aa r X i v : . [ m a t h . N T ] N ov ON ALGEBRA OF BIG ZETA VALUES
NIKITA MARKARIAN
Abstract.
The algebra of big zeta values we introduce in this paper is anintermediate object between multiple zeta values and periods of the multiplezeta motive. It consists of number series generalizing multiple zeta values, thesimplest examples, which are not multiple zeta series, are Tornheim sums. Weshow that convergent big zeta values are periods of the moduli space of stablecurves of genus zero on one hand and multiple zeta values on the other hand.It gives an alternative way to prove that any such period may be expressed asa rational linear combination of multiple zeta values and a simple algorithmfor finding such an expression.
Introduction
Multiple zeta values are number series playing important role in a wide rangeof subjects, see e. g. [Bro09, GM04, IKZ06]. Although the arithmetic nature ofthese numbers is still unknown, one may try to consider all consequences of naturalrelations among these series, that is the algebra of formal multiple zeta values. Onecould expect that there no other rational relations among them, that is the formalalgebra of multiple zeta values is isomorphic to the algebra rationally generated bymultiple zeta values. The long-standing conjecture states that all relations for theboth of them are regularized double shuffle relations ([IKZ06, Conjecture 1]).In the first section, we introduce an algebra of number series called big zetavalues, which generalizes multiple zeta values. One may consider the correspondingformal algebra. The main theorem of the paper states that relations among big zetavalues imply that any big zeta value equals a rational linear combination of multiplezeta values. Thus, the algebra of (formal) big zeta values is another hopefully moreconvenient form of the (formal) algebra of multiple zeta values.In the second section, we describe in some detail relations among big zeta val-ues. An interesting purely algebraic problem is to find all relations among formalmultiple zeta values inside the formal algebra of big zeta values. Stuffle relationsobviously follow from these relations. It is harder but possible to prove shuffle re-lations in terms of number series, see [KMT11]. One may expect that there are noother relations except regularized double shuffle relations as it is suggested by theconjecture mentioned above.The third section is devoted to the proof of the main theorem of the paper,which states that every big zeta value is a rational linear combination of multiplezeta values. Surprisingly enough, the proof does not use all relations among bigzeta values. It does not use the invariance of a big zeta value under a permutationof columns of the matrix defining the value. Generally, a basic matrix may becomenot basic after such a permutation, except in the case of reflection. The invariance
The study has been funded within the framework of the HSE University Basic Research Pro-gram and the Russian Academic Excellence Project ’5-100’. of a big zeta value under reflection is analogous to the duality theorem (see e. g.[SY19]) for multiple zeta values.To prove the main theorem we build an algorithm which allows us to turn any bigzeta value into a rational linear combination of multiple zeta values. The algorithmis of independent interest. It may be considered as a generalization of the stuffleproduct for the following reason. The product of two multiple zeta series is not amultiple zeta series, but a big zeta series corresponding to the direct sum of matricesof these multiple zeta values. Applying the algorithm to it we may expand it backin multiple zeta values. One may see that the result is the stuffle product.In the last section, we connect our construction with integrals of regular differ-ential forms with logarithmic singularities at infinity on the moduli space of stablecurves of genus 0 by the standard simplex in cubical coordinates. This constructionmotivated our definition of the algebra of big zeta values. It turns out that termby term integration of the Taylor series of such an integral is a rational linear com-bination of big zeta series, and conversely, every big zeta series is a rational linearcombination of integrals of such series.Different choices of coordinates on the integration region result in different num-ber series representing the same integral. It gives a bunch of relations among thebig zeta series generalizing the duality relation mentioned above. It would be in-teresting to derive them directly from basic relations.The formal algebra of big zeta values reminds the formal algebra of periods from[BCS10, Mar20] but is more manageable. Relations in it mimic relations in thealgebra of periods: Orlik–Solomon relations (7) correspond to the Stokes theoremand harmonic product relations (10) correspond to Arnold relations. It would beinteresting to find an analogous algorithm, which using integral relations expressesany integral as above as a rational combination of iterated integrals. It would proveor disprove conjectures from [BCS10, Mar20].Being proved that all relations among integrals as above follows from basic re-lations among big zeta number series, one may define purely algebraically the ”in-tegral” of regular differential forms with logarithmic singularities at infinity on themoduli space of stable curves of genus 0 by the standard simplex, which takes valuesin the formal algebra of big zeta values. This ”integral” would have all propertiesof the usual integral such as the Stokes theorem and the Fubini theorem. It wouldfollow that the Drinfeld associator with coefficients in the formal algebra of bigzeta values defined by this ”integral” obey the standard associator relations. Thus,the set of relations among formal multiple zeta values inside formal big zeta valuesalgebra, which are conjecturally regularized double shuffle relations, would implythe Drinfeld associator relations. This is the subject of future research.
Acknowledgments.
I am grateful to H. Tsumura and W. Zudilin for helpfuldiscussions. 1.
Algebra of big zeta values
Denote by e ab a positive A w -root, that is an element ( r i ) ∈ Z w defined by(1) r i = ⎧⎪⎪⎨⎪⎪⎩ a ≤ i ≤ b Definition 1. A d × w -matrix A = ( a ij ) is called basic if it is of rank d , has no zerocolumns and all its rows are positive A w -roots (1). N ALGEBRA OF BIG ZETA VALUES 3
In other words, rows of a basic matrix are coordinates of positive A w -roots, whichare linearly independent and are not contained in any coordinate hyperplane.In the following definition the term ”formal” means that we do not care aboutconvergence of the number series. Definition 2.
For a basic matrix A the big zeta series is a formal series(2) Z ( A ) = ∑ n i ∈ N , ≤ i ≤ d ∏ j (∑ i a ij n i ) The width w of A is called the weight and its height d is called the depth of the bigzeta series.Note that the big zeta series does not depend on the order of rows of the matrix. Example 1. A = ⎛⎜⎜⎜⎜⎜⎝ k d ³¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹·¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹µ . . . k d − ³¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹·¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹µ . . . . . . . . . k ³¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹·¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ . . . . . . . . . . . . ⎞⎟⎟⎟⎟⎟⎠ (3) Z ( A ) = ∑ n i ∈ N , ≤ i ≤ d n k d ( n + n ) k d − ⋯( n + ⋅ ⋅ ⋅ + n d ) k = ζ ( k , . . . , k d ) This is the multiple zeta series, see e. g. [IKZ06].
Example 2. A = ( a + b ³¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹·¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹µ . . . . . . . . . . . . ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ b + c . . . . . . ) (4) Z ( A ) = ∑ n,m ∈ N n a m c ( n + m ) b This is the Tornheim sum, see e. g. [BZ10].Space of formal big zeta series is equipped with a product given by the directsum of matrices. Define the formal big zeta values algebra as the space rationallygenerated by formal big zeta series with this product modulo natural relations.These relations are Orlik–Solomon relations (7) and harmonic product relations(10) below plus the invariance of the big zeta series under permutations of rowsand columns of the defining matrix.Example 1 above shows that formal multiple zeta values lie in the formal big zetavalues algebra. The following theorem states that they generate the whole algebraas a vector space.
Theorem 1.
Any formal big zeta value is a rational linear combination of formalmultiple zeta values of the same weight.
NIKITA MARKARIAN
The proof of Theorem 1 occupies Section 3 below.
Corollary.
The sum of a convergent big zeta series is a linear rational combinationof multiple zeta values of the same weight.
The following proposition gives a convergence criterion of a big zeta series. Italso follows from results of Section 4 below.
Proposition 1.
For a basic matrix A the big zeta series Z ( A ) converges iff A does not contain any rows with only one unit. In other words, it converges iff thecorresponding set of positive A w -roots contains no simple roots.Proof. Let A contains a row with only one unit. Consider the subseries, where allsummation parameters are fixed except the one corresponding to this row. Thissubseries is proportional to the harmonic series without some initial interval. Thusthe series diverges.The big zeta series (2) is dominated by the series ∑ n i ∈ N , ≤ i ≤ d ∏ j ( max i a ij n i ) Divide the summation region in components corresponding to the diagonal stratifi-cation of N d , that is to total orders on the set of rows. At each stratum this seriesis equal to some multiple zeta series (3), and k of all these series is more than theminimal number of units in rows of A . If every row of A contains at least two units,then k of all these multiple zeta values are more than 1. All such multiple zetaseries are known to be convergent, see e. g. [IKZ06]. It follows that the big zetaseries converges too. (cid:3) Relations
For a basic matrix A and a differential operator with constant coefficients D ∈ Q [ ∂ / ∂z , . . . , ∂ / ∂z d ] introduce notation:(5) Z ( A, D ) = ∑ n i ∈ N , ≤ i ≤ d ( D ∏ j ( ∑ i a ij z i ) ) ( n , . . . , n d ) Proposition 2.
For a basic d × w -matrix A and a homogeneous differential operatorwith constant coefficients D , series Z ( A, D ) is a rational linear combination ofseries Z ( A ′ ) for some A ′ s of depth d and of weight w + deg D .Proof. If operator D is of degree one, the right hand side of (5) is equal to arational linear combination of series Z ( A ′ ) , where A ′ are matrices equal to A withone column doubled. Then proof proceeds by induction on deg D . (cid:3) Let V be a vector space over Q and ∆ ⊂ V ∗ be a finite set of non-zero linearfunctions on V , which generates the dual space V ∗ . Denote by G ∆ the subspace ofrational functions on V generated by functions1 ∏ α ∈ κ α n α , where κ ⊂ ∆ is a subset, which generates V ∗ . The space G ∆ is a module over thealgebra Q [ V ] of differential operators with constant coefficients on V . N ALGEBRA OF BIG ZETA VALUES 5
Proposition 3.
The Q [ V ] -module G ∆ is generated by functions ∏ α ∈ κ α , where κ ranges over all subsets of ∆ , which are bases of V ∗ .Proof. The proof is straightforward, see [BV99, Lemma 2]. (cid:3)
Proposition 4.
For a basic d × w -matrix A and a differential operator with constantcoefficients D , series Z ( A, D ) is a rational linear combination of series Z ( S, D ⋅ D ′ ) for some D ′ s, where S ranges over all square submatrices of A of rank d .Proof. Let ∆ be the set of linear functions given by columns of matrix A . Restric-tions imposed on this matrix by Definition 1 guarantee that conditions of Proposi-tion 3 are satisfied. It follows that1 ∏ j ( ∑ i a ij z i ) = ∑ S D ′ S ∏ j ( ∑ i s ij z i ) for some operators with constant coefficients D ′ S , where the summation is takenover all square submatrices S = ( s ij ) of A of rank d . Substituting this to definition(5) we get the statement. (cid:3) Let l , . . . , l k be non-zero linear functions on a vector space V such that ∑ i α i l i = α i . The equality(6) k ∑ i = α i l ⋯ ˆ l i ⋯ l k = V is referred to as the Orlik–Solomon relation ([OT92, 3.5]). Proposition 5 (Orlik–Solomon relations) . Let A be a basic matrix and v , . . . , v k be a set its distinct columns such that ∑ i α i v i = for some ≠ α i ∈ Q . Then for adifferential operator with constant coefficients D (7) k ∑ i = α i Z ( ˆ A i , D ) = , where ˆ A i is matrix A with column v i excluded.Proof. Substituting (6) in definition (5) we get the statement. (cid:3)
Another family of relations among big zeta series is given by subdivision of the setof summands of the series into groups with different total orders of the summationarguments.Introduce some notations. Consider maps of polynomial algebras(8) m ∶ Q [ z , z ] ∋ p ( z , z ) ↦ p ( z + z , z ) ∈ Q [ z , z ] m ∶ Q [ z , z ] ∋ p ( z , z ) ↦ p ( z + z , z ) ∈ Q [ z , z ] m ∶ Q [ z ] ∋ p ( z ) ↦ p ( z + z ) ∈ Q [ z , z ] Denote by(9) m ∗ ∶ Q [ ∂ / ∂z , ∂ / ∂z ] → Q [ ∂ / ∂z , ∂ / ∂z ] m ∗ ∶ Q [ ∂ / ∂z , ∂ / ∂z ] → Q [ ∂ / ∂z , ∂ / ∂z ] m ∗ ∶ Q [ ∂ / ∂z , ∂ / ∂z ] → Q [ ∂ / ∂z ] the maps linear dual to (8). NIKITA MARKARIAN
For a set of w − vectors { v i } and a d × w -matrix M denote by [ v ; . . . ; v n ; M ] the matrix M with rows v i added somewhere. Recall that a big zeta value doesnot depend on the order of rows. Thus for a matrix and vectors as above the valueZ ([ v ; . . . ; v n ; M ] , D ) is well defined for a basic matrix [ v ; . . . ; v n ; M ] , the propertyof being basic also does not depend on the order of rows. Proposition 6 (Harmonic product relations) . Let A be a basic matrix containingrows e ij and e ( j + ) k (see (1)). Denote by A ′ the matrix A with these rows excluded.Then for a differential operator with constant coefficients D acting on the space ofrows of A (10)Z ( A, D ) = Z ([ e ik ; e ij ; A ′ ] , m ∗ D ) + Z ([ e ik ; e ( j + ) k ; A ′ ] , m ∗ D ) + Z ([ e ik ; A ′ ] , m ∗ D ) , where m ∗ i acts trivially on the subspace generated by rows of A ′ and acts as in (9)on the subspace generated by the added rows.Proof. Denote by n and m the summation parameters corresponding to rows e ij and e ( j + ) k in Z ([ e ij ; e ( j + ) k ; A ′ ] , D ) = Z ( A, D ) . Divide summands of this series inthree groups: ones with n > m , ones with n < m and ones with n = m . One maysee, that they correspond to three summands of the right hand side of (10). (cid:3) Proof of theorem 1
Denote by T d the upper triangular d × d -matrix with units on and above the di-agonal. One may see that for any homogeneous differential operators with constantcoefficients D i of degree i , the expression(11) w ∑ i = Z ( T i , D w − i ) is a rational linear combination of multiple zeta values (3) of weight w .On the other hand, by Proposition 4 any multiple zeta value of weight w may bewritten in the form (11). It follows that Theorem 1 may be reformulated as follows. Theorem 1 ′ . Any formal big zeta value of weight w is a rational linear combinationof the ones of the form Z ( T d , D w − d ) for some homogeneous differential operatorswith constant coefficients D i of degree i .Proof. The proof proceeds by induction on the depth. For depth 1 the statementis clear. Suppose that it is proved for depths less than d .By Proposition 4, every big zeta value may be written as a rational linear com-bination of series of the form Z ( S, D ) , where S is a basic d × d -matrix. We needto show that all these series are linear combinations of ones of the form Z ( T d , D ′ ) modulo series of a lower depth. The strategy of the proof is to replace iterativelyZ ( S, D ) with linear combinations of big zeta series whose matrices are closer andcloser to T d .To define what means ”closer”, introduce an order on the set of indices of a d × d -matrix as follows11 < < . . . < d < < . . . < d < . . . < ( d − ) d < dd We say that two matrices are equal up to place ij if they have equal entries atplaces not bigger than ij . We say that two matrices are equal exactly up to theplace ij if they are equal up to this place and are not equal up to the next bigger N ALGEBRA OF BIG ZETA VALUES 7 place. A matrix A is closer to matrix C than matrix B if A is equal to C exactlyup to place ij , B is equal to C exactly up to place i ′ j ′ and ij > i ′ j ′ .The proof consists in the iterative application of two procedures. Procedure turns a basic matrix into a linear combination of upper triangular basicmatrices. Given a basic matrix, rearrange its rows so that the number of the first nonzeroentry in the row decreases from the top to the bottom. Consider rows with thefirst non-zero entries in the row. If there are more than one such rows, take anytwo and apply Proposition 6 to them. We get three matrices. The one of the lowerrank may be thrown away by the induction assumption. Take two other matrices.Rearrange their rows. The number of rows with non-zero entry at the first columnof both matrices is one less than this number of the initial matrix. Repeat thisoperation until the number of such rows equals one. Then turn to rows with thefirst non-zero element at the second place. And so on.
Procedure turns an upper triangular basic matrix, which is equal to T d exactly upto place ( i − ) j , into a linear combination of basic matrices, which are equal to T d up to place ij . Given such a matrix, let k be the number of the last non-zero entry of its j -throw, ” ∗ ” means an unknown entry: i j k ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ i ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ j Apply Proposition 6 to rows i and j . The matrix given by the last term in (10) isof depth d − T d up to place ij . Thus, consider the matrix given by the first term. NIKITA MARKARIAN
Add to this matrix a column as follows, the added column is marked with anarrow:(12) ↓ i + j + k + ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ i i + ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ j − j (it was an ambiguity in arranging i -th and ( i + ) -th rows, we choose this variant).One may see that columns with numbers from i to j + i -th column removed (thatis the initial matrix) is a linear combination of matrices, which are (12) withoutsome column numbered from i + j + i ′ -th column was removed. Then in the resultingmatrix rows with numbers i ′ and i ′ + i ′ s place and 1 at the i ′ s place.As in Procedure 1, apply Proposition 6 to these rows and rearrange the rows of thematrix so that the number of the first nonzero entry in the row decreases from topto bottom. One may see that we get a matrix, which is equal to T d up to place ij .To prove the statement of the theorem, firstly apply Procedure 1 to matrix S .It gives an upper triangular matrix. Find the first place, where it differs from T d ,denote it by ij . Apply Procedure 2. The result is a matrix, which is closer to T d .Apply to it Procedure 1. One may see that it does not change the first i rows andthe first j columns of the matrix. It follows that it is equal to T d at least up toplace ij . Find the next place where it differs from T d . Again apply Procedure 2and so on. (cid:3) Integral representation
Denote by M ,w + the moduli space of w + M ,w + fixthree of w + ∞ . Coordinates ( t , . . . , t w ) of other points of theconfiguration are called simplicial coordinates on M ,w + . N ALGEBRA OF BIG ZETA VALUES 9
Proposition 7.
The algebra of regular differential forms on M ,w + with logarith-mic singularities at infinity is generated by -forms (13) ω ij = dt i − dt j t i − t j ≤ i < j ≤ w + ij ≠ w + , where t i are simplicial coordinates, t = and t w + = . The only relations amongthem are Arnold relations: (14) ω ij ∧ ω jk + ω jk ∧ ω ik + ω ik ∧ ω ij = Proof.
See e. g. [Bro09, 6.1]. (cid:3)
Introduce cubical coordinates:(15) x = t x = t / t . . . x w = t w / t w − The standard simplex(16) ∆ w = { > t > ⋅ ⋅ ⋅ > t w > , t i ∈ R } in cubical coordinates turns into the cube(17) ◻ = { < x i < , x i ∈ R } Let A = ( a ij ) be a basic d × w -matrix such that the series Z ( A ) converges, seeProposition 1. Integrating the Taylor series of the integrand term by term we geta relation(18) Z ( A ) = ∫ ◻ ⎛⎝ d ∏ j = ∏ k x a jk k − ∏ k x a jk k ⋅ w ∏ i = dx i x i ⎞⎠ where the integration region is given by (17). Proposition 8.
The integrand of (18) is a regular differential form on M ,w + with logarithmic singularities at infinity.Proof. Denote by l ( i ) + r ( i ) the numbers of the first and the last unit inthe i -th row of matrix A . Substituting (15) into the integrand of (18) we get thedifferential form(19) d ∏ j = t r ( j ) t l ( j ) − t r ( j ) ⋅ w ∏ i = dt i t i where t = G an unoriented graph with vertices V ( G ) = { , , . . . , w } , with edges E ( G ) corresponding to rows of A , vertices i − j are connected by an edgeiff A contains row e ij (see (1)). G has no cycles, otherwise rows corresponding toedges forming the cycle would be linear dependent. Thus, G is a disjoint union oftrees. Make these trees rooted, choosing the vertex labeled by the minimal numberas a root. Equip G with an orientation by orienting any edge into direction ”fromthe root” of the tree to which this edge belongs. Call an edge ”right” if its source’slabel is less than its target’s one, otherwise call it ”wrong”. Rewrite (19) as(20) ∏ ( ij )∈ E ( G ) , ( ij ) is right t j t i − t j ⋅ ∏ ( ij )∈ E ( G ) , ( ij ) is wrong ( t i t i − t j − ) ⋅ w ∏ i = dt i t i where ( ij ) connects vertices i and j , j > i .Expand brackets. Every vertex of G has at most one incoming edge. It followsthat each of the summands has in the numerator at most one t i for any i , which can-cels with the same term in the denominator. Thus, each summand is proportionalto a product of differential forms (13). (cid:3) We left to the reader to check that for matrix satisfying conditions of Proposition1 the differential form given by this proposition is convergent on the standardsimplex, convergence conditions are described in [Bro09, BCS10].
Proposition 9.
Differential forms appearing as integrands in (18) for any, notonly convergent basic matrix A , generate the space of regular differential forms oftop degree on M ,w + with logarithmic singularities at infinity.Proof. Let(21) w ⋀ k = ω i ( k ) j ( k ) = w ⋀ k = d log ( t i ( k ) − t j ( k ) ) be a top degree form on M ,w + with logarithmic singularities at infinity. ByProposition 7, these forms generate the space of top degree forms with logarithmicsingularities at infinity. Consider the subset of indexes k for which j ( k ) ≠ w + e ( i ( k )+ ) j ( k ) for this set of indexes. Theserows are linearly independent, otherwise functions ( t i ( k ) − t j ( k ) ) would be linearlydependent, what would follow vanishing of the product of their d log ’s, this is aconsequence of the Arnlod relations (14). Thus, the matrix is basic.Consider the differential form, which is the integrand of (18) associated with thismatrix. By the proof of Proposition 8, these form is equal to the initial one up tosome differential forms of type (21) with a bigger number factors ω i ( w + ) . Applyingdecreasing induction on the number of such factors we get the statement. (cid:3) It follows that the space of convergent big zeta value of weight w is the space ofperiods of the pair ( M δ ,w + , M δ ,w + ∖ M ,w + ) , notation M δ ,w + is introduced in[Bro09]. Theorem 1 states that this space is generated by multiple zeta values ofweight w . It is in good agreement with the fact that all periods of these pair aregiven by periods of the fundamental group of a projective line without three points([Del89]), that is multiple zeta values, what is proved in [Bro09]. In contrast withthis proof, which states that the weights of multiple zeta values are not bigger than w , we get multiple zeta values exactly of weight w . It is natural because in (18) weintegrate a differential form with logarithmic poles. References [BCS10] Francis Brown, Sarah Carr, and Leila Schneps. The algebra of cell-zeta values.
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