Field extensions, Derivations,and

Matroids over skew hyperfields

Rudi Pendavingh

Technische universiteit Eindhoven

March 1, 2018

Rudi Pendavingh (TU/e) Fields, Derivations, Matroids March 1, 2018 1 / 22

Introduction Overview

Field extension L/Kxe ∈ L for e ∈ E

Matroid ofσ-derivations

Frobenius FlockLindstrom

valuated matroid

Matroid Flock

Algebraic matroid M(K , x)

Rudi Pendavingh (TU/e) Fields, Derivations, Matroids March 1, 2018 2 / 22

Introduction Overview

Field extension L/Kxe ∈ L for e ∈ E

Matroid ofσ-derivations

Frobenius FlockLindstrom

valuated matroid

Matroid Flock

Algebraic matroid M(K , x)

Rudi Pendavingh (TU/e) Fields, Derivations, Matroids March 1, 2018 2 / 22

Introduction Overview

Field extension L/Kxe ∈ L for e ∈ E

Matroid ofσ-derivations

Frobenius FlockLindstrom

valuated matroid

Matroid Flock

Algebraic matroid M(K , x)

Rudi Pendavingh (TU/e) Fields, Derivations, Matroids March 1, 2018 2 / 22

Introduction Overview

Field extension L/Kxe ∈ L for e ∈ E

Matroid ofσ-derivations

Frobenius FlockLindstrom

valuated matroid

Matroid Flock

Algebraic matroid M(K , x)

Rudi Pendavingh (TU/e) Fields, Derivations, Matroids March 1, 2018 2 / 22

Introduction Main objects

We consider

K ⊆ L field extension

xe ∈ L for each e from finite set E

Definition (Matroid)

E finite set, ground set, each F ⊆ E dependent or independent

independent sets I ⊂ 2E satisfy

(I0) ∅ ∈ I(I1) if A ∈ I and B ⊆ A, then B ∈ I(I2) if A,B ∈ I and |A| < |B|, then A ∪ {e} ∈ I for some e ∈ B \A

Two different matroids arise from K , x

Rudi Pendavingh (TU/e) Fields, Derivations, Matroids March 1, 2018 3 / 22

Introduction Algebraic independence

K [XF ] := K [Xe : e ∈ F ] for F ⊆ E

Definition (Algebraic matroid M(K , x) )

ground set: E

F ⊆ E dependent: ⇐⇒ there is a nonzero q ∈ K [XF ] so that q(x) = 0

Support of q ∈ K [XE ]:

q := smallest F ⊆ E so that q ∈ K [XF ]

Lemma

Suppose e ∈ E and q, r ⊆ K [XE ] irreducible polynomials so that

q(x) = 0 = r(x), q 6= r , e ∈ q ∩ r .

There exists an s ∈ K [XE ] so that s(x) = 0, and e 6∈ s ⊆ q ∪ r .

Rudi Pendavingh (TU/e) Fields, Derivations, Matroids March 1, 2018 4 / 22

Introduction Derivations

Definition (Derivations)

A K -derivation of L ⊇ K is a map D : L→ L so that

D(a) = 0 for all a ∈ K

D(a + b) = D(a) + D(b)

D(ab) = D(a)b + aD(b)

Suppose D is a K -derivation of K (xE ). If q ∈ K [xE ] is such that q(x) = 0, we have

0 = D(q(x)) =∑e∈E

∂q

∂xeD(xe)

Then

d(q) :=

(∂q

∂xe: e ∈ E

)⊥ (D(xe) : e ∈ E ) =: D(x)

Lemma

D is a K -derivation of K (xE ) if and only if d(q) ⊥ D(x) for all q ∈ K [XE ] so that q(x) = 0.

Rudi Pendavingh (TU/e) Fields, Derivations, Matroids March 1, 2018 5 / 22

Introduction Derivations

If D is a K -derivation of K (xE ), then

d(q) :=

(∂q

∂xe: e ∈ E

)⊥ (D(xe) : e ∈ E ) =: D(x)

By the lemma, we have:

{D(x) : D a K -derivation of K (xE )} = {u : u ⊥ d(q), q ∈ K [XE ], q(x) = 0}

This is a linear space, which determines a certain linear matroid:

Definition (Matroid of derivations M ′(K , x) )

ground set: E

F ⊆ E dependent: ⇐⇒ ∃ nonzero q ∈ K [XE ] so that q(x) = 0 and {e : d(q)e 6= 0} ⊆ F

Rudi Pendavingh (TU/e) Fields, Derivations, Matroids March 1, 2018 6 / 22

Introduction The two matroids

Definition (Algebraic matroid M(K , x))

ground set: E

F ⊆ E dependent: ⇐⇒ ∃ nonzero q ∈ K [XF ] so that q(x) = 0

Definition (Matroid of derivations M ′(K , x))

ground set: E

F ⊆ E dependent: ⇐⇒ ∃ nonzero q ∈ K [XE ] so that q(x) = 0 and {e : d(q)e 6= 0} ⊆ F

Lemma

F dependent in M(K , x) =⇒ F dependent in M ′(K , x).

Example

K has characteristic p, E = {a, b}, xa, xb ∈ L ⊇ K so that q(xa, xb) = xa − xpb = 0

M(K , x) has one circuit, {a, b} = q

M ′(K , x) has one circuit, {a} = {e : d(q)e 6= 0}, since d(q) = (1, 0)

Rudi Pendavingh (TU/e) Fields, Derivations, Matroids March 1, 2018 7 / 22

Introduction Overview

Fields L/K , xe ∈ L for e ∈ E

Matroid ofσ-derivations

Frobenius FlockLindstrom

valuated matroid

Matroid Flock

Algebraic matroid M(K , x)

Matroid flock: m ∈ ZE 7→ matroid of derivations M ′(K , (σme (xe))e), where σ : x 7→ xp

Rudi Pendavingh (TU/e) Fields, Derivations, Matroids March 1, 2018 8 / 22

Introduction Overview

Fields L/K , xe ∈ L for e ∈ E

Matroid ofσ-derivations

Frobenius FlockLindstrom

valuated matroid

Matroid Flock

Algebraic matroid M(K , x)

Matroid flock: m ∈ ZE 7→ matroid of derivations M ′(K , (σme (xe))e), where σ : x 7→ xp

Rudi Pendavingh (TU/e) Fields, Derivations, Matroids March 1, 2018 8 / 22

Introduction Overview

Fields L/K , xe ∈ L for e ∈ E

Matroid ofσ-derivations

Frobenius FlockLindstrom

valuated matroid

Matroid Flock

Algebraic matroid M(K , x)

Matroid flock: m ∈ ZE 7→ matroid of derivations M ′(K , (σme (xe))e), where σ : x 7→ xp

Rudi Pendavingh (TU/e) Fields, Derivations, Matroids March 1, 2018 8 / 22

Matroids over skew hyperfields Matroids and linear spaces

Let V be a linear space, ve ∈ V for e ∈ E .

Definition (Linear matroid)

ground set: E

F ⊆ E dependent: ⇐⇒ (ve : e ∈ F ) linearly dependent

Matroid K -linear space

Basis max. indep. B ⊆ E determinant det(ve : e ∈ B) ∈ K

Circuit min. dep. C ⊆ E vector u ∈ KC s.t.∑

e∈C ueve = 0

matroid: ”linear space without coefficients”

Rudi Pendavingh (TU/e) Fields, Derivations, Matroids March 1, 2018 9 / 22

Matroids over skew hyperfields Skew hyperfields

Definition (Hypergroup (G ,�, 0))

� : G × G → 2G \{∅} an associative hyperoperation

0 ∈ G

x � 0 = {x} for all x ∈ G

For each x ∈ G there is a unique y ∈ G so that 0 ∈ x � y . Write −x := y

x ∈ y � z if and only if z ∈ x � (−y)

Definition (Skew hyperfield (H , ·,�, 1, 0))

(H,�, 0) is a commutative hypergroup

(H?, ·, 1) is group, where H? := H \ {0}0 · x = x · 0 = 0 for all x ∈ H

α(x � y) = αx � αy and (x � y)α = xα� yα for all α, x , y ∈ H

Example (Krasner hyperfield K)

K := ({0, 1}, ·,�, 1, 0) with 1 · 1 = 1 and 1 � 1 = {0, 1}.

Rudi Pendavingh (TU/e) Fields, Derivations, Matroids March 1, 2018 10 / 22

Matroids over skew hyperfields Circuit axioms

X := {e ∈ E : Xe 6= 0}

Definition (Left H-matroid M = (E , C))

(C0) 0 6∈ C ⊆ HE

(C1) if X ∈ C and α ∈ H?, then α · X ∈ C(C2) if X ,Y ∈ C and X ⊆ Y , then Y = α · X for some α ∈ H?

(C3) if X ,Y ∈ C are a modular pair in C and Xe = −Ye 6= 0 for an e ∈ E , thenZe = 0 and Z ∈ X � Y for some Z ∈ C

The underlying matroid of M = (E , C) is M := (E , C) where

C := {X : X ∈ C}

Example

Matroid over K ←→ ordinary matroid

Matroid over skew field K ←→ vectors from K -linear space

Rudi Pendavingh (TU/e) Fields, Derivations, Matroids March 1, 2018 11 / 22

Matroids over skew hyperfields Signatures and Coordinates

Definition (Left H-Signature C of matroid N)

C satisfies (C0), (C1), and (C2), and

C is the collection of circuits of N.

If N is a matroid with bases B, then AN := {(B,B ′) ∈ B × B : |B \ B ′| = 1}.

Definition (Coordinates)

Function [.] : AN → H comprises left H-coordinates for N if

(CC0) [Fa,Fb] · [Fb,Fa] = 1 if Fa,Fb ∈ B(CC1) [Fac ,Fbc] · [Fab,Fac] · [Fbc,Fab] = 1 if Fab,Fac ,Fbc ∈ B(CC2) [Fac ,Fbc] = [Fad ,Fbd ] if Fac ,Fad ,Fbc,Fbd ∈ B, but Fab 6∈ B

Rudi Pendavingh (TU/e) Fields, Derivations, Matroids March 1, 2018 12 / 22

Matroids over skew hyperfields Signatures and Coordinates

Given C ⊆ HE :[Fa,Fb]C := X−1

a Xb for any X ∈ C such that X ⊆ Fab

Given [.] : AN → H:

C[.] := {X ∈ HE : X a circuit of N and X−1a Xb = [Fa,Fb] whenever a, b ∈ X ⊆ Fab}.

Lemma

Equivalent:

1 C is a left H-signature of N, and [.] = [.]C2 [.] are left H-coordinates for N, and C = C[.]

Rudi Pendavingh (TU/e) Fields, Derivations, Matroids March 1, 2018 13 / 22

Matroids over skew hyperfields Signatures and Coordinates

Given C ⊆ HE :[Fa,Fb]C := X−1

a Xb for any X ∈ C such that X ⊆ Fab

Given [.] : AN → H:

C[.] := {X ∈ HE : X a circuit of N and X−1a Xb = [Fa,Fb] whenever a, b ∈ X ⊆ Fab}.

Lemma

Equivalent:

1 C is a left H-signature of N, and [.] = [.]C2 [.] are left H-coordinates for N, and C = C[.]

Rudi Pendavingh (TU/e) Fields, Derivations, Matroids March 1, 2018 13 / 22

Matroids over skew hyperfields Quasi Plucker coordinates

Definition (Quasi Plucker coordinates)

Function [.] : AN → H comprises left quasi Plucker coordinates if

(P0) [Fa,Fb] · [Fb,Fa] = 1

(P1) [Fac ,Fbc] · [Fab,Fac] · [Fbc,Fab] = 1

(P2) [Fa,Fb] · [Fb,Fc] · [Fc ,Fa] = −1

(P3) [Fac ,Fbc] = [Fad ,Fbd ] if Fab 6∈ B or Fcd 6∈ B(P4) 1 ∈ [Fbd ,Fab] · [Fac ,Fcd ] � [Fad ,Fab] · [Fbc,Fcd ] .

Theorem (P.,2018)

N a matroid on E , H a skew hyperfield, [.] : AN → H a map, and C ⊆ HE . Equivalent:

1 M = (E , C) is a left H-matroid such that M = N, and [.] = [.]C .

2 [.] are left quasi-Plucker coordinates for N, and C = C[.].

Rudi Pendavingh (TU/e) Fields, Derivations, Matroids March 1, 2018 14 / 22

Matroids over skew hyperfields Orthogonality

A left H-signature C of N and a right H-signature D of N∗ are k-orthogonal if

0 ∈∑e

Xe · Ye

for all X ∈ C,Y ∈ D such that |X ∩ Y | ≤ k .

Theorem (P., 2018)

N a matroid on E , C a left H-signature of N. Equivalent:

(E , C) is a left H-matroid

there is a right H-signature D of N∗ which is 3-orthogonal to C

Rudi Pendavingh (TU/e) Fields, Derivations, Matroids March 1, 2018 15 / 22

The matroid of σ-derivatives Our eyesore

Definition (Algebraic matroid M(K , x))

ground set: E

F ⊆ E dependent: ⇐⇒ there is a q ∈ K [XF ] so that q(x) = 0

Definition (Matroid of derivations M ′(K , x))

ground set: E

F ⊆ E dependent: ⇐⇒ there is a q ∈ K [XE ] so that q(x) = 0 and {e : d(q)e 6= 0} ⊆ F

Example

K has characteristic p, E = {a, b}, xa, xb ∈ L ⊇ K so that q(xa, xb) = xa − xpb = 0

M(K , x) has one circuit, {a, b} = q

M(K , x) has one circuit, {a} = {e : d(q)e 6= 0}, since d(q) = (1, 0)

We make a ”left Lσ-matroid” to replace both matroids, where σ : x 7→ xp.

Rudi Pendavingh (TU/e) Fields, Derivations, Matroids March 1, 2018 16 / 22

The matroid of σ-derivatives A hyperfield

H a skew hyperfield, σ : H → H an automorphism.

Definition (Skew hyperfield of monomials)

Hσ := ({T∞} ∪ {aT i : a ∈ H?, i ∈ Z}, 1, 0, ·,�)

1 := T 0 and 0 := T∞

0 · aT i = aT i · 0 = 0 and aT i · bT j := aσi (b)T i+j

0 � x = x � 0 = {x} and

aT i � bT j :=

{aT i} if i < j{bT j} if i > j(a + b) · T i if i = j and a 6= −b(a + b) · T i ∪ H? · {T k : k ∈ Z, k > i} if i = j and a = −b

Definition (Ore extension of K )

K [T , σ]: Ring of formal polynomials∑

i aiTi , ai ∈ K , with multiplication Ta = σ(a)T .

There is a homomorphism µ : K [T , σ]→ Kσ s.t.∑

i aiTi 7→ amT

m,m := min{i : ai 6= 0}

Rudi Pendavingh (TU/e) Fields, Derivations, Matroids March 1, 2018 17 / 22

The matroid of σ-derivatives Orthogonal signatures

For q ∈ K [XE ] we define q ∈ K [ZE ] such that q = q(X pme

e : e ∈ E)

and m maximal.

Definition (σ-derivative of q)

dσ(q) : E → Lσ is

dσ(q) : e 7→ ∂q

∂zeTme

where m is such that q = q(xp

me

e

).

Lemma

Let N = M(K , x). Then

Cx := {α · dσ(q) : q decorates a circuit C of N, α ∈ (Lσ)?}.

is a left Lσ-signature of N.

Rudi Pendavingh (TU/e) Fields, Derivations, Matroids March 1, 2018 18 / 22

The matroid of σ-derivatives Orthogonal signatures

Definition (σ-derivation)

Given a K -derivation D of K (xE )sep, Dσ : e → Lσ is

Dσ : e 7→ TmeD(xp−me

e

),

where me = max{m ∈ N : xp−me

e ∈ K (xE )sep}.

Lemma

Let N = M(K , x). Then

Dx := {Dσ · β : D a K (xH)-derivation of K (xE )sep, D 6= 0, H hyperplane, β ∈ (Lσ)?}

is a right Lσ-signature of N∗.

Rudi Pendavingh (TU/e) Fields, Derivations, Matroids March 1, 2018 19 / 22

The matroid of σ-derivatives Orthogonal signatures

Recall:

d(q) :=

(∂q

∂xe: e ∈ E

)⊥ (D(xe) : e ∈ E ) =: D(x)

We likewise have:

dσ(q) :=

(∂q

∂zeTme : e ∈ E

)⊥(Tm′eD

(xp−m′e

e

): e ∈ E

)=: Dσ(x)

Theorem (P., 2018)

N a matroid on E , C a left H-signature of N. Equivalent:

(E , C) is a left H-matroid

there is a right H-signature D of N∗ which is 3-orthogonal to C

Corollary (P., 2018)

(E , Cx) is a left Lσ-matroid.

Mσ(K , x) := (E , Cx) is the matroid of σ-derivations.

Rudi Pendavingh (TU/e) Fields, Derivations, Matroids March 1, 2018 20 / 22

The matroid of σ-derivatives Orthogonal signatures

Recall:

d(q) :=

(∂q

∂xe: e ∈ E

)⊥ (D(xe) : e ∈ E ) =: D(x)

We likewise have:

dσ(q) :=

(∂q

∂zeTme : e ∈ E

)⊥(Tm′eD

(xp−m′e

e

): e ∈ E

)=: Dσ(x)

Theorem (P., 2018)

N a matroid on E , C a left H-signature of N. Equivalent:

(E , C) is a left H-matroid

there is a right H-signature D of N∗ which is 3-orthogonal to C

Corollary (P., 2018)

(E , Cx) is a left Lσ-matroid.

Mσ(K , x) := (E , Cx) is the matroid of σ-derivations.

Rudi Pendavingh (TU/e) Fields, Derivations, Matroids March 1, 2018 20 / 22

The matroid of σ-derivatives Overview

Field extension L/Kxe ∈ L for e ∈ E

Matroid ofσ-derivations

Frobenius FlockLindstrom

valuated matroid

Matroid Flock

Algebraic matroid M(K , x)

Rudi Pendavingh (TU/e) Fields, Derivations, Matroids March 1, 2018 21 / 22

The matroid of σ-derivatives Overview

Field extension L/Kxe ∈ L for e ∈ E

Matroid ofσ-derivations

Frobenius FlockLindstrom

valuated matroid

Matroid Flock

Algebraic matroid M(K , x)

Rudi Pendavingh (TU/e) Fields, Derivations, Matroids March 1, 2018 21 / 22

The matroid of σ-derivatives Boundary matroids and flocks

Definition ( Hσ-flock of rank r on E )

A map F : ZE → { left H-matroids on E of rank r}, such that:

(F1) Fα+1e \e = Fα/e for all α ∈ ZE and e ∈ E .

(F2) Fα+1E = σ∗Fα for all α ∈ ZE .

A left Hσ-matroid M uniquely determines a boundary matroid M0

Theorem (P., 2018)

Let F : ZE → { left H-matroids on E of rank r}. The following are equivalent:

1 F is an Hσ-flock.

2 there is a left Hσ-matroid M so that F : α 7→(Mτ(α)

)0.

Mτ(α) arises from M by rescaling each ground set element e by Tαe

Mσ(K , x)0 is the matroid of ordinary derivations M ′(K , x), and Mσ(K , σα(x)) = M(K , x)τ(α)

Rudi Pendavingh (TU/e) Fields, Derivations, Matroids March 1, 2018 22 / 22