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Field extensions, Derivations, and Matroids over skew ... Pendavingh seminar 1 … · Field...
Transcript of Field extensions, Derivations, and Matroids over skew ... Pendavingh seminar 1 … · Field...
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