Denominator reduction

In chapter 3 we explain the integration with hyperlogarithms in detail. But we anticipate that we will compute the integral of say I₀= ψ⁻² step by step, constructing the partial integrals

I_k :=

 ∞ 0

I_k−1 dα_ek for all 1 ≤ k < |E| (2.4.4) along a suitable construction σ = (e₁, . . . , e_|E|) of G. We will find that I_k = ^_ifiLi

is a linear combination of hyperlogarithms L_i with rational prefactors f_i. This requires linear reducibility, which in particular implies that the denominator of each f_i must factor linearly in the next Schwinger variable α_ek+1.

If indeed a summand of I_k has a denominator D_k = (aα_ek+1 + b)(cα_ek+1 + d) that factors, the integration of α_ek+1 results in a contribution to I_k+1 with denominator D_k+1:= ad − bc. In general these quadratic polynomials do not factorize.

But in their seminal work, Bloch, Esnault and Kreimer showed that during the first four integrations (k ≤ 4), all denominators are products of linear polynomials [22, sec-tion 8]. Brown subsequently introduced these²¹ as Dodgson polynomials [49] with Definition 2.4.7. For any sets I, J, K ⊂ E with |I| = |J |, the Dodgson polynomial is

Ψ^I,J_K := det M (I, J )|_α

e=0 ∀e∈K, (2.4.5)

where M (I, J ) denotes the minor of M obtained by deleting rows I and columns J . Remark 2.4.8. Dodgson polynomials depend on the choices made in the construction of a graph matrix M for G through an overall sign, so we understand that we stick to one particular matrix M and thereby fix the order of its rows (edges) and columns (vertices) throughout. For the same reason, the orientation of the edges (signs in the incidence matrix E ) and the deleted column v₀ ∈ V must stay the same.

In particular all denominators factor into such linear polynomials Ψ^I,J_K with I ∪J ∪K ⊆ {e₁, e₂, e₃, e₄}, as long as k ≤ 4. Therefore the first obstruction to linear reducibility can occur at k = 5 and indeed a new polynomial enters the game at this stage: the five-invariant

5Ψ (i, j, k, l, m) = ± det

Ψ^ij,kl_m Ψ^ijm,klm Ψ^ik,jl_m Ψ^ikm,jlm



(2.4.6) which is associated to a set {i, j, k, l, m} ⊆ E of five distinct edges. Any permutation of these changes the determinant in (2.4.6) only by an overall sign [49]. In the special

21Note that the polynomials in [22] are defined via the cycle (or circuit) matrix instead of the incidence matrix E.

case when I₀ = ψ⁻², it even turns out that I₅ = L/D₅ is a trilogarithm L divided by the common denominator D₅ =⁵Ψ (e₁, . . . , e₅).

A typical five-invariant of a complicated graph has irreducible quadratic components.

But when one of the four Dodgson polynomials in (2.4.6) vanishes, ⁵Ψ (i, j, k, l, m) de-generates into the product of two Dodgson polynomials and we say that {i, j, k, l, m}

splits. Francis Brown [49] proved that when e₁, . . . , e_|E| is a construction with vertex-width three, all denominators are Dodgson polynomials (thus linear). In particular,

5Ψ^e_σ(1), . . . , e_σ(5)^ splits.

Actually, a stronger statement can be made.

Theorem 2.4.9 (Iain Crump [71]). A simple, 3-connected graph G splits if and only if it contains none of {K_3,3, K₅, C, O, H} as a minor. Furthermore, this condition is equivalent to vw(G) = 3.

This means that not only ⁵Ψ (e₁, . . . , e₅), but in fact every five-invariant of G splits when vw(G) ≤ 3. Conversely, the splitting of every five-invariant in a 3-connected graph requires vw(G) = 3. Hence graphs with vw(G) ≤ 3 are extremely special from this viewpoint and in [71] we even find

Conjecture 2.4.10. If vw(G) ≤ 3, then it is linearly reducible with respect to any ordering of its edges and the polynomials in the reduction are all Dodgson polynomials.

We like to mention that the characterization (in terms of forbidden minors) of graphs with vertex-width less than or equal to three is much more complicated when we drop the requirement of 3-connectedness. This very intricate problem was solved in [21].

Quadratic identities

Known factorizations of denominators are consequences of local combinatorics (e. g. the presence of triangles or 3-valent vertices) and the representation of the graph polynomial ψ = det M as a determinant. Already Stembridge [163] observed that the Dodgson identity

det M (ij, ij) · det M = det M (i, i) · det M (j, j) − det M (i, j) · det M (j, i) (2.4.7) proves the factorization of the third denominator

D2= Ψ¹₂Ψ²₁− Ψ¹²Ψ₁₂=^Ψ¹₂^2.

In fact, Dodgson [78] refers to a much more general result as “well-known”: Jacobi’s determinant formula [49]. Its application to the graph matrix can be phrased as

Lemma 2.4.11 (corollary 10 in [170]). If the edge sets I = {I₁, . . . , Ir}, J = {J₁, . . . , Jr}, A, B, K ⊆ E fulfil A ∩ I = B ∩ J = ∅ and |A| = |B|, then

det^Ψ^A∪{I_K ^{i},B∪{Jj}}

1≤i,j≤r= Ψ^A∪I,B∪J_K ^Ψ^A,B_K ^r−1. (2.4.8) These and further identities where studied in detail in [49]. Applications to denomi-nator reduction include impressive computations of point-counts of graph hypersurfaces [60] and explicit graphical criteria for the weight-drop phenomenon [63].

In order to understand these identities combinatorially, we need an alternative de-scription of Dodgson polynomials in the spirit of (2.1.11) for the Symanzik polynomials.

This was worked out in [63] and we present a slightly more general formulation and proof of

Lemma 2.4.12. For equinumerous sets I, J ⊂ E(G) of edges, the Dodgson polynomial Ψ^I,J_G =^

P

ϵ^I,J_P · Φ^P_{G\(I∪J )} with signs ϵ^I,J_P ∈ {±1} (2.4.9)

is a linear combination of spanning forest polynomials. These are indexed by partitions P of V (I ∪ J ) such that (I \ J )/P and (J \ I)/P are spanning trees, where H/P denotes the graph obtained from the edges H after identification of all vertices that belong to the same parts of P . The signs are ϵ^I,J_P = (−1)^NI,J · Ψ^I,J_{(I∪J )/P} where

N_I,J = r + |{(e, f ) : e ∈ I ∩ J , f ∈ I △ J such that e < f }| + ^

e∈I△J

e (2.4.10)

and I △ J := I \ J ˙∪ J \ I stands for the symmetric difference.

Proof. Let I \ J = {i₁, . . . , i_r} and J \ I = {j₁, . . . , j_r} in ascending order (i₁ < · · · < i_r).

If we move the columns I \ J (and rows J \ I) of the graph matrix M (I, J ) such that they end up in the first r columns (rows), without changing the relative order of any other columns (rows), then we pick up

NI,J ≡

r



k=1

(i_k− |{e ∈ J : e < i_k}| − k) +

r



k=1

(j_k− |{e ∈ I : e < j_k}| − k) mod 2

minus signs: i_k is the [i_k− |{e ∈ J : e < i_k}|]’th column of M (I, J ) and must be moved to column k.

Now we expand M (I, J ) with respect to the Schwinger variables as in (2.1.12) and obtain, by the matrix-tree-theorem (2.1.10),

Ψ^I,J = ^

S⊇I∪J

det ˜E(I ˙∪ E \ S) · det ˜E(J ˙∪ E \ S) ·^

e /∈S

α_e (2.4.11)

where the non-vanishing contributions are precisely those for which both S \ I and S \ J are spanning trees. Hence F := S\(I ∪J ) is a spanning forest that contributes to Φ^P_{G\(I∪J )} for the partition P = π₀(F ). Note that in (2.4.11), both minors of the incidence matrix share the identical last rows ˜E_e (e ∈ F ).

Contracting any edge e = {v, w} ∈ F amounts to adding the column w to the column v, followed by an expansion in the row e (where the only non-zero entry then remains in column w). This multiplies both of the determinants in (2.4.11) with the same factor and does not change the overall sign. After all e ∈ F were contracted, any vertices belonging to the same part of P have been identified.

G =

v

v

v

v

e

e

e

I, J {1, 2}, {1, 3} {1, 2}, {2, 3} {1, 3}, {2, 3}

H/P

e2

e3

e2

e3

e₁

e3

e₁

e3

e1

e₂

e1

e₂

P 124, 3 13, 24 123, 4 13, 24 12, 34 13, 24

Ψ^I,J_H/P −1 −1 +1 +1 +1 −1

Figure 2.8.: The grey area of G indicates that in the drawing we only show the part H of G that is formed by the three edges e_i and the four vertices v_i they touch.

In example 2.4.13 we compute the corresponding Dodgson polynomials with (2.4.9). The signs are determined by whether the two edges I △ J of H/P connect both parts in the same direction.

To compute Ψ^I,J_{(I∪J )/P}, choose any part P₀ ∈ P to fix a root of both trees (I \ J )/P and (J \ I)/P . Every edge i_k∈ I \ J has two endpoints in (I \ J )/P , and with ϕ_I(k) we denote that one which is further away from the root as shown in figure 2.8. This gives us two bijections ϕ_I, ϕJ: {1, . . . , r} −→ P \ {P₀} and we have [63]

Ψ^I,J_{(I∪J )/P} = sgn^ϕI◦ ϕ⁻¹_J ^·

r



k=1

E_ik,ϕI(k)E_jk,ϕJ(k)^, (2.4.12)

where the product counts how many of the edges in the trees (I \ J )/P and (J \ I)/P are directed towards the root. Note that when r = 1, the permutations ϕ_I and ϕ_J do not play any role.

Example 2.4.13. We consider I ∪ J ⊆ {e₁, e₂, e₃} for three particular edges of G arranged and oriented as shown in figure 2.8. Also assume that e₁< e₂ < e₃ are indeed the first three edges in the order chosen to define the graph matrix.

For I = {1, 2} and J = {1, 3} (so r = 1) we find that N_I,J = 1 + 2 + 3 + 2 from (2.4.10) is even such that ϵ^I,J_P = Ψ^I,J_H/P where H contains the four vertices v_i and the three edges ei. The only partitions P to consider are 124, 3 and 13, 24. In both cases, e₂ and e₃ connect the two parts of P in opposite directions and thus ϵ^I,J_P = −1 from (2.4.12):

Ψ^12,13= −Φ{1,2,4},{3}− Φ{1,3},{2,4}. (2.4.13) Similarly, we obtain expressions for the Dodgson polynomials

Ψ^12,23= Φ{1,2,3},{4}

+ Φ{1,3},{2,4}

and Ψ^13,23= Φ{1,2},{3,4}− Φ{1,3},{2,4}

(2.4.14) from analyzing the partitions summarized in the table of figure 2.8. Note that N_I,J is even in all these cases.