GRAVER BASES, MATCHINGS IN SIMPLICIAL COMPLEXES AND TORIC - - PDF document

GRAVER BASES, MATCHINGS IN SIMPLICIAL COMPLEXES AND TORIC VARIETIES Anargyros Katsabekis Apostolos Thoma

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Let V = {v1, v2, . . . , vn} be a finite set. An abstract simplicial complex D on the vertex set V is a collection of sub- sets of V satisfying: (i) {vi} ∈ D for every i = 1, . . . , n, (ii) if T ∈ D and G ⊂ T, then G ∈ D. A set T ⊂ D of cardinality m + 1 has dimension m ≥ −1 and is called an m- simplex of D. The 0-simplices of D are called vertices, while the 1-simplices are called edges. The dimension dim(D) of D is the maximum of the dimensions of its simplices.

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Let D be an abstract simplicial com- plex on the vertex set V and J be a subset of Ω := {0, 1, . . . , dim(D)}. A set M = {T1, . . . , Ts} of simplices

• f D is called a J-matching in D if

Tk ∩ Tl = ∅ for every 1 ≤ k, l ≤ s and dim(Tk)∈ J for every 1 ≤ k ≤ s. Set supp(M) = ∪s

i=1Ti ⊂ V.

A J-matching M in D is called a max- imal J-matching if supp(M) has the maximum possible cardinality among all J-matchings. By δ(D)J we denote the minimum card(M) among all maximal J-matchings M in D.

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When D is a simple graph, i.e dim(D) ≤ 1, the notion of {1}-matching in D co- incides with the notion of matching. Also maximal {1}-matching coincides with the notion of maximal matching in D. Finally δ(D){1} equals the matching num- ber of D. Recall that a subset M of the edges of D is called a matching in D if there are no two edges which are incident with a common vertex. M is a maximal matching if it has the maximum possible cardinality among all

• matchings. The cardinality of a maxi-

mal matching in D is commonly known as its matching number.

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• Example. Consider the simplicial com-

plex D drawn in the figure. The set {{v1, v2}, {v3, v4}, {v5, v6}, {v7, v8}, {v9}} is a {0, 1}-matching in D, while the set {{v1, v2}, {v3, v4}, {v5, v6}, {v7, v8, v9}} is a {0, 1, 2}-matching in D. Both of them are maximal, since they cover all the vertices of D.

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4 5 1 2 3 6 9 7 8

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Let k be an algebraically closed field and k[x1, . . . , xm] be the polynomial ring in the variables x1, . . . , xm. A binomial in k[x1, . . . , xm] is a differ- ence of monomials. Given a lattice L

• n Zm, the ideal

IL := ({xα+ − xα−|α = α+ − α− ∈ L}) in k[x1, . . . , xm] is called lattice ideal. Where α+ ∈ Nm and α− ∈ Nm denote the positive and negative part of α, respectively, and xb = xb1

1 · · · xbm m

for

b = (b1, . . . , bm) ∈ Nm.

If rank(L) = k, then there exists a ma- trix M ∈ Z(m−k)×m of rank m − k such that L ⊂ kerZ(M). When L = kerZ(M), the ideal IL is prime and called toric ideal. The va- riety V(IL) is called toric variety.

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Let A = {ai | 1 ≤ i ≤ m} be the set of columns of M, we associate to IL the rational polyhedral cone σ = posQ(A) := {d1a1+· · ·+dmam | di ∈

Q≥0}.

We assume that σ is strongly convex, i.e. {0} is a face of σ. With respect to the grading degA(xi) =

ai of the polynomial ring k[x1, . . . , xm]

the ideal IL is A-homogeneous. The binomial arithmetical rank bar(IL)

• f IL is the smallest integer s for which

there exist binomials F1, . . . , Fs in IL such that √IL = √F1, . . . , Fs. Hence the binomial arithmetical rank is an upper bound for the arithmeti- cal rank ara(IL) of IL, which is the smallest integer s for which there exists polynomials F1, . . . , Fs in IL such that √IL = √F1, . . . , Fs.

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When all the polynomials F1, . . . , Fs are A-homogeneous, the smallest integer s is called A-homogeneous arithmetical rank araA(IL) of IL. For a lattice ideal IL the following inequality holds: ht(IL) ≤ ara(IL) ≤ araA(IL) ≤ bar(IL) ≤ µ(IL). Problem. Find lower bounds for the minimal number µ(IL) of generators, the binomial arithmetical rank and the A-homogeneous arithmetical rank of a lattice ideal.

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Let σ = posQ(r1, . . . , rt) ⊂ Qn be a strongly convex rational polyhedral cone. Where {r1, . . . , rt} is a set of integer vectors,

• ne for each extreme ray of σ.

For a subset E of {1, . . . , t} we denote by σE the subcone posQ(ri |i ∈ E) of σ. The relative interior relintQ(σE) of σE is the set of all positive rational linear combinations of ri, i ∈ E. Suppose that σ is not a simplex cone, i.e. the extreme vectors r1, . . . , rt are not linearly independent.

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The set of cones σE, which are not faces of the cone σ, is not empty and form a poset ordered by inclusion. Let {σE1, . . . , σEf} be the minimal elements

• f this poset, which are called the min-

imal non faces of σ. To the cone σ we associate a simpli- cial complex Dσ with vertices the set {σE1, . . . , σEf} and T ⊂ {σE1, . . . , σEf} be- longs to Dσ if ∩σEi∈TrelintQ

• σEi
• = ∅.

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To the simplicial complex Dσ we can associate the 1-skeleton G(Dσ) of Dσ, formed by the vertices and edges of Dσ. The complement G(Dσ) of G(Dσ) is the graph with the same vertices as

G(Dσ), and {vi, vj} is an edge of G(Dσ)

if and only if {vi, vj} is not an edge of

G(Dσ).

The chromatic number γ(G(Dσ)) of the graph G(Dσ) is the smallest integer k for which there is a function c : V ertices(G(Dσ)) → {1, . . . , k} such that c(vi) = c(vj) if {vi, vj} is an edge

• f G(Dσ).

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• Theorem. For a lattice ideal IL with

associated cone σ = posQ(A) we have: (i) µ(IL) ≥ bar(IL) ≥ δ(Dσ){0,1} = δ(Dσ){0} − δ(Dσ){1}, (ii) araA(IL) ≥ δ(Dσ)Ω ≥ γ(G(Dσ)), (iii) If √IL = √F1, . . . , Fs, then (a) the total number of monomials in the nonzero terms of the polynomials F1, . . . , Fs is greater than or equal to the number of vertices δ(Dσ){0} of Dσ. (b) the total number of A-homogeneous components in F1, . . . , Fs is greater than

• r equal to the chromatic number of

G(Dσ).

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We consider the polynomial ring

k[y1, . . . , yt], by taking one variable for

each extreme vector ri. From the set

Rσ = {r1, . . . , rt} we can construct the

toric ideal IRσ, which is the kernel of the k-algebra homomorphism φ : k[y1, . . . , yt] → k[z1, . . . , zn, z−1

1 , . . . , z−1 n ]

given by φ(yi) = zri. The toric variety V(IRσ) is called ex- tremal toric variety.

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A binomial F(u) := yu+ − yu− in IRσ is called primitive if there exists no other binomial yv+ −yv− ∈ IRσ such that yv+ divides yu+ and yv− divides yu−. The set of primitive binomials of IRσ is finite and is called the Graver basis Gr(Rσ)

• f IRσ.
• Theorem. Set E := {E ⊂ {1, . . . , t} |

∃ F(u) ∈ Gr(Rσ) with supp(u+) = E or supp(u−) = E}, where supp(v) = {i ∈ {1, . . . , t} | vi > 0} for v = (v1, . . . , vt) ∈

• Nt. Then σE is a minimal non face of

σ if and only if E is a minimal element

• f E.
• Theorem. A set T = {σEi, σEj} is an

edge of Dσ if and only if there is a prim- itive binomial F(u) ∈ IRσ with supp(u+) = Ei and supp(u−) = Ej.

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Example. Consider the lattice L = kerZ(M), where M is the 3 × 6 ma- trix with columns the vectors of the set A3 = {rij = 2ei + ej|i, j ∈ {1, 2, 3}, i = j} and {ei|1 ≤ i ≤ 3} is the canonical base of Q3. The cone σ = posQ(A3) is associated to the toric ideal IL ⊂ k[xij]. Every vector of A3 is an extreme vec- tor of σ. The Graver Base of IA3 = IL consist of 3 binomials of the form xijxkj − xjixki, 3 binomials of the form x2

ijxki−x2 ikxji, 3

binomials of the form x2

ijxjk − x2 jixik, 3

binomials of the form x3

jixki−x2 ijx2 jk and

3 binomials of the form x3

kixji − x2 kjx2 ik.

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The simplicial complex Dσ is drawn in the Figure. We can prove that δ(Dσ){0,1} = 5 and δ(Dσ){0,1,2} = γ(G(Dσ)) = 4. In fact bar(IA3) = 5 and araA3(IA3) = 4. For ara(IA3) we have that 3 ≤ ara(IA3) ≤ 4, but it is unknown whether it is 3 or 4.

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{32, 13} {31, 12} {12, 23} {21, 32} {23, 31} {13, 21} {23, 13} {21, 31} {32, 12}

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We can generalize this example letting An = {rij = 2ei +ej|i, j ∈ {1, . . . , n}, i = j}, where n ≥ 3. We can prove that bar(IAn) = 5

  n

3

  + 6   n

4

  and

araAn(IAn) = 4

  n

3

  + 6   n

4

 .

For n = 10 we have that bar(IA10) = 1860 and araA10(IA10) = 1740, while 80 ≤ ara(IA10) ≤ 90. In those 80 up to 90 polynomials that generate the radical of IA10, there should be totally at least 3600 monomials in at least 1740 A10-homogeneous com- ponents.

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