Log-concavity in combinatorics and geometry

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Log-concavity in combinatorics and geometry

Omid Amini

CNRS - CMLS, ´Ecole Polytechnique

March 11th 2021

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1 Log-concave sequences in combinatorics

2 Log-concavity versus convexity

3 Algebro-geometric log-concavity

4 Applications and further results

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Log-concave sequences

Definition

A sequence of non-negative numbers(ai)^∞_i=0 is calledlog-concaveif for any i≥1

a²_i ≥ai−1a_i+1. Example (Binomial coefficients)

The sequence ⁿ₀ , ⁿ₁

,. . ., _n−1ⁿ , ⁿ_n

is log-concave.

Example (Newton’s inequality) Let t1, . . . , tn≥0. The sequence

Sk:= 1

n k

X

i1<···<ik

ti1ti2. . . tik

is log-concave.

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Log-concave sequences

Definition

,. . ., _n−1ⁿ , ⁿ_n

is log-concave.

Sk:= 1

n k

X

i1<···<ik

ti1ti2. . . tik

is log-concave.

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Log-concave sequences

Definition

,. . ., _n−1ⁿ , ⁿ_n

is log-concave.

Sk:= 1

n k

X

i1<···<ik

ti1ti2. . . tik

is log-concave.

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Matchings in graphs

G= (V, E) a graph

A matchingin Gis a collection of edges which do not share any extremity.

Definea_i := number of matchings of sizeiinG.

Theorem (Gruber-Kuntz 1976) The sequence ai is log-concave.

Proof.

The matching polynomial p(^X) =P

iaiXi is real rooted.

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Matchings in graphs

G= (V, E) a graph

A matchingin Gis a collection of edges which do not share any extremity.

Definea_i := number of matchings of sizeiinG.

Theorem (Gruber-Kuntz 1976) The sequence ai is log-concave.

Proof.

The matching polynomial p(^X) =P

iaiXi is real rooted.

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Graph colorings

G= (V, E) a graph. Define

fG :N→N

f_G(k) = number of proper coloringsofG bykcolors.

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Graph colorings

fG :N→N

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Graph colorings

fG :N→N

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Graph colorings

For the Petersen graph:

f_G(2) = 0.

fG(3) = 120.

f_G(k) =k(k−1)(k−2) k⁷−12k⁶+67k⁵−230k⁴+529k³−814k²+775k−352 .

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Graph colorings

Proposition

f_G is a polynomial in k.

Proof.

Define

a_r:= number of proper colorings ofGbyexactly kcolors.

Then

f_G(k) =

k

X

r=0

a_r k

r

.

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Graph colorings

f_G(X) =a₀Xn−a₁Xn−1+a₂Xn−2+· · ·+ (−1)ⁿa_n. f_G(k)6= 0 iff Gis k-colorable. ForGplanar, f_G(4)6= 0.

a₀= 1.

ai are all non-negative.

Theorem (Huh 2010)

The sequence a0, a1, . . . , an is log-concave.

Conjectured by Read-Welsh 1968.

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Spanning forests

G= (V, E) connected graph.

A spanning forestis a subgraph (V, F) for F ⊆E such that (V, F)does not have any cycle.

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Spanning forests

G= (V, E) connected graph.

Define

a_i:= number of spanning forests withiedge.

Theorem (Adiprasito-Huh-Katz 2015) The sequence a0, . . . , an−1 is log-concave Conjectured by Mason 1976.

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Matroids

Matroid: a combinatorial structure axiomatising the linear dependency between finite collection of vectors by getting rid of the ambiant vector space.

Introduced by Hassler Withney 1935 in his study of graphs.

Question

How to recover Gfrom its cycles ?

Today, we see this as Torelli theorem for graphs.

Matroids arise naturally in the study of singularities in algebraic geometry.

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Matroids

Given a collection of vectorsv₁, . . . , v_nin a vector spaceW, we can define the collection I of independent sets I ⊆ {1, . . . , n}

(i.e. such that vectors v_i for i∈I are linearly independent)

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Matroids

Example

v1 = (1,0,0),v2 = (0,1,0),v3= (0,0,1),v4 = (2,1,0).

I = n

∅,

{1},{2},{3},{4},

{1,2},{1,3},{1,4},{2,3},{2,4},{3,4},

{1,2,3},{1,3,4},{2,3,4}o .

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Matroids

Given a collection of vectorsv₁, . . . , v_nin a vector spaceW, we can define the collection I of independent sets I ⊂ {1, . . . , n}.

the collection B of bases.

Example

v₁ = (1,0,0),v₂ = (0,1,0),v₃= (0,0,1),v₄ = (2,1,0).

B= n

{1,2,3},{1,3,4},{2,3,4}o .

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Matroids

Given a collection of vectorsv1, . . . , vnin a vector spaceW, we can define the collection I of independent sets I ⊂ {1, . . . , n}.

the collection B of bases.

rank function r: 2^{1,...,n} −→N∪ {0}.

Example

v1 = (1,0,0),v2 = (0,1,0),v3= (0,0,1),v4 = (2,1,0).

r({1,2,3}) = 3 r({1,2,4}) = 2.

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Matroids

Given a collection of vectorsv1, . . . , vnin a vector spaceW, we can define the collection I of independent sets I ⊂ {1, . . . , n}

the collection B of bases

Each of these satisfies some basic properties. E.g.

B satisfies the exchange axiom.

r is submodular:

∀A, B, r(A) +r(B)≥r(A∪B) +r(A∩B).

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Matroids

A matroid M on the ground set E is defined by any of the following:

the collection I of independent sets I ⊂ {1, . . . , n}

r is submodular:

∀A, B, r(A) +r(B)≥r(A∪B) +r(A∩B).

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Matroids

A matroid M on the ground set E is defined by any of the following:

the collection I of independent sets I ⊂ {1, . . . , n}

r is submodular:

∀A, B, r(A) +r(B)≥r(A∪B) +r(A∩B).

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Matroids

Example

Any connected graph G= (V, E)gives a matroid M_G:

Independent sets are I ⊂E such that(V, I) is a spanning forest.

Bases are all B ⊂E such that(V, B) is a spanning tree.

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For any field K, a collection of vectors v1, . . . , vn in the vector spaceK^r define a matroid. These are called representable.

The majority of matroids are non-representable.

Theorem (Nelson 2016)

Almost any matroid is not representable on any field.

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Characteristic polynomial of matroids

M matroid of rank r on ground set E={1, . . . , n}

Define the characteristic polynomialof M by f_M(x) = X

A⊆E

(−1)^|I|x^r−r(I).

Write f_M(x) =a₀x^r−a₁x^r−1+· · ·+ (−1)^ra_r.

Theorem (Adiprasito-Huh-Katz 2015) The sequence a₀, . . . , a_r is log-concave.

Conjectured by Rota 71, Heron 72 and Welsh 76.

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Characteristic polynomial of matroids

M matroid of rank r on ground set E={1, . . . , n}

Define the characteristic polynomialof M by f_M(x) = X

A⊆E

(−1)^|I|x^r−r(I).

Write f_M(x) =a₀x^r−a₁x^r−1+· · ·+ (−1)^ra_r. Theorem (Adiprasito-Huh-Katz 2015)

The sequence a₀, . . . , a_r is log-concave.

Conjectured by Rota 71, Heron 72 and Welsh 76.

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Convex functions

Definition

f :Rⁿ→Rcontinuous isconvex if for allx, y∈Rⁿ, f(x+y

2 )≤ f(x) +f(y)

2 .

Example

f(x) =x²₁+· · ·+x²_n is convex.

More generally, for symmetricA∈Mn(R), functionf(x) =x^tAxis convex if and only ifA is positive semi-definite.

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Mixed-determinant

A1, . . . , An symmetric matrices in Mn(R).

det(t1A1+· · ·+tnAn) is a polynomial int1, . . . , tn of degreen.

Definition (Mixed-determinant)

Definemdet(A₁, . . . , A_n)as the coefficient of t₁t₂. . . t_n in det(t1A1+· · ·+tnAn).

Example A1=

1 2 2 5

,A2=

2 −1

−1 1

,t1A1+t2A2=

t₁+ 2t₂ 2t₁−t₂ 2t1−t2 5t1+t2

. det(t₁A₁+t₂A₂) =t₁t₂+ 10t₁t₂−2t₁t₂−2t₁t₂+. . . This gives mdet(A₁, A₂) = 7

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Mixed-determinant

Example

IfA₁=· · ·=A_n=A, then

det(t1A1+. . . tnAn) = (t1+· · ·+tn)ⁿdet(A)

=n! det(A)t1t2. . . tn+. . . and so

mdet(A, . . . , A) =n! det(A).

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Mixed-determinant

A, B two positive-definite matrices inMn(R). Define ai = det(A, . . . , A

| {z }

i times

, B, . . . , B

| {z }

(n−i) times

).

Theorem (Alexandrov 1938)

The sequence a₀, a₁, . . . , a_n is log-concave.

Example A=1 2

2 5

,B = 2 −1

−1 1

.

a0 = 2, a1 = 7, a2 = 2

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Mixed-determinant

| {z }

i times

, B, . . . , B

| {z }

(n−i) times

).

Example A=1 2

2 5

,B = 2 −1

−1 1

.

a0 = 2, a1 = 7, a2 = 2

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Mixed-determinant

| {z }

i times

, B, . . . , B

| {z }

(n−i) times

).

Example A=1 2

2 5

,B = 2 −1

−1 1

.

a0 = 2, a1 = 7, a2 = 2

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Mixed-volume

P₁, . . . , P_n convex bodies inRⁿ.

vol(t1P1+· · ·+tnPn) is a polynomial int1, . . . , tn of degreen.

Definition (Mixed-volume)

Definemvol(P₁, . . . , P_n) as the coefficient oft₁. . . t_n in vol(t1P1+· · ·+tnPn).

IfP₁ =· · ·=P_n=P, then t₁P₁+· · ·+t_nP_n= (t₁+· · ·+t_n)P, and so vol(P, . . . , P) = (t₁+· · ·+t_n)ⁿvol(P).

This gives mvol(P, . . . , P) =n!vol(P).

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Mixed-volume

ConsiderP, Q two convex bodies inRⁿ. Define a_i := mvol(P, . . . , P

| {z }

itimes

, Q, . . . , Q

| {z }

(n−i) times

).

Theorem (Alexandrov-Fenschel 1937) The sequence a0, . . . , an is log-concave.

Slogan: log-concavity is related to convexity via mixed-volumes.

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Combinatorial log-concavity

f_G(k) = number of proper colorings of Gbyk colors

=a₀xⁿ−a₁xⁿ⁻¹+a₂xⁿ⁻²+· · ·+ (−1)ⁿa_n. More generally, for a matroid M,

f_M(x) =a0x^r−a1x^r−1+· · ·+ (−1)^rar

the characteristic polynomial of M.

Theorem (Adiprasito-Huh-Katz 2015) The sequence a₀, . . . , a_r is log-concave.

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Algebraic-topological context

X: atopological object.

Suppose we can associate to X an algebraA(X) decomposed as A(X) =A⁰(X)⊕A¹(X)⊕ · · · ⊕A^d(X) such that

eachAⁱ(X) is a real vector space of finite dimension.

the multiplication map sends (a, b)∈Aⁱ×A^j to ab∈A^i+j. Example

X compact smooth manifold, andAⁱ(X)theith cohomology ofX.

X=S²,A=R⊕(0)⊕R.

X=T torus, then A=R⊕R²⊕R. X=Sⁿ, then A=R⊕(0)⊕ · · · ⊕(0)⊕R.

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Algebraic-topological context

The decomposition

A(X) =A⁰(X)⊕A¹(X)⊕ · · · ⊕A^d(X) usually has nice properties.

A⁰(X)'A^d(X)'R.

(Poincar´e duality) The multiplication

Aⁱ(X)×A^d−i(X)→R=A^d(X) is non-degenerate.

Example

X anorientedcompact smooth manifold, Aⁱ(X) theith cohomology group of X,A satisfies the Poincar´e duality.

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Algebraic-topological context

The decomposition

A(X) =A⁰(X)⊕A¹(X)⊕ · · · ⊕A^d(X) usually has nice properties.

A⁰(X)'A^d(X)'R.

(Poincar´e duality) The multiplication

Aⁱ(X)×A^d−i(X)→R=A^d(X) is non-degenerate.

Example

X anorientedcompact smooth manifold, Aⁱ(X) theith cohomology group of X,A satisfies the Poincar´e duality.

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Algebro-geometric context

Additionally, in nice geometricsituations, one can definepositiveelements ω ∈A¹(X).

(Lefschetz property) multiplication byω^d−2i gives an isomorphism ω^d−2i :Aⁱ(X)−^'→A^d−i(X).

In particular,

ω^d:A⁰(X)−→^' A^d(X) ω^d−2:A¹(X)−→^' A^d−1(X) (Hodge-index) the bilinear form

A¹(X)×A¹(X)−→R (α, β)→αβω^d−2 has precisely one positive eigenvalue.

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Algebro-geometric context

In particular,

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Algebro-geometric context

In particular,

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Algebro-geometric context

A(X) =A⁰(X)⊕A¹(X)⊕ · · · ⊕A^d(X).

(1) A⁰(X)'A^d(X)'R.

(2) (Poincar´e duality) multiplication Aⁱ(X)×A^d−i(X)→Ris non-degenerate.

(3) (Lefschetz property) for positive ω∈A¹(X),ω^d−2i :Aⁱ −→^' A^d−i. (4) (Hodge-index) the bilinear form

(. , .) :A¹(X)×A¹(X)−→R, (α, β)→αβω^d−2 has a unique positive eigenvalue.

Corollary

For positiveα, β ∈A¹(X), we have det

(α, α) (α, β) (α, β) (β, β)

≤0.

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Algebro-geometric context

A(X) =A⁰(X)⊕A¹(X)⊕ · · · ⊕A^d(X).

(1) A⁰(X)'A^d(X)'R.

(2) (Poincar´e duality) multiplication Aⁱ(X)×A^d−i(X)→Ris non-degenerate.

(3) (Lefschetz property) for positive ω∈A¹(X),ω^d−2i :Aⁱ −→^' A^d−i. (4) (Hodge-index) the bilinear form

(. , .) :A¹(X)×A¹(X)−→R, (α, β)→αβω^d−2 has a unique positive eigenvalue.

Corollary

For positiveα, β ∈A¹(X), we have det

(α, α) (α, β) (α, β) (β, β)

≤0.

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Log-concavity in the algebro-geometric setting

For α, β two positive elements in A¹(X), define ai :=αⁱβ^d−i ∈R'A^d(X).

Theorem (Khovanski-Teissier 80s) The sequence a0, . . . , adis log-concave.

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Example

X projective complex manifold of dimensiond.

Aⁱ(X) =H²ⁱ(X)the 2ith cohomology group of X.

X satisfies all the above properties.

For positive classes α, β∈H²(X), we get a log-concave sequence β^d, αβ^d−1, . . . , α^d−1β, α^d.

Theorem (Okounkov 1996, Kaveh-Khovanski, Lazarsfeld-Mustata 2013)

There are convex bodies P andQ in R^d associated to α andβ such that

αⁱβ^d−i= mvol(P, . . . , P

| {z }

itimes

, Q, . . . , Q

| {z }

(d−i)times

).

The convex bodyP is called the Newton-Okounkov body ofα.

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Example

X projective complex manifold of dimensiond.

Aⁱ(X) =H²ⁱ(X)the 2ith cohomology group of X.

X satisfies all the above properties.

For positive classes α, β∈H²(X), we get a log-concave sequence β^d, αβ^d−1, . . . , α^d−1β, α^d.

Theorem (Okounkov 1996, Kaveh-Khovanski, Lazarsfeld-Mustata 2013)

There are convex bodies P andQ in R^d associated to α andβ such that αⁱβ^d−i= mvol(P, . . . , P

| {z }

itimes

, Q, . . . , Q

| {z }

(d−i)times

).

The convex bodyP is called the Newton-Okounkov body ofα.

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Combinatorial log-concavity via convex bodies

Conjecture

There exist convex bodies Pα and Q_β so that log-concavity of matroids is explained by log-concavity of mixed volumes.

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Other log-concavity conjectures

Welsh conjecture on log-concavity of the values of the Tutte polynomial on the diagonal.

Log-concavity conjecture of Amelunxen-B¨urgisser for intrinsic volumes.

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Hodge theory for tropical varieties

Smooth tropical fans provide local charts for smooth tropical varieties.

Theorem (A.-Piquerez)

Any smooth projective tropical variety verifies all the nice properties.

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Computational complexity of counting bases of matroids

Counting number of spanning trees in a graph is easy (Matrix-Tree theorem). What about counting forests ?

Theorem (Anari-Gharan-Vinzant 2018)

The number of bases a matroidM of rank r can be approximated within a factor of O(2^r)in polynomial time.

The proof is based on log-concavity of thebasis partition function on basis. Letxe for e∈E be variables. Define

P(x₁, . . . , x_n) = X

B∈B

Y

e∈B

x_e.

Theorem (Anari-Gharan-Vinzant 2018) The application P :Rⁿ+→R+ is log-concave.

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Computational complexity of counting bases of matroids

Counting number of spanning trees in a graph is easy (Matrix-Tree theorem). What about counting forests ?

Theorem (Anari-Gharan-Vinzant 2018)

The number of bases a matroidM of rank r can be approximated within a factor of O(2^r)in polynomial time.

The proof is based on log-concavity of thebasis partition function on basis. Letxe for e∈E be variables. Define

P(x₁, . . . , x_n) = X

B∈B

Y

e∈B

x_e.

Theorem (Anari-Gharan-Vinzant 2018) The application P :Rⁿ+→R+ is log-concave.

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Computational complexity of counting bases of matroids

Theorem (Anari-Liu-Gharan-Vinzant 2018)

The number of bases a matroidM of rank r can be approximated within a factor of (1±) with probability larger than1−δ in polynomial time.

Proof based on log-concavity of the basis partition function and mixing properties of random walk on higher dimensional simplicial complexes.

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Negative dependance for random forests

Conjecture (Kahn 2000 - Grimmet-Winkler 2004) For all pair of edges e, f ∈E,

P(e, f ∈F)≤P(e∈F)P(f ∈F) on uniform random forests F.

Theorem (Br¨anden-Huh 2019)

P(e, f ∈F)≤2P(e∈F)P(f ∈F).

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Some References

Adiprasito-Huh-Katz,Hodge theory for combinatorial geometries, Ann. Math. 2018.

Amelunxen-B¨urgisser,Probabilistic analysis of the grassman condition number, Found. Comp. Math. 2015.

Amini-Piquerez,Homology of tropical fans, preprint 2021.

Amini-Piquerez,Hodge theory for tropical varieties, preprint 2020.

Anari-Gharan-Vinzant,Log-concave polynomials I, FOCS 2018.

Anari-Gharan-Liu-Vinzant,Log-concave polynomials I, STOCS 2019.

Borcea-Br¨anden-Ligget,Negative dependence and the geometry of polynomials, JAMS 2009.

Br¨anden-Hu,Lorenzian polynomials, Ann. Math. 2020.

Bobkov,Isoprimetric inequality for log-concave probability measures, Ann. Prob. 1999.

Gruber-Kuntz,General properties of polymer systems, Comm. Math. Physics 1971.

Huh,Milnor numbers and chromatic polynomial of graphs, JAMS 2012.

Kahn-Neiman,Negative correlation and log-concavity, Random structures and algorithms, 2010.

Kaveh-Khovanskii,Newton-Okounkov bodies, Ann. Math. 2012.

Lazarsfeld-Mustata,Convex bodies associated to linear series, Ann. de l’ENS 2009.

Milman-Rotem,Mixed integrals and related inequalities, J. Func. Anal. 2013.

Nelson,Almost all matroids are non-representable, Bulltin London Math. Soc. 2018.

Okounkov,Brunn–Minkowski inequality for multiplicities, Inventiones 1996.

Saumard-Wellner,Log-concavity and strong log-concavity, Statis. Surv. 2014.