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Polynomial invariants of geometric structures

Ruud Pellikaang.r.pellikaan@tue.nl

Castellon2016 May 4

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Characteristic polynomial

• Knots and links and its (multi variable) Alexander polynomial

• Complement of projective curve and its Alexander polynomial

• Arrangement and its (two variable) characteristic polynomial

• Milnor fibre of an isolated singularity andits characteristic polynomial of the monodromy

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Zeta function

• Number field and its zeta function

• Algebraic variety over a finite field andits (two variable) zeta function, characteristic polynomial of Frobenius

• Error-correcting codes and its (two variable) zeta function

• Graph and its zeta function

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Weight enumerator

• Code and its weight enumerator

• Code and its extended weight enumerator

• Code and its generalized weight enumerator

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Chromatic, Tutte and characteristic polynomial

• Graph and its chromatic polynomial

• Matroid and its characteristic polynomial

• Matroid and its two variable Tutte polynomial

• Matroid its (two variable) zeta function

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Content

1. Error-correcting codes

2. Weight enumerator

3. Arrangements and codes

4. Arrangements and weight enumerators

5. Extended weight enumerator

6. Matroids

7. Tutte-Whitney polynomial

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References

• T. Britz, Higher support matroidsDiscrete Mathematics 307, 2300-2308, (2007)

• T. Britz, Code enumerators and Tutte polynomialsIEEE Transactions on Information Theory 56, 4350-4358, (2010)

• Relinde JurriusCodes, arrangements, matroids, and their polynomial linksPhD thesis, Technical University Eindhoven, August 2012

• Relinde Jurrius and Ruud PellikaanCodes, arrangements and matroidsin Series on Coding Theory and Cryptology vol. 8Algebraic geometry modelling in information theory, pp. 219–325, 2013

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(Generalized) weightenumerator

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Let C be a code of length nAw denotes the number of codewords in C of weight wA(r)

w denotes the number of subspaces of C of dimension r weight w

The weight enumerator is:

WC(X, Y ) =n∑

w=0

AwXn−wY w.

The r-th generalized weight enumerator is:

W(r)C (X, Y ) =

n∑w=0

A(r)w X

n−wY w.

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Theorem

Let C be a [n, k] code over Fq

Then

WC⊥(X, Y ) = q−kWC(X + (q − 1)Y,X − Y )

ProofSeveral proofs are known

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Proposition

Let WC(X, Y ) be the weight enumerator of CThen the probability of undetected error on aq-ary symmetric channel with cross-over probability p is given by

Pue(p) = WC

(1− p, p

q − 1

)− (1− p)n.

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Arrangements and codes

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Let C and D be linear codes in Fnq

Then C is called permutation equivalent to Dif there exists a permutation matrix Π such that Π(C) = DIf moreover C = D, then Π is called an permutation automorphism of C

The code C is called generalized or monomial equivalent to Dif there exists a monomial matrix M such that M(C) = DIf moreover C = D, then M is called a monomial automorphismof C

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An arrangement in Fk is ann-tuple (H1, . . . , Hn) of hyperplanes in Fk

The arrangement is called simpleif all the n hyperplanes are mutually distinct

The arrangement is called central if 0 ∈ Hj for all jIf the arrangement is central one considers the hyperplanes in Pk−1(F)

A central arrangement is called essential if ∩jHj = {0}Projective systems and essential arrangements are dual notions

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Let G = (gij) be a generator matrix of anondegenerate code C of dimension kSo G has no zero columns

Let Hj be the linear hyperplane in Fkq with equation

g1jX1 + · · · + gkjXk = 0.

AG is the arrangement (H1, . . . , Hn) associated with G

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There is a one-to-one correspondence between:

1. generalized equivalence classes ofnondegenerate [n, k] codes over Fq

2. equivalence classes of essential arrangements ofn hyperplanes in Pk−1(Fq)

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PropositionLet C be a nondegenerate code over Fq with generator matrix GLet c be a codeword c = xG for the unique x ∈ Fk

q

Then n − wt(c) is equal to the number of hyperplanes of AG throughx

ProofNow cj =

∑i gijxi

So cj = 0 if and only if x ∈ Hj

Hence

n− wt(c) = |{ j | cj = 0 }| = |{ j | x ∈ Hj }|

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Arrangementsand

weight enumerator

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Aw the number of codewords of weight w equalsthe number of points that are on exactly n− w of the hyperplanes ofAG

In particular An is equal to the number of points that is inthe complement of the union of these hyperplanes in Fk

q

This number can be computed by the principle of inclusion/exclusion:

An = qk − |H1 ∪ · · · ∪Hn|

= qk +n∑

w=1

(−1)w∑

i1<···<iw

|Hi1 ∩ · · · ∩Hiw|.

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H2

H4

H5

H6

H1

H4

H3

H6

H5

H7 H7

H1

H3

H2

G =

1 0 0 0 1 1 10 1 0 1 0 1 10 0 1 1 1 0 1

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Let C be the code over Fq with generator matrix GFor q = 2, this is the simplex code S2(2)The columns of G represent also the coefficients of the lines ofAG

Assume q is evenA0 = 1A4 = 7(q − 1) there are 7 points on exactly 3 linesA6 = 7(q − 1)[(q + 1)− 3] = 7(q − 1)(q − 2) there are 7 linesand (q + 1) − 3 points of such a line is exactly on one of theseA7 = q3 − A0 − A4 − A6 = (q − 1)(q − 2)(q − 4)So

WC(X, Y ) = X7+7(q−1)X3Y 4+7(q−1)(q−2)XY 6+(q−1)(q−2)(q−4)Y 7

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Assume q is odd, then similarly

WC(X, Y ) =

X7+6(q−1)X3Y 4+3(q−1)X2Y 5+(q−1)(7q−17)XY 6+(q−1)(q−3)2Y 7

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The following method is based on Katsman-TsfasmanLater we will encounter another method:matroids and the Tutte polynomial by Greene

DefinitionFor a subset J of [n] := {1, 2, . . . , n} define

C(J) = {c ∈ C|cj = 0 for all j ∈ J}l(J) = dimC(J)

BJ = ql(J) − 1

Bt =∑|J |=t

BJ

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The encoding map x 7→ xG = cfrom vectors x ∈ Fk

q to codewordsgives the following isomorphism of vector spaces⋂

j∈JHj∼= C(J)

Furthermore BJ is equal to the number of nonzero codewords cthat are zero at all Hj in J

BJ = |C(J) \ {0}| =

∣∣∣∣∣∣⋂j∈J

Hj \ {0}

∣∣∣∣∣∣

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LemmaLet C be a linear code with generator matrix GLet J ⊆ [n] and |J | = tGJ is the k × t submatrix of G existing of the columns of G indexed by JLet r(J) be the rank of GJ

Then l(J) = k − r(J)

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LemmaLet C be an Fq-linear code of dimension kLet d and d⊥ be the minimum distance of C and C⊥, respectivelyLet J ⊆ [n] and |J | = tThen

l(J) =

{k − t for all t < d⊥

0 for all t > n− d

and

Bt =

{ (nt

)(qk−t − 1) for all t < d⊥

0 for all t > n− d

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PropositionBt relates to the weight distribution as follows:

Bt =n−t∑w=d

(n− wt

)Aw

ProofCount in two ways the number of elements of the set

{ (J, c) | J ⊆ [n], |J | = t, c ∈ C(J), c 6= 0 }

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TheoremThe generalized weight enumerator is given by the following formula:

WC(X, Y ) = Xn +n∑

t=0

Bt(X − Y )tY n−t

ProofUse the previous propositionthe fact that Bt = 0 for t > n− dchange the order of summation anduse the binomial expansion:

Xn−w = ((X − Y ) + Y )n−w

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PropositionThe following formula holds:

Aw =n∑

t=n−w(−1)n+w+t

(t

n− w

)Bt.

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PropositionThe weight distribution of an MDS code oflength n, dimension k and minimum distance d = n− k + 1

Aw =

(n

w

)w−d∑j=0

(−1)j(w

j

)(qw−d+1−j − 1

)for w ≥ d = n− k + 1

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Extended weight enumerator

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Let G be the generator matrix of a linear [n, k] code C over Fq

Fq is a subfield of Fqm

Consider the code C ⊗ Fqm over Fqm

by taking all Fqm-linear combinations of the codewords in CThis is called the extension code of C over Fqm

G is also a generator matrix for the extension code C ⊗ Fqm

Hence C ⊗ Fqm has dimension k over Fqm

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Remember:DefinitionFor a subset J of [n] := {1, 2, . . . , n} define

C(J) = { c ∈ C | cj = 0 for all j ∈ J }l(J) = dimC(J)

LemmaLet C be a linear code with generator matrix GLet J ⊆ [n] and |J | = tGJ is the k × t submatrix of G existing of the columns of G indexed by JLet r(J) be the rank of GJ

Then l(J) = k − r(J)

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l(J) = k − r(J) by the previous lemmar(J) is independent of the extension field Fqm

Therefore

dimFqC(J) = dimFqm

(C ⊗ Fqm)(J)

This motivates the usage of T as a variable for qm in the next definition

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Remember:Let C be a linear code over Fq

BJ = ql(J) − 1

Bt =∑|J |=t

BJ

Extend: Definition

BJ(T ) = T l(J) − 1

Bt(T ) =∑|J |=t

BJ(T )

Note that BJ(qm) is the number of nonzero codewords in (C ⊗ Fqm)(J)

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Remember:

WC(X, Y ) = Xn +n∑

t=0

Bt(X − Y )tY n−t

Define the extended weight enumerator by

WC(X, Y, T ) = Xn +n∑

t=0

Bt(T )(X − Y )tY n−t

Is well-defined for any linear subspace C of Fn over any field F

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TheoremThe following holds:

WC(X, Y, T ) =n∑

w=0

Aw(T )Xn−wY w

A0(T ) = 1, and Aw(T ) =n∑

t=n−w(−1)n+w+t

(t

n− w

)Bt(T )

for 0 < w ≤ n and

Bt(T ) =n−t∑w=d

(n− wt

)Aw(T )

Proof is similar to the proof relating the Aw’s and Bt’s

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PropositionThe weight distribution of an MDS code oflength n and dimension k is given by

Aw(T ) =

(n

w

)w−d∑j=0

(−1)j(w

j

)(Tw−d+1−j − 1

)for w ≥ d = n− k + 1

ProofSimilar to the proof for Aw

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PropositionLet C be a linear [n, k] code over Fq

ThenWC(X, Y, qm) = WC⊗Fqm

(X, Y )

The number of codewords in C ⊗ Fqm of weight wis equal to Aw(qm)

ProofSubstituting T = qm in Bt(T ) givesBt(q

m) which is equal to the Bt of C ⊗ Fqm

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TheoremLet C be an [n, k] code over Fq

Then

WC⊥(X, Y, T ) = T−kWC(X + (T − 1)Y,X − Y, T )

ProofSubstituting T = qm gives the MacWilliams identity for C ⊗ Fqm

WC⊥(X, Y, qm) = q−mkWC(X + (qm − 1)Y,X − Y, qm)

which holds for all mNow Aw(T ) is a polynomial in T with coefficient in ZGiving infinitely many identities for the weight distributions ofC ⊗ Fqm and C⊥ ⊗ Fqm = (C ⊗ Fqm)⊥

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The following formula will be useful later inidentifying the extended weight enumeratorwith the Tutte polynomial

PropositionLet C be a linear [n, k] code over Fq

WC(X, Y, T ) =n∑

t=0

∑|J |=t

T l(J)(X − Y )tY n−t

ProofUse the description of WC(X, Y, T ) in terms of the Bt(T ) andthe definition of Bt(T ) in terms of the l(J)

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Matroids

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Matroids were introduced by Whitney in axiomatizing and generalizingthe concepts of independence in linear algebra and cycle in graph theory

DefinitionA matroid M is a pair (E, I) consisting ofa finite set E and a collection I of subsets of E such that:

(I.1) ∅ ∈ I .

(I.2) If J ⊆ I and I ∈ I , then J ∈ I .

(I.3) If I, J ∈ I and |I| < |J |, then there exists a j ∈ (J \ I) such thatI ∪ {j} ∈ I .

A subset I of E is called independent if I ∈ Iotherwise it is called dependentCondition (I.2) is called the independence augmentation axiom

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If J is a subset of E, then J has a maximal independent subsetif there exists an I ∈ I such that I ⊆ J andI is maximal with respect to this property and the inclusion

If I1 and I2 are maximal independent subsets of Jthen |I1| = |I2| by condition (I.3)

The rank or dimension r(J) of a subset J of E isthe number of elements of a maximal independent subset of J

An independent set of rank r(M) is called a basis of MThe collection of all bases of M is denoted by B

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Let n and k be non-negative integers such that k ≤ nLet [n] = {1, . . . , n}Let In,k = {I ⊆ Un,k||I| ≤ k}

Then ([n], In,k) is a matroid and it is denoted by Un,k

It is called the uniform matroid of rank k on n elements

A subset B of [n] is a basis of Un,k iff |B| = kThe matroid Un,n has no dependent sets and is called free

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Let G be a k × n matrix with entries in a field F

Let E be the set [n] indexing the columns of GLet IG be the collection of all subsets I of Esuch that the columns of GI are independentThen MG = (E, IG) is a matroid

A matroid that is isomorphic with an MG is calledrepresentable over the field F

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Suppose that F is a finite field andG1 and G2 are generator matrices of a code CThen (E, IG1

) = (E, IG2)

So the matroid MC = (E, IC) of a code C is well defined by(E, IG) for some generator matrix G of C

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Let M = (E, I) be a matroidLet B be the collection of all bases of M

Define B⊥ = (E \B) for B ∈ Band B⊥ = {B⊥|B ∈ B}

Define I⊥ = {I ⊆ E|I ⊆ B for some B ∈ B⊥}Then (E, I⊥) is called the dual matroid of M and is denoted by M⊥

The dual matroid is indeed a matroid

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Let C be a code over a finite fieldThen (MC)⊥ is isomorphic with MC⊥ as matroids

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Tutte-Whitney polynomial

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DefinitionLet M = (E, I) be a matroidThe Whitney rank generating function RM(X, Y ) is defined by

RM(X, Y ) =∑J⊆E

Xr(E)−r(J)Y |J |−r(J)

and the Tutte-Whitney or dichromatic Tutte polynomial by

tM(X, Y ) =∑J⊆E

(X − 1)r(E)−r(J)(Y − 1)|J |−r(J)

HencetM(X, Y ) = RM(X − 1, Y − 1)

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PropositionLet C be a [n, k] code over Fq

Then the Tutte polynomial tC of the matroid MC of the code C is

tC(X, Y ) =n∑

t=0

∑|J |=t

(X − 1)l(J)(Y − 1)l(J)−(k−t)

Proof

tC(X, Y ) =∑J⊆E

(X − 1)r(E)−r(J)(Y − 1)|J |−r(J)

Now r(E) = k, t = |J | and l(J) = k − r(J)

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TheoremLet C be a [n, k] code over any field FThen the Tutte polynomial tC of the matroid MC of the code Cand the extended weight enumerator WC(X, Y, T )determine each other

tC(X, Y ) = Y n(Y − 1)−kWC(1, Y −1, (X − 1)(Y − 1))

and

WC(X, Y, T ) = (X − Y )kY n−k tC

(X + (T − 1)Y

X − Y,X

Y

)Second identity proved by Greene (1976) in case T = Fq

Similar result holds the generalized weight enumerators W (r)C (X, Y )

by Britz (2007,2010)

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TheoremLet tM(X, Y ) be the Tutte polynomial of a matroid MLet M⊥ be the dual matroidThen

tM⊥(X, Y ) = tM(Y,X)

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TheoremLet C be a [n, k] code over Fq

Then

WC⊥(X, Y, T ) = T−kWC(X + (T − 1)Y,X − Y, T )

Proof Use

• tM⊥(X, Y ) = tM(Y,X)

• MC⊥ = (MC)⊥

• tC(X, Y ) and WC(X, Y, T ) determine each other

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The following polynomials determine each other:

WC(X, Y, T ) extended weight enumerator of C

{W (r)C (X, Y )|r = 1, . . . , k} generalized weight enumerators of C

tC(X, Y ) dichromatic Tutte polynomial of matroid MC

χC(S, T ) coboundary or two variable char.pol. of geometric lattice LC

ζC(S, T ) two variable zeta function of C by Duursma

ButWC(X, Y ) is weaker than WC(X, Y, T )