Lind, Speyer-Sturmfels, Payne, D, ... ] Tropicalising a variety, fundamental theorem K algebraically closed, v nontrivial, I ⊆ K [ x 1 , . . . , x n ] , then V R (Trop( I )) := � h ∈ Trop( I ) V R ( h ) = { v ( x ) | x ∈ V K ∗ ( I ) } . Structure theorem: If V K ∗ ( I ) is irreducible of dim d , then V R (Trop( I )) is the support of a finite, balanced, weighted poly- hedral complex of dimension d .
Tropical ideals [Maclagan-Rinc´ on] 4 - 3 Proposal to axiomatise the algebra side of tropical geometry. Notation: f ∈ K [ x 1 , . . . , x n ] or f ∈ R [ x 1 , . . . , x n ] � write [ f ] x α for the coefficient of x α in f . Observation: if f, g ∈ I ⊆ K [ x 1 , . . . , x n ] with [ f ] x α = [ g ] x α , then h := f − g ∈ I has the following properties: • [ h ] x α = 0 and • for all β � = α , v ([ h ] x β ) ≥ min { v ([ f ] x β ) , v ([ g ] x β ) } with equality if the two valuations are different. Definition: an R -subsemimodule J ⊆ R [ x 1 , . . . , x n ] is a tropical ideal if x i ⊙ J ⊆ J and for all f, g ∈ J and x α with [ f ] x α = [ g ] x α there exists an h ∈ J such that [ h ] x α = ∞ and for β � = α : [ h ] x β ≥ min { [ f ] x β , [ g ] x β } with equality if distinct.
Tropical ideals [Maclagan-Rinc´ on] 5 - 3 Equivalently: J ≤d := R [ x 1 , . . . , x n ] ≤d is a tropical linear space related to a valuated matroid [Dress-Wenzel], for each d ; and x i ⊙ J ≤ d ⊆ J ≤ d +1 . Note that Trop( I ) is a tropical ideal. Key results [ Maclagan-Rinc´ on ] • tropical ideals satisfy the ascending chain condition (but are not finitely generated) and • a tropical ideal J defines a finite polyhedral complex � h ∈ J V R ( h ) equipped with weights, called its tropical variety . Motivating question: Which weighted polyhedral complexes arise in this manner? It is not known if balancedness is necessary, nor do we have a notion of prime tropical ideal.
Main result 6 - 4 Theorem [ Draisma-Rinc´ on ] The Bergman fan of U 2 , 3 ⊕ V 8 , with weight 1 on the top- dimensional fans, is not the tropical variety of any tropical ideal. • U 23 is the uniform matroid of rank 2 on 3 elements, with Bergman fan: ( R ≥ 0 e 1 ∪ R ≥ 0 e 2 ∪ R ≥ 0 e 3 ) + R (1 , 1 , 1) ⊆ R 3 . amos matroid of rank 4 on 8 elements, so its • V 8 is the V´ Bergman fan is a 4 -dimensional fan in R 8 . The direct sum of these fans is a 6-dimensional fan in R 11 , and is not the tropical variety of any tropical ideal in 11 variables.
Tensor products of matroids? [Las Vergnas] 7 - 4 K -vector V , W Given spaces and nonzero vectors v 1 , . . . , v m ∈ V and w 1 , . . . , w n ∈ W , get vectors ∈ V ⊗ W and three matroids: M on [ m ] , N on v i ⊗ w j [ n ] and P on [ m ] × [ n ] . P has the following properties: • for each i ∈ [ m ] , j �→ ( i, j ) is an iso from M to P | { i }× [ n ] ; • for each j ∈ [ n ] , i �→ ( i, j ) is an iso from N to P | [ m ] ×{ j } ; and • rk ( P ) ≥ rk ( M ) × rk ( N ) . Question: for general matroids M and N on [ m ] , [ n ] , does a tensor product P with these properties exist? Theorem [ Las Vergnas ] No, e.g., not for M = U 2 , 3 and N = V 8 .
Proof sketch of our theorem 8 - 4 Theorem (D-Rinc´ on) The Bergman fan of U 2 , 3 ⊕ V 8 , with weight 1 on the top- dimensional fans, is not the tropical variety of any tropical ideal J . • Call the first three variables x 1 , x 2 , x 3 and the last eight vari- ables y 1 , . . . , y 8 . Reduce to the case where J is homogeneous and saturated with respect to � x 1 , x 2 , x 3 , y 1 , . . . , y 8 � . • Show that M ( J 1 ) | { x 1 ,x 2 ,x 3 } = U 2 , 3 and M ( J 1 ) | { y 1 ,...,y 8 } = V 8 . • Show that M ( J 2 ) , M ( J 2 ) | { x i x j } , M ( J 2 ) | { y i y j } have ranks: � 2 + 4 + 1 � 2 + 1 � 4 + 1 � � � = 21 , ≤ = 3 , ≤ = 10 , 2 2 2 so M ( J 2 ) | { x i y j } has rank ≥ 21 − 13 = 8 = 2 · 4 . Find: M ( J 2 ) { x i y j } is a tensor product of U 2 , 3 and V 8 , a contradiction. �
Summary and outlook 9 - 4 • Not all tropical linear spaces are the tropical varieties of (How about Bergman ( V 8 ) ?) tropical ideals. • Challenge: describe properties of varieties of tropical ideals, e.g. balancedness? • Challenge: are there notions of prime tropical ideal, irre- ducible tropical variety, and tropical-algebraic matroids? Thank you!
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