Feynman integrals and hyperlogarithms


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Feynman integrals and hyperlogarithms
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Panzer, Eric
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Institut des Hautes Études Scientifiques (IHÉS)
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Many Feynman integrals evaluate to multiple polylogarithms and their special values like multiple zeta values. One particularly successful approach to understand this phenomenon is due to Francis Brown and uses iterated integrals called hyperlogarithms as a basis for the arising transcendental functions. I will go through the computation of a simple example in detail in order to explain this method, which combines calculus, algebra and combinatorics. The underlying geometric idea of linear reducibility will become clear as well. This hands-on talk aims to be accessible to students of both physics and mathematics.
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and then you nothing can and I think you're election think you're much too you stayed around in the afternoon this last fall primarily for the organizers to this opportunity to speak at the edges and I will try to connect with friends is a story in the morning and after this so the previous told some other approaches and ideas on how to take a family jewels on on the Parramatta representation with France's showed you in the morning and actually France's put forward at the time the new approach to the computation of time intervals which uses a hyper with them throughout and is 1 attempt and 1 way to understand the prevalence of multiple . women's and CDs in in the calculation of many minivan diagrams sold 1 of the most financially knows that this is the only 1 that nations would have lowered and then the motivation for this particular approach is 1 of the goal is to understand why there are so many firemen and you also have disagree with that I know expressible entire multiple body logarithms investors already showed the slashes in the morning and just recall briefly cities these of functions it again I'm like like multiple series of distressed are now the index by amount of integers and want to Indian they depend on several complex variables and wanted to define them my at the summation over nested integers K 1 2 kg and then using sometimes does the model indexed by the summation in Israel and the parameters divided by the summation in this eastern raced to certain powers as stated by the index of models .period all them all the scenery some examples and was 2 over these functions show up environment calculations and it's still 2 days of mysterious why there's so many of them where this is the case but you also know we remember not ball Simon diagrams of intervals all of this fall the noise basic examples and they have many conjectures of Raftery I'm pretty sure that that that therefore much more complicated form and you think of the whole world of time and I've enzymatic probably this is a set of measures 0 and then whatever kind of measure you put on the set of firemen grasp at the surprise is that for most things jump relevant for visited calculations actually you get already very farmers just the Samoan islands but still understanding why they all this form is very difficult and we will ever tried to demonstrate the 1 approach with which it is in some cases makes this a very clear why this is the case I tried to be very explicit and very down-to-earth survival has undertaken explicit example that we've seen already and computer from beginning to end so anything is unclear in the meantime please interrupt me because it should all be 1st a 2nd year calculus I hope that it's still interesting because it's see what's actually the important the important ingredients like this approach works so the idea everyone followers take a parametric representation and integrate out watching a parameter at a time the recall that we had parameters associated with each stage of the graphical Avery needed fishing intrigued many at the parametric representations and I will consider an exquisite example with 3 slopes which was the 1st nontrivial example that France's mentioned in his office so what was the wheel it is this crap it's we'll have 3 spokes call these edges 1 2 3 4 5 6 there's think of it as a graph fighter before the year is 4th with 4 external legs but because the the division for dimensions it has some residue the driver not by iron this craft was just being too Over major G divided by the square of the graph .period only the 1st 2 months of polynomial what did this mean all evaluators and you will find so we just have to integrate from 0 to infinity polishing up parameters well not actually all of them because reminders this for it's a projected interval so then made fine began and restriction which were happen never choose the payments in the form that I had to sit out a 6 2 0 fell into gave 1 of exciting spread out me but I set up a successful bid to 1 source said this the interval we want a computer In the recorded by definition this semantic putting on the society was the sum of all spending trees and spanning tree contributes among unknown know which is a part of all it is not the spanning tree so In particular With victory each individual variable this means good that that the 1st integrations are essentially elementary what this work we want computer amplitude and Letters to the 1st annual Over the 1st Variable from 0 to infinity of so over half of 1 contagious cleared just remind you that this is a linear knowing that it's just say that I call PCI 1 the coefficient of up to 1 that's going on and so on the constant part in this interval is clearly attributed to do and just as the primitive of the integration Smith evaluators there'd unity and just get 1
over the product of tap 1 this was easy the next stage we have to leave the rational functions the latest compute the next interval of of the 2 set up a 1 7 0 0 1 I know the sire coefficients of the region .period homage was in each individual variables this point of still union the next variables so we can do is similar the composition the other and I just continue this labeling schemes like 0 1 2 the coefficient of of the 2 since I 1 which is quite normal depending on the other variables and set up a 1 0 0 2 the constant part with the specter a 2 in the bottom of this just shorthand for for particular spanning trees which contain the edge to but it's 1 of which contained neither of which wanted to inform him of of the need to so so I started doing it when already complete square so just get the coefficient of of 1 times the constant part so this is a tragedy .period normal but Victor rising into a linear forms In those who feel the ISS these Lisa .period almost a year about 110 or 1 article visions of respect other ones available that would animals and the remaining members of the report In this case the justices in the world "quotation mark including company selling eponymous innumerable they in each variable I started apologizing in which they not Nantucket efficient so it's going in all the other rebels this important .period sir I only if the new transient here said on 1 of the 2 of its and the constant part I so like this interval intergalactic part affection the composition so I get the prefect of from the Fatah affectionate composition the chicken worked out very easily and then he left the interval of so I hope we can all believe this formula and now we can just compute this enabled tons of rubble with them said evaluated their at infinity when this thing it just becomes law with them
call sign attitude or 1 side of we have to and in the denominator we have the other 2 proficiency want to and settle what we see is that the Canaries ended with the 1st 2 variables and at that stage the the partial interval if computers survives along with them off some coefficients of the grappling all milk and has a prefect which is given by this point from the partial fraction decomposition now the idea is to continue with this injury 1 run after the other but in general now in a difficult situation because this is not generally a gratuity .period all mothers as before because multiplied to the new things and if you want to go into it again we would like this actually continues factorizing these linear forms fissile here so that a look with this actually is in this case so what is the scion up 1 or 2 I want to integrate after 3 next so that's the compose this with respect to the 3 to 1st of all this is the coefficient of science which also offer 1 too is set to 0 so it is the sum of all the spanning trees which do not contain into 1 but which regional contain edge to and you you think about this actually means and this is the autograph .period the graphic indeed meet edge 100 contract edge to the so-called contraction addition formula with a takeover of the edge won that contract edge to what happens is that 6 and 4 become parallels 6 and 4 instead 3 and 5 in the business this is actually the scuffling along that I want to see how it depends on of 3 just for the fun of it so I know this is any travels the provision of of the 3 of us again the spanning trees which could not contain half of these like industry leader which 3 that essentially the 1 grass and we know from 1 graft the some of all the entries in the news and then we have a contribution thousand 3 0 after contract after 3 this me the rest going on Of the sunrise which With the edges at 4 5 and 6 years in mind this was cemented when Norman commendable you've scenery in France's top noting where you have a share of the yes side of the company ,comma was picked up at 3 in the critical vision of of of 3 and whatever that as that of the 2 0 sway contract after 3 and then everything becomes there are no organized in the same game the other on-site leading edge to and contract then 5 and 6 become parallel and 3 and 4 remain inorganic and indeed of 3 also that I have to pony up to 5 . 6 remaining include and I have the same man I contracts 3 I still get the same sunrise on I don't bother writing the also have other 1 to this means that ideally both edge 1 managed to what happens then is that you have this lonely 6 hanging around 10 the edges 4 5 remaining this is just 1 of government has told you that just have to sum up all the variables include Gators polynomial turned last raised worries that other 1 and 2 2 0 so we contract both edges of 1 of the 2 which means that the X 3 is now connect both endpoints of 3 amount connected because it conflicted 1 into this limit on so we have
this verdict edged 3 is like a South Loop and then we have the sunrise of 4 5 and 6 years but the south of nearly I was standing trees amino spanning tree cont can contain history be with and would have a new and spent reasonable out so actually know spanning tree contains 3 which means that half of the actually mother prices .period Olmert's Advisor of the point and then the thing that remains just again this summer as part of again so we will soon see the reason why I wrote of his way to what is actually this denominator we want understand this kinetic . 0 million so what is it well let's collect their terms quadratic enough of 3 so we get terms quadratic 3 from the problem to minus the product these 2 so from this product and powerful because of 6 times of a fast without and then we subtract the police to which this gives sunrise then we have terms junior another 3 that year the terms in 3 come from multiplying the sign with such a chore and also hear from the subtraction from the turning out 3 is this times this 60 all these terms are divisible by sunrise rough .period on and we get on the focus of a 5 and throughout the city from these then subtract up 4 1 4 5 then finally have a constant part which just the square of this on .period now you
see that actually these Council and what is this .period unloaded on him such as such as a modified without the only thing that remains in here the office spread so what you find is with the family with of the unit that the models it's always the same I was lazy it always has been a member of the and its emerging phonetic among the 3 variables will follow at only 4 overs remaining and I explicitly look what happened after 3 and for so we observed where the miracle happening and think this thing and he accompanied square but you can see over there that's just ethnically tense after 6 months to December when normal speed and this thing inside the squares of is again lineage viable because you know that sides obviously manager this is a complete square Of course there must be some explanation for this and this is an example all also the so-called Thompson identities In these identities follow from the fact that the .period only connection as a determinant of a matrix and then there's whole theory animosities between them intervals of adoption .period Armitage's Sunday also friends and family it's worked out great detail no you might know was the Dobson also known as Lewis Carroll until then I don't know so this is the 1 of Alice in Wonderland and won the point is that the securitizations are extremely important for the fact that we see his mother .period over them themselves so so many places visit this would not happen if this would just be a generic forget it .period all then ordered to continue the debate you would have to take the roots of this .period novel and introduce algebraic functions already but because we have a complete square delivered by parts yes so that they can do it by hard-to-get one-term River just knew it competes Granby keep the log just evaluated the 3 0 at infinity and the other German states into still but not became rational and do again separation that part part of actually composition than the again get deadlocks that the point is we don't get a dialog with him so the next integration because of his company's fare he remained vote of trouble with arms and elementary functions and the 3 others say that the the period but the residue of this craft you get doesn't yet because of this there is no possibility several terms which can you write them with the cemetery and you actually see that I can write 3 times the same into gold the amateur in terms of far the now for and then the sunrise polynomial and lowered its forecast of a 5 . 4 per cent 6 over son asked you is easy exercise of his calculation that we get this expression the point is that because of this ,comma scary didn't get a dialog with them at this stage now I want to write solid explicitly so remember we have 3 variables left upper floor 5 of the 6 but a successor to 1 so ridiculed the two-dimensional interval never just rename the variables so after 6 years 1 after 5 firefight called Why don't before reporters said this is the end will to infinity the wild Y and you can do the part affection the composition here To ordered for you here this expression Bloomberg said so this comes from the sunrise .period all of his new the substitutions under the party faction the composition and then you just get this is not called good and that's why didn't just 1 and what is going on just fly away from here the White on the computation of interval to do this to damage interwoven over them and he again see mean that there were already got place in the the 1st the thing that I shouldn't be too but also in this innovation apostasy to here we have to do new part affection compositions are and also there actually made use of stops identities for the the observation is that again everything is doing everything factor into the net function the next integration variables and so you know into it that everything is linear and then with the Kauai and also the arguments yelling and said so Everything looks very and that was at this stage now we have to introduce more general and special functions because it cannot computers into when terms of physical model with national fighters anymore so we do need a dialog on this but the question is how do you represent these column or Withams dusty raised some representation because you don't want to work with the sum he wanted to have an iterated interval representation strong you want to explore the
catering digital representation 12 month employer with loans from other pollutants are not only sums but it can also consider intervals and so for this term uh let us take a set of points in the complex claim to be financed and you think of all the call these these elements of the set singularities all letters and I want to write them special symbol so they introduce symbols a signal which would indicate is actually representing a differential form for all these letters this is an abstract possibility then we can find special functions for each word in the south of the and the model the loss of the rumors is the capital thing all those this RG 1 where you so sorry and this is a different thing man sorry Seoul and redefined the hype along with them Alan W. of Z associated to word w in a star that somebody just a sequence of these letters by the following rules :colon 1st of all if had a string of only 0 0 hours this test should be the policy of the problems and normalization with Victoria then we also warned that if you take a hyper lower which has worked which begins with some letter then there comes some of remaining at that this 1st letter tells you the differential behavior so this should be 1 always that might signal times I have overcome associated to To the tale of the word this is just the reverse way of writing down iterated interval that it does not fix the constant of integration of course the treaty required now such that the limit and that was 0 all of this happened Williams 2 0 0 unrest the word of Florence monument to the end the string of just them to be lower than that of course diverge quiet Monday the euro but for the other Apple weapons defined this way we should think of them as interim intervals from 0 to Z some effect and some local area for example yes wrote that exactly what's going to happen them for example and what happens if you just take 1 letter and this is just being from 0 to Z 150 18 of the miners Sigma and because the Internet given by the French behavior but also has to banish the slothful standing on their own there has to banish identical 0 so it's this situation duel and this is just the novel of them that led to signal the miners in the world then there is only 9 0 only 1 what the point is this does this look like Wellington break from 0 to Z the the wanted where are the 1 minus 0 and this is the last integration differentiated the specter that we have to get get on this benefit and role this is integration of the 2 2 2 my .period and if you recall prefers the dimension that injured during were opposition from allied to his actually minus allied to offer z and quite generally when you have now have a bunch of zeros moments the is story I'm changing orders now the people familiar with the medieval inside out so I take a work but I have to make distinctions about the letter 0 not because it's treated differently in the set that suppose positive which ends a non-zero letter becomes a bunch of zeros and the comes among director and bunch of zeros and so on so I have deed non-zero letters and each of them a cumbersome zeros In this is actually the same as minus 1 to the D times that the mother .period that these industries evaluated that the ratio
of the non-zero letters so we get all the model .period wouldn't just in a particular representation this is the point Of course the the benefit of writing them in this way is that it's true belief that it is trivial to integrate representation because just defined them as the iterate intervals so if you just continue our example there let's just be right is lower than here as an entry into this form so we just have to look at media the arguments of the with so and interval there to infinity can we didn't mind slump that has long don't lie lower this is nothing new and -minus wine and the lower them we're addressing alleges that he was minus 1 and I have and the denominator the moment which has it's similar this Z minus Y 1 5
Mahmoud should have the right to French regales if you think if am before fix these constants correctly which not depend on them but in all of that is taken up with the Meehan said as at 2 0 0 that is good wine the numerator and also just declined in all matters of this vanishes they'd that it 0 and is also defined vendor so this is the correct expression another integrators but definition just means that penned and only 0 and an army of minors while wanted fly to primitive of these and I evaluated infinity From this language this just means None the following so I introduce shortened from imitation L I want to write linear combinations of words and the arguments let us define this finding out an extension but now it is better people and this combination of sorted nice express the integration in terms of an interagency will evaluated at infinity but still depends implicitly on the remaining integration variable Y which are still have to integrate so if you want to continue In this way the 1st have to understand this function as an entry of wine I hope this is clear the ideas that each stage in this crossed process we want to express the interval the into as an interagency will namely have a them in the next integration level and fostered on enough time to do this in all glory detail we have to take the shortcut but it's actually quite simple this so there is a which should improve some suppose that we take that have along with them some word and some arguments that might be infinity like in this case and computer it's still a relative in the case of releasing miles an hour considers functions so that they're not faithfully competed for total diverted and United proves there the police only for this clear some of all the letters and you take the word made needs 1 of those that and then you have an explicit over would make company relatives after consecutive differences of approved there's a differentiated the Clinton rule and integration by parts but it's really very simple exercises do this
I there's a mountain region of this former so there's a stigma 0 appearing here defined to be president and his amendments 1 declined the this corresponded the bond returns of integrated section is a totally symmetric ,comma now with this if you applied if he applied at all derivative mounted such an expression you see this this grading buried this recursive structure with so special about the polyps coming into place here because in our view the pilot rooted on the right inside the only have no wait upon which have 1 that an explicit with Madrid no nowadays these letters here are rational functions of why Soviet computer to go with me derivative of rational functions we just have to factor in the rational function to Minya and his zeros and poles suspect wine together happily with them at the differential forms finding so this looks good and for this piece that's in this case would still be lower than remains again just applied the machinery recursive even example here but it won't work it out now I just started so this function here you and if you do all this it's just um you can write it a hyper lower than with respect to why so the point is that not all the letters are independent of wine so you can convert this represented this and this is I with make there's no magic in this process but now the final to register all right because you just have to integrate it to modify this was 1 of a wide delivery from 0 to infinity so this just means that we have to put another only 0 in front finally arrived that said the residue after the with his photograph that we started with it's 3 times um this thing so have an alternate juror duo of wait 3 know that moment hazardous form there is defending that still might look mysterious but what is it this is an interagency on the minus 0 and minus 1 so we need to know that his mother was attacked you it also think of it as a period of time 0 6 and if you want to write this model with it and I just can you just use them this transformations for example they were associated which relate infinity to 1 in some way so they can apply to this transformation could send said presented with this 1 which means that boundaries the Apollon earlier Infinity number at 2 1 0 stays 0 and it can compute the pull of the differential forms under this nervous transformation December exercise and applied his all in you get that this is the same as the injury will only this course coming up their 0 1 this is now I have a lower them evaluated 1 which is essentially a mother visited you this little thing the regularization have to do because of 0 and you can use the shopper product and that if you use this you control that this is the same as some analysts you have to modify the salt into this little thing which I don't have to explain that it's really not that difficult it can be right in this 1 and just take the definition of the model 4 over them how they relate to the saddled with arms maybe this formula I can and see that this is essentially a life free this is what I want ,comma 2 of his 3 times 3 of 1 closer alignment ,comma 2 of what could 6 times that of the so I spent an awful amount of time on explaining this the 1st Montreal example but I hope it was at least understandable and something the amazing aspect of this calculation is that it gets you ridiculously far In practice so at least in some families for it in for example the single scalar Masters fighter before under rules that France was mentioning this is a case where we have I would think that the best knowledge concerning the the expansion the number of Harken baguette and so computer a lot of intervals this is really an outstanding compared to other schematic configurations can be much more complicated and so we know we already have troops Soviet elliptic things this massive sunrise Thailand's but in these masses diagrams following these lines 1 can get quite hot and the basic idea is exactly the same soul let us forget what it's computed the Pasha Liman and meaning because I more friends called and and that still depend on the remaining rebels from paper this 1 after final battle send Americans into infinity the 1st cable and let's say we just computer such
residue so compute the and it was 1 of the times at each step we just integrate inexorable um the prerequisite for doing this it's that we can express the sideshows have women's right I was already telling you how to manipulate have women's but in many cases the complicated once the fund was not expressible as with and it is on the sharing of parameters and maybe it's not even a sensible interval and can form but in the case of his works we can justify this procedure which outlined to you so the priorities for this to work is that all singularities these functions it is a multi valued function of these variables so there is a there are some devices that some variety which ascribes to this can pick up momentum but we that all of these singularities and on India In the next variable navigate spot because if this is the case then at a statue and I wouldn't based on on this land another lemon which tells you how you get the boundary Constance this and I would imagine this situation can transform a representation such given such an impressive form into a representation is explicit and wife and then we can just find a primitive by pretending that of doing intervention apart and we can compute the financial so the message is that that's all this only working very special situations but it works and the whole procedures automated is implemented and computer programs so we can try to focus our attention on the actual geometry which as its behind this computation so the actual into itself and what happens here 1 of the precise .period Norwood which appear this is not so important the only thing that is important is what kind of singularities on there and how would they depend on the variables so I just want briefly this gets the idea behind this so In the beginning now on the didn't integrate any variable yet so we just have 1 of sites squared the only similarity it is a science I can only have a singularity inside ventures after 1 integration this would happen after 1 iteration much personal habits and there was still almost 2 and suffer what integration that way we needed this just that 1 of example won't settle the perihelion I have to potential difference singularities so we have set up a 1 and said all along and we also had they had to offer to integrations which was this combination we had this denominator um times along with them so happily 1 vote to set up a on top of 1 2 7 0 1 2 so here we have singularities well I will be used for normal Spanish but the bottom line is that the center similarities to take before reduce this then this was also a linear for a nominal rent which I 1st saw this it was spread so and that although in this case of the Veracruz actually fulfilled all the central singularities all union the next observation not everyone understand firemen diagram and all the amplitudes said you might associate of this time around it is enough to just look at this point males and how these singularities developing new integrated more and more variables so there's a name for this which is sung in friends introduced under the so-called putting normal reduction which is very interesting and very important but of course I only have 5 minutes left scientists as stated that their final exams and the name of putting on a reduction To compute upper bounds on this said all similarities they have a name that called on all varieties the bottom line is that once 1 has understood this .period on reduction for particular graft 1 knows that all the amplitudes 1 Creekwood associated with it will be my head as I eyewitness president of mother .period weapons of a particular time so note the actual into an it does not matter a we can any Indian intervention which is compatible with these singularities he started City computer .period on reduction for this graph just with the 1st 2 months of 14 on then we can also make statements about intervals generalized into of this form the bombing but integrate but we raise cited some higher power In case and it would summon polynomial In the numerator so that's that everything is homogeneous or more generally you know I stood received 1 staircase began also general firemen and teachers theory can also such a polynomial but now allowed tho both grasp polynomials excited when normal praised arbitrary integer hours subject Florida the condition that it is well-defined projected annual and so in this case the start of the variety and it contains both going on which general makes things more complicated but it does have a graph where religious masses and you know that excited pajamas contracted and he actually did not get very far but in many applications for example about matters orders perfume as they can
still play the same game and get statements which are valid so I don't have time to expand as I wouldn't want to explain to you what what the outcome of this thinking he said so we have reduced um the study of the actual amplitude and intervals to the larger on geometric task of understanding how these sets of 4 enormous which ascribed the similarities hungry behave and so we have the advantages 19 which of course comes from the ground but some of take a detour we make things more complicated ought to understand the attitude we understand something much more complicated his study hall is partial intervals depend on all the on all the Shia parameters and so here we have a change from we get here by the intervals Pottinger intervals it is 1 and in case of genetics also depend on them but we don't even need to knowing anything but this explicit functions the the only need to know the singing the praises of could singularities so actually we get information about these looking at the sets of polynomials and if you want to understand this moved sequence of singularities you will have to take care of these results in some of these Dobson identities which I mentioned earlier the point is that not appear always but they depend on the comment or structure of the grass to the question whether such a denominator you get from perfection the composition is factorize into the factors it's something which relates to common Torres structures in the action grass In the grass feeds into as well and the bottom line is that it can get statements like the 1 France's briefly mentions the theorem mayhem let GE has excluded now were actually able to give another example the 1 minute left only consider family of the boxes these are the intervals the forest earnings but you can make them arbitrarily and you can make to let's massive on side you can have a triple boxes of what we call a triple box and you can have Hollytree and mobile boxes this 2 massive legs so we have an infinite family firemen graft then there's gene it's such a graft then the actual amplitude associated this draft as a function of the former mentor the 2nd there is another point over and you can actually say what kind of mother .period and so you can write down an upper bound on the on the differential forms which make up the situation well in other words you can specify what the arguments of his mother .period can be in the worst case and there's another class of grass which France's studies which upwards through 3 graphs the Indians no those could contain grass for example the three-eighths of a disease likely spring the and again you can generalizes to have 3 external mentor and because these grasses a special control structure the very rigid and you can use this structure to prove factorization identities of these thoughts and polynomials which then tells you that the actual amplitude is a model upon a widow of a particular type and practice you can also use this to be explicit computations I end here and thank you for your attention yeah please this region as was the he also and I have united anything about image I'm sorry for that what is you you mentioned this in your own the of the head and you use the complexion of life but some of this season removed and then get most of of the sun during the day and he said what we used his plan for a variety of roles the when that is the soul of your case you won't use the Gatehouse of there and factorize militantly the stone confirmed about an hour and that's the most of it will be the involvement of the 2nd edition of the New across almost all of the people in the region that an expansion in like yeah I to get somebody to you and get the support some journalists who will because recent functioning using diesel goes only so much you can say something to the but I think so essentially wanted remembered everybody going asthmatics and make this another case mean envisage that this whole business of expansion by regions for example latest just take individual firemen around we found was a moment people some mascots people in this kind of thing you always doing in this picture it the unified interpretation somewhere because this interval which is made up of this this cycle normal in this polynomial yet a different masses a moment and I look at what happens when 1 of those dominated the others so essentially you'd you would hope that a quick can expand the into ground in this limit and then just come to the endurance intervals themselves the only problem is that if you do this expansion might introduce divergences which manager to give but this is not the case then the you ride so you using this case securitization of this .period normal in this particular genetic limit what it what it hears that I look to the factorization of the polynomial which does not involve any Canadian variance but you also affect relations were numbers for the for the Zeppelin when they're all very important also in the mid make approaches that been mentioned today Seoul yeah certainly very much related what you also realize immediately when you do this is that they such an expansion you simplify the situation drastically because you replace this complicated byproduct of too much simpler polynomials and then essentially the the .period reduction also breaks down into 2 independent pieces because the variable separates so yeah so we do have cases where fit for for example in such an expansion everything's dinner reducible and you can compute the coefficients and other mob Apollo's wistful before actual functions much more complicated and not all what happened so there's also on approach to try to get closer to something very complicated which should understand this is citing the same you so you have reach viewers the nuclear really some old men of the view that produces fuel some of its power over how stable the interest computers have been met with you want some action solutions some clever way to the human nature could combine the somehow I'm saying I must say in all year of course the hope is that there should be something and you would hope to find but so far I'm aware of in some attempts but it is not yet
clear how should that the problem is that it's you have all these integration by parts operations in the ideal world you could just take the representations for different Ireland's with the enumerators but the more 1 numerate and look at this interval in 1 go and then you should we would hope that you see the cancelations of the leading increase whatever the problem is but even tho this is the sort of canonical representation for abided by around so the unique form because of all these integration by parts identities so you you can imagine situations where you take a softer tone that they don't do that then all visible cancelations at Albany do part integration right different form than their manifest in the the soul I know that some people have tried this in another applications the 2 to try to combine the forbearance and and see why certain cancelations happen gage theories but as far as I'm aware this is still very much a work in progress and the dormant don't yet know but the right approach the problem is really that the representations Martinique and you would need a way to find the right representation in order to see it easy slow but probably finding that representations as complicated solving the problem where you what's the change representation from the steps he said the changes which available the future some of give but only look at 1 and 2 at a time it now and then and antennae that's right and changing the representation of major don't it's function just right different ways which makes it essentially trivial to compute the jewel in this new representation so this is this is the law .period what I should mention briefly is says that Of course this is a very simple-minded approach in the sense that I just take the Singaporean leaders that shot 50 years old was 60 years old at least and try to do the interval undesirables and it sort of and this is 1 area that we actually get very far in many cases but also ,comma examples where where we know that interval is modified over the many do not see it in this way because nothing and reusable so you can a situation as something of a very complicated but it's just a sign that actually these variables of the bad variables and it was all of small that it's extremely important to find the right variables and also for the trio of amplitudes if you want to be efficient writing down and you have a similar situation the integration process of course Anthony unclear what the perfect representation in each case would be so this is a completely open field I would say at the moment and the many different ideas 1 could try out and tried to to follow that I Kansas said moment I examples many examples where we have to twist this representation go to another representation some changes of variables and then it's clear that it's appalling over them or something related but not in the original not for a full-blown religion or longer the the use of in some cases usually blocked the neighbors to marshal the form you go the export of senior leaders this is the best of times of relations here but that the problem is In the question came was a 5th different intervals known to combine them conceded that some cancelations if you bring them and she said it is elects amassed intervals and sure right everything in terms of his nascent it was such independent then you should see it on the phone but it's not necessary manifest in the In this kind of representation of what could be used for sure if good rest the arrange the the creation of party and I it was like it was useful to the review on I did not think it was the beginning of the year the rest of the Institute of should disclosed on whether this interval and it has been computer thing to a 3 years ago In the end the onto the masses case and that the probe was always 1 of the kinetics that so if you are get away with the spokes and distributed at a three-point function and sleep with 1 moment and 2 0 but take the other 3 tremendous in this class of voters 3 grafs at end of Tandja's that this function you can compute explicitly in in terms of multiple problems to Alderson epsilon close to or even dimensions with arbitrary powers we on the on the propagators it soon as you go to 4 offset certain moment I don't even know what the rights representation for the cannot exist in a even if it takes the 1 who books 74 offshore momentum it is only 1 mentioned but if you're performers excellent I mentioned not really not know what to do because it's just too ,comma the Guimaraes complicated so the only case I know where this is completed before Eastern momentum has also cheating because it was all for Cermeno on shall only function of 2 variables and then was computed by hand often was shown to be bottle and this is actually 1 of these examples where men had to admit the sharing of parameters you do not get lineages ability in the sense that I mentioned somewhat after 1 one-dimensional conducted .period normal so it seems mysterious but then there's a tender such a canoe which makes thing factorize men can find out whether interceded but it's 1 of the examples where I would like to have a better reason 2 0 the wife the total with but is it can compute Iowa descendants of the the reason is that when you are told so the excellence it's year-round and these strong Citadel expansion that's among the only thing you do is you introduce mobile after putting on notice which are of course in the space of a trading duels over these lands in the north of any race is this thing that he had the actionable doesn't really matter so I can Nevada Power's but it can also expand the powers and allow for women's you can also have lower than of individual sharing parameters In the game and arbitrary powers of those this is not playing any role for the function theory of course makes impact accommodation complicated takes more time but it does not change conceptually what happens to true to digital you hear on the news to those who have lot the year so I'm I was quite sloppy actually with way reality finding a reusability cell it didn't even mention the name today but he said didn't get to write it down severe idea that it's really just something depending on the polynomials so take the almost many see what is the worst thing that could happen if a trigonometry said he did not exactly if it is to not want to look at the action with the state take whatever underwent on the similarities the 1 originally was said was that I could have you matched quite general arguments especially at the vibration of this variety prove these things these sets are up abounds on the similarities and I hope I might somehow clear how this actually arrives in the computation because whenever they want to compute such you allow this is an all-women's I have this day out right in this capital with for and the letter-writer tear this this year's tie-up up along after 1 must turn along so the letter will be something like this from only 0 as a sigh 1 divided by itself so after the integration I have of course singularities and these voters will have less to the the pot a fraction the compositions but these are all taken care of by the so be tending the powers editing changes the actual representation at the point but it doesn't change the fact that it is a hyper lower them with these as an upper bound of letters and denominators which appear this is the 1 in in way make it more complicated because you have to look at things in a singer parameters but a comedian physical of course majority begin the very last step than it did the last bit of integration but on the other hand you yet struck from the actual and and you look at all and that would Michigan as I described the same time because this is something which depends on this geometry Of the rough surfaces right thank you and


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