Construction of the Lindstr om valuation of an algebraic matroid - PowerPoint PPT Presentation

Construction of the Lindstr¨ om valuation of an algebraic matroid Dustin Cartwright University of Tennessee, Knoxville August 3, 2017 Dustin Cartwright Construction of Lindstr¨ om valuation

Algebraic matroids Given K ⊂ L = K ( x 1 , . . . , x n ) a field extension, algebraic matroid of K ⊂ L is: ◮ Independent sets are sets I such that { x i | i ∈ I } are algebraically independent ◮ Bases are the maximal independent sets (called transcendence bases in field theory) ◮ Circuits are the minimal dependent sets ◮ Rank function rk( S ) is the transcendence degree of the extension K ⊂ K ( x i | i ∈ S ) ◮ Geometrically: If L = Frac( K [ x 1 , . . . , x n ] / I ), then rk ( S ) is the dimension of the projection defined by K [ x i | i ∈ S ] ∩ I Algebraic matroids are hard! Dustin Cartwright Construction of Lindstr¨ om valuation

Differentials: Linearizing algebraic relations ◮ If K ⊂ L = K ( x 1 , . . . , x n ) is a field extension, the vector space of differentials Ω L / K is a L -vector space whose dimension is the transcendence degree of K ⊂ L , generated by elements dx 1 , . . . , dx n ◮ Geometrically: If L = Frac( K [ x 1 , . . . , x n ] / I ), then Ω L / K is the dual to the tangent space of V ( I ) at the generic point ◮ If K has characteristic 0, the algebraic matroid of K ⊂ L is the same as the linear matroid of the differentials Ω L / K with vectors dx 1 , . . . , dx n ◮ On the other hand, if K has characteristic p , and L = Frac K [ x 1 , x 2 ] / � x p 1 − x 2 � , then: ◮ dx 1 is non-zero, dx 2 = 0, so only basis of linear matroid of Ω L / K is { 1 } ◮ Bases of the algebraic matroid of K ⊂ L are { 1 } and { 2 } Frobenius function x �→ x p is weird in characteristic p Dustin Cartwright Construction of Lindstr¨ om valuation

Tropicalization of a vector space ◮ Set-up: k field with valuation val: k × ։ Γ ⊂ R , V a k -vector space, x 1 , . . . , x n ∈ V ◮ Given: w = ( w 1 , . . . , w n ) ∈ Γ n ◮ Scale: t w 1 n x n , where t w i ∈ k , val( t w i ) = w i 1 x 1 , . . . , t w n ◮ Generate: R -submodule of V generated by t w 1 x 1 , . . . , t w n x n , where R is the valuation ring of k ◮ Reduce: tensor with R / mR to get in w ( V ), where m is the maximal ideal of R ◮ Tropicalization: Trop( V ) ∩ Γ n is the set of w ∈ Γ n such that reductions t w 1 x 1 , . . . , t w n x n ∈ in w ( V ) are all non-zero ◮ The tropicalization is equivalent to the valuated matroid of V Dustin Cartwright Construction of Lindstr¨ om valuation

Rough idea “Tropicalization” for fields: ◮ Scaling = ⇒ Frobenius ◮ Reduction = ⇒ Differentials Dustin Cartwright Construction of Lindstr¨ om valuation

“Tropicalization” of field extensions ◮ Set-up: K field of char. p , L = K ( x 1 , . . . , x n ) ◮ Given: w = ( w 1 , . . . , w n ) ∈ Z n ◮ Scale: F − w 1 x 1 , . . . , F − w n x n , where F is Frobenius: Fx = x p , in l K ( x 1 / p l , . . . , x 1 / p l ˜ L = � ) n 1 ◮ Generate: K ( F − w x ) := K ( F − w 1 x 1 , . . . , F − w n x n ) ◮ Reduce: Vector space of differentials Ω K ( F − w x ) / K generated by differentials dF − w 1 x 1 , . . . , dF − w n x n ◮ Tropicalization: Trop( L / K ) ∩ Z n is the set of w ∈ Z n such that differentials dF − w 1 x 1 , . . . , dF − w n x n are all non-zero ◮ Trop( L / K ) is the tropicalization of a unique valuated matroid, called the Lindstr¨ om valuated matroid of K ⊂ L (Bollen-Draisma-Pendavingh) Dustin Cartwright Construction of Lindstr¨ om valuation

Local and global structure If V is a k -vector space, x 1 , . . . , x n ∈ V , then Trop( V ) is ◮ Globally: The recession fan of Trop( V ) is equivalent to the linear (non-valuated) matroid of x 1 , . . . , x n ∈ V ◮ Locally: At w ∈ Γ n , the link of Trop( V ) is equivalent to the linear matroid of in w ( V ) If K ⊂ L = K ( x 1 , . . . , x n ) is a field extension, then Trop( L / K ) is ◮ Globally: The recession fan of Trop( L / K ) is equivalent to the algebraic matroid of K ⊂ L ◮ Locally: The link of w ∈ Z n is equivalent to the linear matroid of Ω K ( F − w x ) / K Dustin Cartwright Construction of Lindstr¨ om valuation

A bridge example: monomials ◮ A : a d × n integer matrix ◮ L = K ( z 1 , . . . , z d ), x i = z A 1 i · · · z A di 1 d ◮ The Lindstr¨ om valuated matroid of K ⊂ L is the same as the valuated matroid of the columns of A in Q d with the p -adic valuation ◮ Prior example of K [ x 1 , x 2 ] / � x p 1 − x 2 � is a monomial example with � � A = 1 p Dustin Cartwright Construction of Lindstr¨ om valuation

Circuits of the Lindstr¨ om matroid As before: K ⊂ L = K ( x 1 , . . . , x n ) ◮ If C is a circuit of the algebraic matroid of K ⊂ L , then { x i | i ∈ C } is a minimal dependent set, and there exists a polynomial relation f C ∈ K [ x i | i ∈ C ], unique up to scaling ◮ Write � c u x u 1 1 · · · x u n f C = n u ∈ J C where J C ⊂ Z n ≥ 0 , u i = 0 if i / ∈ C , and c u � = 0 ◮ Define: ∈ ( Z ∪ ∞ ) n � � C ( f C ) = . . . , min { val p u i | u ∈ J C } , . . . where val p is the p -adic valuation ◮ The valuated circuits of the Lindstr¨ om valuation are the vectors C ( f C ) + λ 1 as C ranges over circuits of the algebraic matroid, and λ ∈ Z Dustin Cartwright Construction of Lindstr¨ om valuation

Valuation of the Lindstr¨ om matroid As before: K ⊂ L = K ( x 1 , . . . , x n ) ◮ Let B be a basis of the algebraic matroid, meaning a maximal independent set of variables ◮ The extension K ( x i | i ∈ B ) ⊂ L can be uniquely factored as K ( x i | i ∈ B ) ⊂ K ( x i | i ∈ B ) sep ⊂ L where the first extension is separable (roughly: like characteristic 0) and the second is purely inseparable (defined by taking p th roots) ◮ The Lindstr¨ om valuation is: v ( B ) = log p [ L : K ( x i | i ∈ B ) sep ] ∈ Z ≥ 0 Dustin Cartwright Construction of Lindstr¨ om valuation

Cocircuits of the Lindstr¨ om valuation As before: K ⊂ L = K ( x 1 , . . . , x n ) ◮ Let H be a hyperplane of the algebraic matroid, meaning a maximal set with rk( H ) = rk( { 1 , . . . , n } ) − 1 ◮ Define: C co ( H ) = . . . , log p [ L : K ( x i | i ∈ H ∪ { i } ) sep ] , . . . � � ◮ The valuated cocircuits of the Lindstr¨ om valuation are: C co ( H ) + λ 1 as H ranges over the hyperplanes, and λ ∈ Z Dustin Cartwright Construction of Lindstr¨ om valuation