 Thank you very much. So I continued with MRS factorization already presented by Gerard and but particularly I will give some application of this MRS factorization. This application concerns PolyZeta, polynomial homogenous polynomial connection between PolyZeta which I try to explain in the recent report published in Cuffland test mathematics. See now I would like to present it in live review, what I've written. The plan of my talk is follow-up. This is the third part, part one and two. I will give some work on the course of polygagapism and polypsum in which I can derive some property about PolyZeta. And to establish a general polynomial relation among PolyZeta, I mean the technology of non-computing general series. So I will also include some important result, analytical result of that. Before to draw out some consequence of the particular property of PolyZeta, I will give some computational example about this particular, in particular the Pritch's equation, the Pritch's equation give a relation among the actual truth of PolyZeta. Here we have something like the up-end theorems on analysis, but of this time I try to establish for non-computing general series of PolyZeta, of the other evidence in Asia. On the description of the image and the cabinet of the polymorphism Zeta. I will explain more why is the polymorphism. First of all, what is the PolyZeta? The PolyZeta is just a special values of the Zeta function on several variables. It is defined by this, what is this next example. And this sum is convergent in the domain hr, where r here is some positive. And when this convergent PolyZeta, this sum can be considered as a limit of infinity of harmonism and the limit of z to the one of polymorphism. Harmonism is considered as a payload coefficient of the polymorphism over one minus and the polylocartism is defined by well defined on the e-skin. And now, sorry, PolyZeta, the area power of PolyZeta is spun by the values of polylocartism of one. Here, the minty indices are intercoms with sr, so 3 than 1, and r is all positive. And all the values of the PolyZeta are infinity. Here again, the minty indices are intercoms, s, and 1. In this case, when the minty indices are integer, we can see that this minty indices can be considered as a sequence of some monolith generated by integers, which can be considered as injection with growth over the alphabet y. Again, it can be considered as an intersection with growth ending with x1 over the finite alphabet x0 and x1. Now, we can see that the Zeta is sum mapped over monolith. It's not just sum, but it will be mapped on the monolith with a loop in C. And in the case of minty, integrand minty indices, when the generator can be recorded by growth ending with x1, because this can be decoded by iterative and recurrent. And the iterative and recurrent in this iterative and recurrent can observe the order of the growth and the order of the iterative and recurrent. The iterative and recurrent is a long path x0, z0, zeta, and with respect to the different form of the other row and of the other row. So in this, we can already see the role of x and y, at least to consider. The window was of x, x, and the words over y. x will keep the order x1, fifth and x0, and y1, fifth and x0, etc. And here, the cartographical x in red in the slide is in red, when we do not see x or y. And we will consider, of course, polynomial and formal power series over x with coefficient on some cognitive thing containing Q. And we will consider also the Bianca-Pra concatenation and co-product of the shuffle of the Bianca-Pra over alpha y. We keep it as a concatenation and the dual flow of the shuffle. She already gave the definition of the shuffle and was the shuffle product of the shuffle, so I don't need to read one more again. In the follows, I will consider the ring, often consider the ring is some differential ring of holomorphic function as a simply connected domain and keep it the differential partial. It can be extended over formal power series of x with coefficient on the holomorphic function by derivation, third factor, coefficient by coefficient. Now, complete the definition of polylogarism. I put polylogarism x0k as iterated with the path 1 to z associated to the word x0k. It's just a logarithm power k over factorial k. And by this one, the map le bullet now becomes the morphism of Bianca-Pra from shuffle of Bianca-Pra to another morphism of polylogarism. It is morphism of Bianca-Pra and it is injective. So this Bianca-Pra polylogarism, I admit the polylogarism exists by leading words and I can take basis. Now, if we modify the polylogarism by taking polylogarism over one minus z, this function is expanded as a Taylor expansion and the coefficient of Taylor expansion is nothing as a molyx. And this map, again, is the morphism of Bianca-Pra to the shuffle of Bianca-Pra to the adama shuffle of polylogarism. Here, the morphism is so this Bianca-Pra, I admit the pl, and it exists by leading words for the adama shuffle. There is an equivalent between the modified polylogarism and the Taylor expansion. So the morphism, as a lecture from Stefan Bianca-Pra to the Bianca-Pra of Aconicsum is injective. And now, Aconicsum, and it exists by leading words for I can take basis of the Bianca-Pra of Aconicsum. Aconicsum is three points. The zeta now is really the polymorphism from the shuffle of Bianca-Pra or from the shuffle of Bianca-Pra to the Bianca-Pra of Aconicsum to the Bianca-Pra of polyzeta. So the polymorphism zeta, I would say polyzeta, polymorphism zeta. Bianca-Pra have already gave a reason why we adopt the name zeta or polyzeta. Now, we have a second reason that it is polymorphism zeta. This we just need to have their value after the I got 5 generator. Here the leading words for I got 5 generator for polyzeta. This function is partially defined, so we can extend it as a character. Now we precise also for the generator x0 equals 0 as a logarithmic one. For the two next generator for alphabets and alphabets 0, the value of polyzeta after version. But here we take the value 0 because the finite part of the singular expansion of logarithm 1 minus z in this comparison scale is 0. The same for Aconicsum, the asymptotic expansion on this comparison scale gives 0 as a constant term. Those are the half, we have a definition of the z-module generated by polyzeta at vk. We can also introduce the aspect q vector space of polyzeta of longer gate or of way gate. And we can see that zk is q tensor over z of ak. And Zagir have constructed that the dimension of ak satisfies this regression equation. It leads to the problem to know if this map, this row is objective or that is a cart algebra. So let us see now to study these two characters. I now to go to recurve the urban light program by considering the diagonal series on the co-catenation sufferer behind the path. This diagonal series sum of double view of urban world of the view by the unity resolution of unity. This also is a sum of all worlds S double view, P double view. P double view is the Poincaré-Mirkoff with different worlds after overlapping the path. And S double view is its dual basis. These two, your basis are multiplicative. So we can factorize the real linear worlds so called recurve by Gerard and S factorization. It is the same way for the co-catenation sufferer behind the path. The diagonal series d y is sum over double view for all worlds on y star is sigma l times sum of all worlds over all worlds of the x y star sigma l pi is sigma double view tensor pi double view. Here again pi one is Poincaré-Mirkoff by this of the developing Agapra. LPL here is the PL for basis of the Lee-Agapra of premitage element. And we obtain here the modify the sufferer modify of MFS factorization. Now we use two diagonal series to form at first the general series of polygraphism is just the image of the LLX tensor identity of the general series the X and the polygraphism is the image of ash bullet tensor identity d y. And using the factorization we can factorize y at z and h n is y. And we can the definition that suffer either the value of one of L regular here the factorization are co-version because all the co-efficient on local co-ordinate of this series are polyseta co-version they are known the version term is in the local co-ordinate. The same for the second one this is factorizer the co-version difference and again here the local co-ordinate of this co-ply series are co-version. And this series help satisfy the non-completive difference series this difference series again and satisfy also the as a particular condition at the and with the change of variable X over A over 2 pi I and X1 again minus by B over 2 pi I is nothing as the equation three and the series z suffer is nothing as the different associative. Now let's consider the third and last for me the non-completive series z gamma where here the co-efficient gamma double view is one if the double view is right work of not the just the final part of the asymptotic expansion of the asymptotic view on this now is in the comparison scale and up to B and up to A and look at it up to B okay and this again is a constant gamma bullet is a character so we just need to have the value you just need to have a look at the basis here we take the basis sigma k and so so gamma gamma that is a character this series is a co-ply series and we can be factorized by MMS stiffer modify MMS fragmentation and we do up there on the left of these co-ply series the factor exponential of gamma by one and lie on the left by zeta stiffer zeta stiffer is this series okay the co-efficient of exponential the coefficient of the exponential my one is zero but in this series the co-efficient is gamma the left coefficient for the need is the follow i will enter into the mono zeta is an ordinary general series out by y1 up to N is the one um variable series with co-efficient on olomorphic function and it's the the law co-efficient of hp y1 n my one n is a harmonic sum so we i can consider this no competitive this is an ordinary general in series and using the definition of p1 p y upper right it's nothing has a political x1 over k over uh one minus z with is this expression so we get mono zeta is this function and using the newton-jirard identity or equivalently this star kin star t y k is exponential of this exponential of the legend function the long function co-efficient is exponential sum the long function this co-efficient series give uh taking in this formula the finite part on the left and the right number of the symbol we get here b1 b y1 is nothing as the function one over gamma one plus y1 here you see that the coefficient of y1 is gamma again is gamma but for the prime y1 in the mono mode of a card the coefficient of y1 is 0 we will use this non-continental series later now the non-continental series of iterative and current or iterative and current associated to worth of the view is nothing as the chain series of the differential for obliac therefore an integral one along the path z0 to z and this series is could be like this is also differential solution of the more complicated function equation now we take the transformation option z map to 1 minus z this change of variable will map on by pullback on the differential for obliac therefore n number one by g star g therefore equal minus obliac one and g star obliac one equal minus obliac one so the change series associated to along the path g z0 to z is by definition is a different series of this integral using the change of variables is nothing is the substitution on the chain series along the path z0 to z by the morphism sigma x0 again minus x1 sigma x1 equal minus and now the change series of this path is related leading to the non-continental series of polygonism by this formula it's the same for along the path g z0 to z okay and use as a particular behavior of l at zero we can deduce that when z0 go to zero this change series is equivalent to sigma and z exponential x1 logarithm at z0 and we can deduce when z0 go to zero the formula linking non-continental series of polygonism at one minus z is equal sigma of l z z shuffle this morphism is evolutive so we can put again and we get this correlation now use the capitalization MFS of the numerical teaching of polygonism we get the second identity so we can deduce the we have here asymptotic we have here as one of n z is equal exponential minus x1 logarithm one minus z figure and taking the Taylor expansion we can also deduce the behavior at infinity the finality of h n equivalent to constant n by one another term we get this abel theorem okay express the limit of this the series of polygonism multiplied by some renowned factor the counter term this is a counter term of all direction of one is the same the limit of n what do you actually mean of amulism h is a non-competitive series of amulism multiplied by its counter term counter term of that of that fashion. Now, in the last quality, we take the finite part and we get the each equation. This equation makes the relation between the Schroff-Rangafar and the Schroff-Rangafar of polyseta. Use the factorization of MPS, which is equivalent to this formula. Here is Z gamma, here is Z stuff. In the second identity, the regular step, use B times Y. Y is the first previous equation, use B Y1 in the B Y1 figure, gamma, but not first. So, after that, when we identify the local coordinate of this equation, we obtain a relation among polyseta. Polyseta. Do you remember this formula? Here, gamma figure, gamma, the coefficient of Y1 is gamma. Okay. Now, I can remark that in this asymptotic expansion, we can say that zeta shuffle of the reduce is the coefficient of zeta shuffle. It is a finite part of the singular expansion of Ni w in this scale, the same way. Zeta shuffle double view is the coefficient of zeta shuffle, which is the finite part of H omega in this comparison scale. And at the present, I think it is the only case we can justify how to realize divergence polyseta simultaneously at X1 and Y1 for the shuffle and what is the only one case we can justify analytically and analytically. Now, let's us back to the differential equation here. That is just the first of the differential equation. The Galois differential equation is nothing as the half loss of the group is the group of exponential C of where C is some least series. So, we can clone solution of the A by we can lie on the left, on the right of N by some exponential of least series. And so, we can clone the Shikha z by multiply on the right by the exponent of C. And we have again the orthopedic expansion of the clone solution at one and that they log coefficient actually. And the Abel-like theorem is again, is a sense form, is exactly the clone theorem Abel-like theorem. Now, if we introduce the DNA is the group obtained by multiplying on the left of exponential C. So, C is some least series on the coefficient A and the coefficient of x0 and each one on exponential C is equal to 0. In this case, here again we obtain the equivalent on the clone of zeta gamma as the clone of zeta shuffle by this formula called Pritch equation again and it is equivalent to this clone from I dot it. We have already saw that the local coordinate of the clone shuffle z shuffle and clone of z shuffle we are put over numbers polynomial on co-version with coefficient in A. When we need to identify the local coordinate, we get correlation among political polynomial correlation with coefficient in A. But these coefficients are these relations are free of constant gamma if gamma is not belong to the ring of our coefficient. Let us see how to use this formula for example for the Pritch equation this single equation. If we identify the coefficient y1 uppercase double use we get the generalization of gamma y1 uppercase. This is a closed formula to keep polynomial on the co-version single zeta values and gamma the constant gamma earlier concerned gamma is quite here it's not equal to 0. It's the same for the other formula gamma y1 uppercase double use. The formula is more complicated but we can implement it in and to obtain the expression by exact coefficient by coefficient. Here again we can see there are polynomial on co-zeta and gamma but when we use this same formula the Pritch equation we make the polynomial identify the local coordinate we get the polynomial relation among political these equations are homogeneous in weight since all the coordinates are of the right and this is by the reverse and written words are totally of the operator and we can recover by weight to get this relation. And I insist again that the own polynomial relation among political are independent of constant gamma is the same is the same for the closed Pritch equation. Now since zeta is the morphism when we can go through the there on the left hand of the of the equality we get and we do appear here the polynomials qrl ql will generating inside kernel of zeta this polynomial in bracket are homogeneous in weight each polynomial are and actually it's by the reverse and we can take the ideal shape I think by ql thoughtless f y for the stuff one and f x for shuffle we read here we can see the inclusion now we test the symbol equal by row we get here nothing else the some reflecting system on the local coordinates here we can see that for each n on the written words zeta of sigma n profile to the polynomial on all the zeta and xh by written words okay and on the right of the these are reflecting rule they are the irreducible poly zeta because all these steps reply to each other so they don't cannot be readable so they are irreducible and we can see for the gen weight by weight the set of irregularity at infinity and we put their irritable by the human for all p of zeta irritable and the irritable we can see in this formula that they fall on a higher picture system for the higher part of what is it okay now we take the image of this image because now zeta we already see that zeta is subjective so we can take the image of the section we get the reflecting system over sigma n okay here and irritable is obtained by the inverse image of the section of zeta we have to get again the chain between and irritable we okay the sigma n refers to himself is this equivalent to say zeta the qn is qn written pattern are several of notes so let's explain summary summary like that the identification of local coordinate and give us the two family the first family is irreducible zeta values okay it's obtained by the chain and the inverse image by the section of zeta so that zeta restricted on q the expression of q by irritable to zeta this map this restriction is rejected rejected and I give also the set of the polynomial of the polynomial of qn generally beaten fx okay in which for any n the qn is homogeneous in weight the weight is nothing as the weight of the written words n so we have to see that there is a equivalent to say that q is zero of sigma n refers to sigma n of by definition sigma n is irreducible in the case of qn is not again so qn is a polynomial the little detail is nothing as sigma l sigma l is a the element of eigen factor values it's not belong to add irreducible so it is transcendent on the eigen factor generated by l qn is equal sigma n plus epsilon n epsilon n l is a polynomial on qn direction okay so we can say that the eigen factor generated by sl for l conversion can be decomposed by fx plus q and irreducible this sum is a direct sum of weight of space of course so we have already see that fx is included in the cabinet of zeta and now if we take any polynomial on k zeta not the three of course constant that no without constant that so q can be decomposed on q1 and q2 this is the coefficient and by reduction by fx q1 belong to fx so we can see now the image of zeta is nothing as zeta is generated by irreducible poly zeta and the cabinet of zeta is exactly the ideal fx by that we can conclude that z is a factor can be good conclude that z is kind of factor because z is a gamma of zeta is obtained as a Gaussian of zeta by the kata ideal okay so that is kata and now if we take the any homogenous polynomial and the product belong to the cabinet of zeta homogenous in weight each polynomial cn is of different weight because we have this equation so c could not satisfy the eigen solution with coefficient in q but for any s in irreducible s is homogenous in weight it satisfies the previous scheme so we can say that zeta s is constant over q so I think I I finish I must stop my time is up thank you very much for your understanding thank you min for interesting talk questions yes I have a question yes if you go through slide 23 yes then you say that at the middle of point one you say that the map going from the algebra generated by lindon irreducible to z is uh objective yes can you explain a little more no that's the restriction of zeta because zeta zeta is subjective okay yes the the element l r are obtained as image inverse image of that irreducible by the image and there's also section so the restriction of zeta the the map zeta is the the polymorphism of zeta is defined the shoes are even there I just between them there are some irreducible zeta I just take the image inverse image by the section and the so that this an r is an inverse image of zeta irreducible to solve for this middle bijection and uh so you constructed a section yeah did you construct it formally or is it uh reasonable intuition yeah the contraction is nice okay from the relation among poly zeta okay yes zeta is the morphism so I can recover on the left own poly zeta and after that I use the fact that zeta is subjective yes I just factor the relation each relation and exceeds by this polynomial qrn okay I just forget everything to take the annexation by qrn the section you uh are describing as a unique the section this can not unique and the section can but the section is not unique yeah okay I just take one this one is um it's for my covariance I take this one but from every projection you can always build the section the section yeah I can find the section to just by this way but if it is subjective it you you need this section to be unique at the moment I don't need this hypothesis that it is unique I just take at this step I don't suppose that it's unique I can at least it is a it is a good track it is a good proposal to with many many things to check together of course and with people who would like to because it is easy to implement as I see and if you consider the relations you have by identifying coordinates yes did you recover the relations up to order I don't know up to a certain order yes recover the relations that you have with double shuffle and yes of course I formally to prove uh prove that the this equation the relation obtained by the identification of this equation um implies the double shuffle relation and at the weight jump we have already verified that the whole relation obtained by software satisfies the uh okay I am not a specialist of the subject but uh the zaggy paper uh in the zaggy paper I don't remember which one uh which year but the the the direct sum was formal direct sum yeah the the paper of zaggy zaggy is this one in this paper in Birkhauser oh yes yes this paper yeah the direct sum of zaggy concern okay I show back okay but it was very descriptive in style so as he didn't say that it was formal I think it was formal at the time because it is still conjectural yes I don't I don't ask uh don't get because in the paper exact exact in this paper he seemed to say that is give this direct sum is some something is uh abstract and formal direct sum why is that here is something is uh closely closely constructed over an arithmetic constant is I don't know what he means that this sum is direct but this sum directly I already saw that no it is direct because it is formal so you take the slices okay and you put them in the direct sum uh in the in zaggy paper because they are quite defined okay maybe Dominique I speak under the control of Dominique Malchon who knows the subject much more than me so maybe we can uh because in two minutes it has to be talked by Natalia okay so maybe we can discuss this later yeah okay thank you thank you again