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

Construction of the Lindstr¨

• m valuation of an

algebraic matroid

Dustin Cartwright

University of Tennessee, Knoxville

August 3, 2017

Dustin Cartwright Construction of Lindstr¨

• m valuation

Algebraic matroids

Given K ⊂ L = K(x1, . . . , xn) a field extension, algebraic matroid of K ⊂ L is:

◮ Independent sets are sets I such that {xi | 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(xi | i ∈ S)

◮ Geometrically: If L = Frac(K[x1, . . . , xn]/I), then rk(S) is the

dimension of the projection defined by K[xi | i ∈ S] ∩ I Algebraic matroids are hard!

Dustin Cartwright Construction of Lindstr¨

• m valuation

Differentials: Linearizing algebraic relations

◮ If K ⊂ L = K(x1, . . . , xn) is a field extension, the vector space

• f differentials ΩL/K is a L-vector space whose dimension is the

transcendence degree of K ⊂ L, generated by elements dx1, . . . , dxn

◮ Geometrically: If L = Frac(K[x1, . . . , xn]/I), then ΩL/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 ΩL/K with vectors dx1, . . . , dxn

◮ On the other hand, if K has characteristic p, and

L = Frac K[x1, x2]/xp

1 − x2, then:

◮ dx1 is non-zero, dx2 = 0, so only basis of linear matroid of

ΩL/K is {1}

◮ Bases of the algebraic matroid of K ⊂ L are {1} and {2}

Frobenius function x → xp is weird in characteristic p

Dustin Cartwright Construction of Lindstr¨

• m valuation

Tropicalization of a vector space

◮ Set-up: k field with valuation val: k× ։ Γ ⊂ R, V a k-vector

space, x1, . . . , xn ∈ V

◮ Given: w = (w1, . . . , wn) ∈ Γn ◮ Scale: tw1 1 x1, . . . , twn n xn, where twi ∈ k, val(twi) = wi ◮ Generate: R-submodule of V generated by tw1x1, . . . , twnxn,

where R is the valuation ring of k

◮ Reduce: tensor with R/mR to get inw(V ), where m is the

maximal ideal of R

◮ Tropicalization: Trop(V ) ∩ Γn is the set of w ∈ Γn such that

reductions tw1x1, . . . , twnxn ∈ inw(V ) are all non-zero

◮ The tropicalization is equivalent to the valuated matroid of V

Dustin Cartwright Construction of Lindstr¨

• m valuation

Rough idea

“Tropicalization” for fields:

◮ Scaling =

⇒ Frobenius

◮ Reduction =

⇒ Differentials

Dustin Cartwright Construction of Lindstr¨

• m valuation

“Tropicalization” of field extensions

◮ Set-up: K field of char. p, L = K(x1, . . . , xn) ◮ Given: w = (w1, . . . , wn) ∈ Zn ◮ Scale: F −w1x1, . . . , F −wnxn, where F is Frobenius: Fx = xp, in

˜ L =

l K(x1/pl 1

, . . . , x1/pl

n

)

◮ Generate: K(F −wx) := K(F −w1x1, . . . , F −wnxn) ◮ Reduce: Vector space of differentials ΩK(F −wx)/K generated by

differentials dF −w1x1, . . . , dF −wnxn

◮ Tropicalization: Trop(L/K) ∩ Zn is the set of w ∈ Zn such

that differentials dF −w1x1, . . . , dF −wnxn are all non-zero

◮ Trop(L/K) is the tropicalization of a unique valuated matroid,

called the Lindstr¨

• m valuated matroid of K ⊂ L

(Bollen-Draisma-Pendavingh)

Dustin Cartwright Construction of Lindstr¨

• m valuation

Local and global structure

If V is a k-vector space, x1, . . . , xn ∈ V , then Trop(V ) is

◮ Globally: The recession fan of Trop(V ) is equivalent to the

linear (non-valuated) matroid of x1, . . . , xn ∈ V

◮ Locally: At w ∈ Γn, the link of Trop(V ) is equivalent to the

linear matroid of inw(V ) If K ⊂ L = K(x1, . . . , xn) 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 ∈ Zn is equivalent to the linear matroid

• f ΩK(F −wx)/K

Dustin Cartwright Construction of Lindstr¨

• m valuation

A bridge example: monomials

◮ A: a d × n integer matrix ◮ L = K(z1, . . . , zd), xi = zA1i 1

· · · zAdi

d ◮ The Lindstr¨

• m valuated matroid of K ⊂ L is the same as the

valuated matroid of the columns of A in Qd with the p-adic valuation

◮ Prior example of K[x1, x2]/xp 1 − x2 is a monomial example

with A =

• 1

p

• Dustin Cartwright

Construction of Lindstr¨

• m valuation

Circuits of the Lindstr¨

• m matroid

As before: K ⊂ L = K(x1, . . . , xn)

◮ If C is a circuit of the algebraic matroid of K ⊂ L, then

{xi | i ∈ C} is a minimal dependent set, and there exists a polynomial relation fC ∈ K[xi | i ∈ C], unique up to scaling

◮ Write

fC =

• u∈JC

cuxu1

1 · · · xun n

where JC ⊂ Zn

≥0, ui = 0 if i /

∈ C, and cu = 0

◮ Define:

C(fC) =

• . . . , min{valp ui | u ∈ JC}, . . .
• ∈ (Z ∪ ∞)n

where valp is the p-adic valuation

◮ The valuated circuits of the Lindstr¨

• m valuation are the vectors

C(fC) + λ1 as C ranges over circuits of the algebraic matroid, and λ ∈ Z

Dustin Cartwright Construction of Lindstr¨

• m valuation

Valuation of the Lindstr¨

• m matroid

As before: K ⊂ L = K(x1, . . . , xn)

◮ Let B be a basis of the algebraic matroid, meaning a maximal

independent set of variables

◮ The extension K(xi | i ∈ B) ⊂ L can be uniquely factored as

K(xi | i ∈ B) ⊂ K(xi | i ∈ B)sep ⊂ L where the first extension is separable (roughly: like characteristic 0) and the second is purely inseparable (defined by taking pth roots)

◮ The Lindstr¨

• m valuation is:

v(B) = logp[L : K(xi | i ∈ B)sep] ∈ Z≥0

Dustin Cartwright Construction of Lindstr¨

• m valuation

Cocircuits of the Lindstr¨

• m valuation

As before: K ⊂ L = K(x1, . . . , xn)

◮ Let H be a hyperplane of the algebraic matroid, meaning a

maximal set with rk(H) = rk({1, . . . , n}) − 1

◮ Define:

Cco(H) =

• . . . , logp[L : K(xi | i ∈ H ∪ {i})sep], . . .
• ◮ The valuated cocircuits of the Lindstr¨
• m valuation are:

Cco(H) + λ1 as H ranges over the hyperplanes, and λ ∈ Z

Dustin Cartwright Construction of Lindstr¨

• m valuation