On an arithmetic criterion for holonomicity
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Dimitrov, Vesselin
I will present in detail a new twist in the subject of arithmetic algebraization theorems. It comes out of a joint work in progress with Frank Calegari and Yunqing Tang on irrational periods, and bears also a relation to a variation by Zudilin around the classical Polya-Bertrandias determinantal criterion for the rationality of a formal function on the projective line. Time permitting, I will sketch an application to an irrationality proof of the 2-adic avatar of $\zeta(5)$.
 He's speaking about arithmetic, columnar and algebraic dynamics. Thank you very much. It is a great pleasure to be able to participate in this wonderful conference and workshop. I learned a lot from the previous talks. I hope my topic will fit into some of the things presented at the workshop. I begin with a motivation from the field of algebraic dynamics that first got me to think about arithmetic rationality criteria of the kind that I will discuss. Then there is a more recent development, which exposes a joint work with Frank Kalegari and Yunqing Tang for the second part. Around arithmetic holonomicity in the theory of g-functions and an application that I hope to outline. We consider the familiar setting from algebraic dynamics where I take an iteration, some endomorphism of alpha in space. That's a dynamic system about the simplest non-trivial kind. We do not have an extent to a projective space. Endomorphism is given by polynomials in the coordinate. This is a polynomial map. I consider a second coordinate. I'll call it lambda. That denotes a regular function, just morphism or polynomial. We ask for the growth behaviors of the height. This will be over a global field K. We take any orbit, the intention is it should be pretty arbitrary, and just record the values of the coordinate lambda or the function lambda along the orbit. We ask how can this grow and also the sequence of degrees if it is not eventually periodic. We start with a very obvious observation, just set theoretically. If you have an iteration from zd to zd, let's see how the dimension will come into play in the bounds we can expect and improve. Just by the pigeonhole principle, this is the north cut trivial bound, if the orbit is not eventually periodic, then just by arranging the points, arranging the orbit by increasing height, arranging points by bounded height, there are noting that there are not many points of height at most. This is where I will use the logarithmic height. Then we have a very obvious a priori bound. There is a dichotomy that either the orbit must be finite or the logarithmic height roughly grows at least n to the power 1 over d in the sense of taking the lim soup along the orbit. I like to start here because we will also mention some interesting work on this subject to which my criteria also relate. The arithmetic criteria will enter in the following way. Instead of doing this set theoretically, we can just, so let's now give them up an algebraic structure. I have a polynomial in the variables. For simplicity here, I assume it takes zd to zd to zd. We may reduce module of primes and do the same observation that the reduced orbit acts on a finite set of cardinality p to the d, the number of fp points, mod p points. It must be pre periodic and its pre period and period are both bounded by this. A very convenient way to use this is on the level of the generating function. Now if lambda is another polynomial this time function to z, then we record our orbit history on a formal power series which will be a generating function with integer coefficients in this case. It has various mod p rationality conditions that the reduction of mod p of this power series we have seen must be rational just by pigeonhole because we have finite iteration on a finite set and the degree of this rational function is at most p to the d. Here is a criterion of this kind that will immediately lead lower bound on the growth of the orbit. I think I, well essentially all of this goes back to a short paper from 1981 published in Acta Ritmetica. I actually have their evidence below on my slides. Also in the full generality I, in higher dimension as well, I outline, I have a short preprint on archive with this criterion. It just follows although Daniel considered the language of linear current sequences looking at the coefficients of the power series. But on the level of power series it reads we have f of x, a formal power series over a global field k. And for every place v at which we can reduce mod v, let's consider the degree of the reduction if that is, if the reduction is a rational function. And if the reduction does not exist or is not a rational function I'll just take the convention that dv is just infinity. And then we have the criterion that this formal power series is in fact a rational power series just as soon as these inequalities fulfilled. And now for this talk the quantity on the right will be ubiquitous. This is a standard notion of height for formal power series. I'll call it the height of the power series. I use this subscript k just to keep the general case in mind because but maybe you want, maybe it is good to restrict to k to the rational field q then I will omit the index just include the global function field case as well where we have no ground field. And then the condition is this inequality on the residual characterities of the finite fields, finite residue fields. It roughly says that if we have enough mod p rationality, rationality in a quantitative sense against the growth of the coefficients of the function ultimately all comes down to the product formula in the global field then the function is rational. In fact this is obviously a characterization the converse is trivial. So this is completely characterizes the rational functions in an arithmetic way. The proof is a typical use of Ziegels lemma and I'll just indicate quickly because we will, the theme will occur in the more in the more recent criteria. We have, we can use m equations in m variables for these choices and there was this parameter kappa in this in the criterion that will come from exactly Ziegels lemma as the Dirichlet exponent. And then we construct this is typical transcendence extrapolation proof with auxiliary construction. We start with an auxiliary construction which is the approximate relation. We solve a linear system of equations for a polynomial q with integer coefficients and small height. We have degrees of p and q bounded by this and height bounded by that. And then extrapolate the coefficients using the multi-information. This is like the trivial Bezou bound in the projective space over the residue fields for OV. Just the bound on the intersection multiplicity at zero by the total degree. And then we use the product formula and the details are quite easy. Again here is the reference to the original paper actually it's 1982. And I also mentioned that this is in fact can the optimal version is not what I wrote. The following the generalization of the conjecture of Roussa on pseudo polynomials. The shark version should delete somehow the parameter kappa that I will not discuss but that came artificially in Ziegels lemma. In the in the applications of the arithmetic columnistic theorem this will be the key point however in the analogous setting that I will turn to. And this is used I think I sketched how this should imply a lower bound on the orbit goals for iterations of our fine space. Because the iteration has a model as well as the lambda over the a-syntegers for a finite set of places as the full height history is just controlled by by just the fine veil height of the individual coordinates. The whole orbit is S integral we have the prime number theorem that tells us exactly how much mod piationality we have. You see the dimension occurring here because of as one over the index exponent. Because I know the reduction mod V is a rational function bounded of degree bounded by that. So this is ultimately how how it enters. And finally we get the following the rationality criterion does much better essentially improves the previous the trivial counting bound by north code than exponential. It turns out it follows that either our orbit recorded by the coordinate lambda I mean by the observable by the function lambda is either so this either piecewise and a polynomial is wise in that the large enough n partitioning to finitely many arithmetic progressions and on each of them these coordinates is a polynomial in n or else we have this where now notice that my the height function was logarithmic and I've taken the log of the logarithmic height on and also I have included the individual lambda here that is still not trivial from the north code bounded for the full orbit growth. So this is the exponential improvement in particular and all right I think I repeated the last display and also I'll just mention that one can use a specialization argument to the global field to also get an implication from this arithmetic statement this arithmetic method to a purely algebraic statement about the growth of degrees but in restricted situation again we move and then lambda is just a polynomial map then we know that the sequence of these degrees again I think what makes it non-trivial is that I have first taken lambda and then composed with the nth iterate is either eventually periodic or unbounded and if it's unbounded there is some explicit but horrible lower bound on the growth. So before I turn to the holonomicity versions of such criteria I will I want to point out some related work by different methods that there is the theorem of Ulle that contains exactly this kind of bound on the degree but for the full for the full degree there is no lambda involved and it is I think more general that it applies not only to morphisms of affine space but any at the morphism of an affine variety and also even for the lemins of the height there is this work of Kantat and Xie giving divergence of lemins if the sequence of degrees is not eventually periodic and in the setting of rational meromorphic algebraic dynamics where we have any dominant rational self-map of a quasi-projective variety then again we have the same observation starting from Northcott on the limb soup of this kind of growth but this is for the full height and it does not include the lambda the orbit is not sampled on the fixed function you cannot look at just one coordinate and I would like to point out that it seems this same bound if you put lambda of fn of p should be also fulfilled in the following with exactly this gap of one over the dimension of x just as soon as the orbit is not piecewise polynomial in n and sorry and that is I apologize that this is how it should read this not quite right I should have the orbit to be not piecewise polynomial or arithmetic progressions and however there is recent work that I'll indicate on the next slide that does show the positivity of this incomplete generality and these are the references the I think please correct me if I get it wrong but I think there is no so there is some explicit bound but it's worse than one over the dimension of the variety I think this should be expected and there was the paper by Belkioka and Satyano in the case where f is an endomorphism by a periodic analytic method as opposed to this one which is spot p but where p varies I think in full generality and even more recent paper oh I apologize I do not okay I think on the slides I sent you there I have updated the publication the full reference for this paper which is in the case of a rational self map now I turn to two more general criteria and I will introduce the height a bit of notation that I already mentioned this is for any go over any global field and there is also another standard notion that I will call the tau tau of f is a what remains of the height after you take account of all the convergence radii so that means that the full height will decompose into an analytic piece so to speak which will involve which will be just the log plus of one over the convergence radius of the power series and so this would be the case if the coefficients were as in to go for a set of places but in general think of the think of the logarithm series for example the convergence radii are all one and the height is one by the prime number theorem so in general there remains another part that is the true denominators part true arithmetic part if you if you that makes sense of the height so technically it's defined by this formula that you first take a finite set of places then you consider the same growth rate of the height but after it's depleted you remove the contributions from the finite set of places and finally you see how small can you make that if by increasing the finite set of places so the the moral of this is that tau is zero if the coefficients are a syntagere for a finite set of places and the condition tau equal to zero should be read as a slightly weaker quantitative version of a syntagere coefficient. Assyntagere coefficients appear in the most celebrated classical rationality criterion the one of Emil Borrell that work revised in his work on the congruent zeta function he used it to give the to prove the rationality of the zeta function of any variety over a finite field and we will include this so the first result on this will be refinement of the theorem of Borrell and work that also will be the sharp strength strengthening of theorem from Andres book g functions and geometry the broad work criterion will be the case but tau is equal to zero essentially a syntagere coefficients and the improvement will be essentially by completely deleting the parameter kappa that was artificial and we saw earlier but in this in this context I will I will not include mod p conditions at all I'll just just limit the discussion to the attic the attic analytic conditions so as in the broad work criterion the vatic conditions are that well we consider the largest radius of meromorphia of the formal power series f we can do this for every place v of the global field and that means that we can express f of x as a quotient of two convergent power series on the vatic disc of given radius rv so I guess I should write sorry I should write mv of f or mv is the largest radius and then the criterion for this the first new theorem on this on the subject but it is a sharp rationality criterion it states that the f is rational just as soon as this whole that the sum of all the logarithms of all the meromorphic adi is greater than this tau invariant that I introduced so when the tau when the coefficients were a-symmetric the right hand side is zero and that's just the familiar condition of braille and vortice you have meromorphic on disks or poly disks whose product of radii strictly exceeds one but so what improves so this this will be just a small improvement after all I will use Andrei's own method but show how a multivariable use of his methods allows to bypass an artificial constant that he had so his theorem proved had a weaker condition certain absolute constant times the tau invariant and this is sharp because conversely now for any collection of such radii of real numbers there is a continuum not just uncountably many but in fact the cardinality is the continuum of a formal power series that are even convergent on those disks and where this this turns out on equality so as soon as we have a strict inequality we have a countably many and in fact all of them are just rational functions this is a complete characterization of rational functions for the so sorry thank you for the question so in I should say yes I I want to consider the radius of meromorph so the lag is the rv is what what we defined the mv of f and the remainder now I turn to holomicity and the results that follow are joint promote work in progress with Frank Kalegai and Yunqing Tang all possible errors in these examples I hope everything is fine but everything that that is not fine is entirely my own responsibility we have also filtered invariants that so I defined the invariant tau of f but for any every integer we can consider a sequence of decreasing numbers by just looking at the string of not not the entire initial string of coefficients of the power series but all coefficients in degrees between these two two multiples of n and then we will have an improvement of the criterion that that is sometimes useful here are some examples the polylogarithmic series have a tau invariant equal to k but indeed this sequence of filtered invariants is strictly decreasing to zero the tau r of the kth polylogarithmic series is just the asymptotic density of the primes up to n that divide some number in the range that we use that range comes up in in the Zigo lemma proof in a similar similar way to the profile sketch of the first criterion and however this modification will have tau equal to k for all k and r so the sequence will not decrease so we we work in a more general setting that this follows andres approach that we can call simultaneous simultaneous neomorphic uniformization we don't instead of asking for f of x to be neomorphic on a disk of a given radius we will ask for both x of z and f of x of z to be to be neomorphic on on a disk in the z plane and I will use notation for the polydisk of radius rv in this sense that the it is an equalized polydisk we take the same radius for all the coordinates and the obvious extension of my notation with these invariants in the multivagant setting using the total dp and as we look at the growth rate of the vector of coefficients of tot of a bounded total dp I apologize I should follow the chart okay there is nothing new and here is the setting that we all the remaining criteria are placed we look at the radius for each place of v and a holomorphic mapping that's just a convergent power series for every place into the into cv to the d it's from the d dimensional polydisk with this normalization that is all I ask of the map this is like a substitution but it should be normalized to have a unit derivative at the origin otherwise it has just transcendental coefficients and also we ask that it should be this is very important for for the criterion it should be trivial for all but finally many v x v of z is just the identity for all but finally many v I will call this an adelic template because I will be interested to to fill the solutions of this template and by that I mean the solutions will be the formal power series in the variables with this property that when you do the substitution f of x of z at every place and then look at the z plane or in the dimensional polydisk then we get a germ of a meromorph function on that polydisk but I also again stress that I allow this f of x of z to be meromorphic in z but I insist and that is will be a crucial restriction because Andres method is more general I insist x of z to be holomorphic in z and then the basic problem will be to to describe all the solutions of this for given data this is the starting datum and then to find all the formal power series with these properties subject to denominator constraint for example on the invagans tau of f of the more precise denominator information for f in this language the rationality theorem is expressed like like this if we take let's this limit to dimension one here so and then we take x of z to be z of every place with some radius rv and then all the formal solutions of this inequality time variance more than the sum of the logs of all the radii precisely the rational power series we complement this by dropping this restriction that the substitutions are all the identity and now we are we can make any substitution which is holomorphic such that f of x of z is meromorphic on at every place v and now if this and then we also relax the condition a bit to the sum of the logs of all the radii the total arithmetic degree that we have to be strictly larger than the tau infinity filtered invariant so this is the limit of all the tau r if this is confusing then just put tau of f here that would be a special case and so the simplest version of the theorem states that such formal power series are necessarily defined power series they are holonomic functions and they are solutions of linear differential equations over the function field k of x I state a special case of the main delfantin criterion in Andrei's book with a mild invariant to the filter invariant that I point out and in our notation I must stress it is very important to additionally assume the set of places in the criterion must be finite otherwise actually the proof breaks down and it treats this way suppose we have these substitutions additionally I will have a second bit of notation to denote the maximum sv of the absolute value of x on the disc on the polydisk so we suppose we have holomorphic mappings that send the polydisk radius rv to the polydisk radius sv of course s must be at least r just by Schwarz's lemma and if x equals z then we have the equality and then Andrei's delfantin criterion gives linear relationship over the function field it states that if there is a parameter kappa that arises similarly in serial lemma the slides actually the extended slides indicate the full proof of this criterion taken from Andrei's book but I will just indicate the choices of the parameters involved if this holds for some kappa then the template problem simultaneous uniformization problem has at most n minus 1 independent solutions over the function field again it is a similar kind of inequality if you like it's also possible to include mod p conditions reducing also the power series that some of the primes like we had in the first criterion the parameter sorry yes I missed what this tau k when you have m functions for the tau oh yes it is defined just by taking the totality of all the coefficients of the entire vector of the mod x to the n truncations for all the power series that's a point in some huge projective space and then we take the height so defined in the same way and I just want to convey the intuition for the parameters involved this is how Andrei's in a condition looks like we will be interested to remove the dependence in kappa that is completely obstructs applications but the essential condition really is this the absolute potential positivity condition without it we have just uncountably many formal solutions with it we have very easily just countably many formal solutions and we would like to see some structure about them the proof is by transcendence methods we just like before we use Ziko's lemma to construct an auxiliary linear form in the functions where the qi will be polynomials and the combination we just want to give an approximate linear relation the combination will vanish highly at the origin and the polynomials will have heights bounded proportionally to kappa times h kappa will be the deflex exponent in the Ziko lemma and the height of I mean the height of the vector of formal functions fi and then of course there will be a parameter of a parameter that will go to infinity I mean the proportionality parameter the degrees of the polynomials will be bounded by this proportional to this and that is where this term will come from from the parameter count but we can't really get rid of that kappa it comes there is an optimal choice that it's a mess that to equalize the two terms with kappa and finally this term will come from estimating the polynomials on on the polydisk so we need a growth information both on the x of z as well as on the functions f of x and then I have essentially the full indications of the proof on the on the next few slides I think I should just leave okay just just display Ziko's lemma the the full statement works like this we have a parameter count we have to solve that many linear equations in that many coefficients and as soon as n exceeds the number of equations we will have a solution then we make an additional make this choice to to get a bound on the height and this is where kappa will come from but and the rest will be actually that is taken from Andres book so let me from now but move to the move on to the proofs of the new parts the rationality criterion is a multivariate application of Andres criterion again here we take the basic case where x of z is trivial we introduce the auxiliary new variables like in Roth's method and that will improve the parameter count in Ziko's lemma so I will denote the block of variables by by this x and out of the original univariate power series we build a bunch actually quite a few of them super exponentially many of them of new power series the crucial point is to take the split variable disjoint disjoint variable product of f evaluated on products of pairs of polynomials so I arrange this ranges over subset of pairwise disjoint pairs ij of indices their number is clearly exponential in this square and the crucial point is that is super exponential in d because we if we take all right if we take tau of a power of f it just in just one variable this increases and can get as large as the original tau plus term like like this but unless tau is equal to zero but for disjoint variables we always have this if you take a new power series a power series in x1 multiplied by power series in x2 and so forth the tau never increases so additionally when we do change f of x to f of x times y the height as well as the radii will scale by the same factor completely proportionally by factor of one half so we have this as well and then andre's condition becomes the following and we can now firstly take kappa to zero that will kill this term and then we will take d to infinity and it will because m was super exponentially large in d and will also kill this term and what survives is just the essential condition that we have proved already supplies rationality but this was in the case where all the substitutions were just trivial x of z equal to z the proof also immediately shows the first part the qualitative part of the holonomistic theorem actually that already follows from andre's criterion because we can take enough sufficiently many derivatives in andre's criterion that never changes the invariant style or the height of the power series and we can just kill the the additional terms we can take kappa to zero and then sufficiently many derivatives will make the third term in andre's criterion also go to zero however we have the following quantitative estimate on the order of the differential equation this is how it reads not only are the solutions of our problem the formal power series f satisfying these substitutions on the relativistic conditions not only are they holonomic but also but in fact they satisfy a linear differential equation I take the inhomogeneous form there exists a differential operator l of order r minus 1 such that l applied to f is just a polynomial where and the bound on r can be taken it is explicitly in terms of all the invariants that we have as soon as r satisfies this inequality then then the rank of holonomicity will be bounded by will be in fact smaller than r and a variant will be if we restrict the denominator types more precisely let's consider rational coefficients for the power series and suppose we have a real number s and an integer t such that all the coefficients when multiplied by the least common multiple of the integers up to sn and then raised to the t power are as integral where the finite set of places can can be allowed to depend on f we do not fix it in advance so this is the situation where tau is bounded by s times t but we have a specific behavior for the denominators then the version of the criterion states that the same condition entails a finite dimension okay a dimension bound by this quantity on the linear span of all the solutions so they just range from not only they are holonomic but the totality of them has a finite dimensional span there are just essentially finitely many operate differential equations involved the proof is works similarly by additional variables where we replace the original variety that was the affine line by the dth power and we will take the asymptotic the number of variables to vote infinity this time we cannot use a super exponential number of new power series because that no longer applies if x of z is not the trivial substitution this will no longer be analytic on the radius of square root of r the best we can do is to just take the disjoint variable products just of the power series and their derivatives and that way we can generate exponentially many exponentially many auxiliary solutions in d variables out of the original univariate polynomial the key input is that when we take disjoint like split variables product like this f of x times g of y and the variables are disjoint these two invariants the height and tau will not increase and the the proof then works very similarly suppose for contradiction we have these derivatives are linearly independent over the function field k of x and then for some are satisfying um andre's condition then we will consider the r to the d new power series out of the original ones by just taking derivatives that does not change and does not change the infargence and multiplying in split in disjoint variable products and then and then this choice of the substitution just calling just the Cartesian power of the univariate one in this procedure the term from the ego's lemma the kappa will disappear if we let d to infinity and then uh the resulting inequality is exactly the contradiction is exactly the opposite that is I will uh I wanted to stress that the proof involves uh crucially the holomorphi of x and z we can uh andre's criterion is nonetheless more general it applies to uh meromorphic substitutions as well so this improvement is not always possible to just remove the parameter kappa incomplete generality there was a not obviously related paper by Zudelin in a journal basically in a different field apparently constructive approximation theory a couple of years ago and the title of the paper is a determinant approach to irrationality where he essentially gave rationality criterion in of the of a similar kind and this is a particular case of our setting his criterion looks like this uh we have a formal power series with rational coefficients uh we suppose it is um analytic say holomorphic on a slit plane c minus this branch curve from minus infinity to minus r but because of the Riemann map uh for for that simply connected domain that is explicit here that condition is exactly equivalent to convergence when you make this substitution so that that places his uh this in our setting uh and also to uh suppose f has unit uh piadika radii i mean convergent on the piadic unit disc for all points p and finally that the total arithmetic degree the sum of logs of all those radii is bigger than three halves times the tau invariant then Zudelin proves that uh f is a rational function um and there are examples like the logarithm series for example log of one plus x uh is um uh not rational it has to be holonomic because because of the holonomist the theorem uh because um uh and because tau of the power series f is equal to one and that is smaller than log four uh and uh this shows that uh there must we cannot have an algebraic uh conclusion in general we we can uh uh at best have holonomicity uh and it remains an open problem to find the best constant in Zudelin's criterion it is somewhere between 1.5 and log four um there is another uh case uh where uh we can solve the uh problem precisely and that will be similar criterion to Zudelin's um to this to precisely describe uh which holonomic functions arise in in a special case of of this setting uh let me directly just take uh point out uh this case take the prime two and uh this hypergeometric function that it just reduces to essentially the arc sine function uh that has um a two-adic radius uh as as high as four um this this combination although the arc sine uh although um if we uh so this is equivalent to taking this function uh and this is a holonomic function it satisfies uh linear differential equation of order one in homogeneous uh and with singularities only at zero one and infinity the two individual factors this and this have two-adic radii as small as a quarter their product has a two-adic radius as high as four sixteen times as big um the Taylor series of the function has denominated has denominator type is we take the least common multiple of the first n denominators it divides this it is essentially this uh and so does this function uh modifying h by the substitution x over x minus one um so we are in a position to fully determine all functions with with these properties so i will just state another case uh where we have a precise characterization of the solution space uh it states like the following we take a power series uh with rational coefficients such that it converges on the two-adic disc of radius four uh and on the p-adic disc of unit radius second uh the if we take the nth coefficient multiplied by the least common multiple of the first two n plus one integers uh we get uh an s integer uh where uh s can depend on the function f and third uh they're up to multiplying the power series by some polynomial non-zero uh there is a linear differential equation l of f equal to zero for some linear differential operator with singularities uh close enough to either zero one or infinity uh which were the singularities for this function then uh any such function is just a rational linear combination with q of x combination of those three we have a three-dimensional space of solutions with the basis uh given given like this uh there are not many cases where we can get an exact characterization and uh i think i should uh all right so my slides also include the proof uh outline of this as well as the application to rationality of the of the two-adic zeta value that i uh did not have time to discuss um so it is what it was with uh such applications in mind that this was uh these criteria are developed um i think uh i i get to the uh i'm out of time at this point so i should um i want to thank everybody for for your attention on the wonderful conference okay uh feel free to unmute yourself and and applaud some people are putting little hands in whatever that is yeah that's cool uh are there any questions can you say how your um how your thing results would relate to uh Sonye's results and strength for for ruch's conjecture where um oh the one uh yeah if i go back i think i can do it just just like that on the first slide basically the two slides are exactly of the same strength actually i would say uh if you recall this paper he had um constant like uh e to the i think three minus two root two something like this he proved that if we have um a linear recurrent uh if sorry if we have a sequence of integers uh such that the reduction mod p is uh satisfied a linear recurrence delay equation mod p of uh a length less than p uh and the growth rate uh was smaller than this then uh the sequence satisfies a linear relation so the and if you optimize kappa in this display it gives exactly the same constant i would say this is the same result uh as i stated the one variable proof uh optimizes to Sonye's quantitative result exactly all improvements come from for more variables and similarly in Andrei's book there is a related constant for uh i i stated um his uh sorry um his rationality theorem is stated under the conditions this some of the law talks of mermorphic radii exceed basically the same kind of constant i wrote 12 but in fact he he the optimal constant is i think x of two times that the same three minus two root two something like this times the tau invariant but uh again it is possible to mix the two types of criteria if that is useful i would be very happy to know if there are more applications if they come up useful for other things uh you can add mod p conditions and um uh the attic analytic conditions at the same time can i ask a question sure hi i i'm uh i'm you probably i don't know if this is a bad question or not but i would be interested in actually trying to use this criteria or trying to apply it so it can it be is it practical to try to make it effective you gave an example on a function which was somehow already known and it was hyper geometric but what what kind of format of function do i need to be actually try to make it effective if i actually like i would really love to try to use this yeah so i think you have in mind the the holonomistic criteria with oh yeah the whole yeah sorry sorry i should jump ahead sorry i had it in my head in the about the teaching five minutes so i uh right right no that's a great question that's exactly what we uh we we try to do in some cases there are potential applications where if you can describe completely the solution space we just give um semi semi effective result with some dimension bound but we really have a finite dimensional space and we can't describe it effectively this is somehow uh i apologize for the comparison but i think it's a little similar to to isiga rope where you have a bound on the number of solutions but finding them effective is hard by the method that you need some anchors some extremely good examples like in bombilla effective three methods to start with uh and this was just an example where uh the arc sign example or this hyper geometric case where it was possible yeah because we uh we have an excellent two-adic radius uh in this case and we started with three solutions uh that are real solutions and they they somehow are enough um to to bootstrap to to know solutions you you can find some indications on the slides that are posted on the workshop but the short answer is we would very much love to do it in more more interesting cases okay thank you but can you see it in some sense it's just being a very general like paulia kirlson type theorem or something like this is it you mean again um are you interested uh referring to the holonomist twist not just uh yeah i mean you don't yeah you're getting logs log one plus x and things like this mm-hmm yes already in zutilin i think zutilin's work was the first extension of this kind because if you yeah we have seen that holonomicity holds even without this coefficient of 1.5 but then it we really have holonomic solutions not rational and we have a criterion for holonomicity up to given given order that is the what is used in in the application to transcendence we don't know the best constant here in this gap but i don't i don't know if we can apply the holonomistic criterion to uh orbit growth lower bounds in a similar way like irrationality the irrationality criterion gives uh for example a height lower bound on holonomic functions uh for example the height must be positive if uh the function is not rational of the special form some polynomial over one minus x to the power uh i guess uh cyclotomics product like that i don't know if the holonomistic criterion has similar strength stronger consequence are there other questions for vessel if not let's thank vessel in again i've got to figure out how to stop recording somehow i uh let's see it's still recording right now um there's supposed to be a button um oh wait a minute maybe i see the button now here we go i was on the wrong page stop um um so let me go back into
 He's speaking about arithmetic,  columnar and algebraic dynamics.  Thank you very much. It is a great pleasure to be able to participate in this wonderful  conference and workshop. I learned a lot from the previous talks. I hope my topic will fit  into some of the things presented at the workshop. I begin with a motivation  from the field of algebraic dynamics that first got me to think about arithmetic rationality  criteria of the kind that I will discuss. Then there is a more recent development,  which exposes a joint work with Frank Kalegari and Yunqing Tang for the second part.  Around arithmetic holonomicity in the theory of g-functions and an application that I hope to  outline. We consider the familiar setting from algebraic dynamics where I take an iteration,  some endomorphism of alpha in space. That's a dynamic system about the simplest non-trivial kind.  We do not have an extent to a projective space. Endomorphism is given by polynomials in the  coordinate. This is a polynomial map. I consider a second coordinate. I'll call it lambda.  That denotes a regular function, just morphism or polynomial. We ask for the growth behaviors  of the height. This will be over a global field K. We take any orbit,  the intention is it should be pretty arbitrary, and just record the values of the coordinate lambda  or the function lambda along the orbit. We ask how can this grow and also the sequence of degrees  if it is not eventually periodic. We start with a very obvious observation, just set theoretically.  If you have an iteration from zd to zd, let's see how the dimension will come into play in the  bounds we can expect and improve. Just by the pigeonhole principle, this is the north cut  trivial bound, if the orbit is not eventually periodic, then just by arranging the points,  arranging the orbit by increasing height, arranging points by bounded height, there are  noting that there are not many points of height at most. This is where I will use  the logarithmic height. Then we have a very obvious a priori bound. There is a dichotomy that  either the orbit must be finite or the logarithmic height roughly grows at least n to the power  1 over d in the sense of taking the lim soup along the orbit. I like to start here because  we will also mention some interesting work on this subject to which my criteria also relate.  The arithmetic criteria will enter in the following way. Instead of doing this set theoretically,  we can just, so let's now give them up an algebraic structure. I have a polynomial in the variables.  For simplicity here, I assume it takes zd to zd to zd. We may reduce module of primes and do the  same observation that the reduced orbit acts on a finite set of cardinality p to the d, the number  of fp points, mod p points. It must be pre periodic and its pre period and period are both bounded by  this. A very convenient way to use this is on the level of the generating function. Now if lambda  is another polynomial this time function to z, then we record our orbit history on a formal power  series which will be a generating function with integer coefficients in this case. It has various  mod p rationality conditions that the reduction of mod p of this power series we have seen must be  rational just by pigeonhole because we have finite iteration on a finite set and the degree of this  rational function is at most p to the d. Here is a criterion of this kind that will immediately  lead lower bound on the growth of the orbit. I think I, well essentially all of this goes back to a  short paper from 1981 published in Acta Ritmetica. I actually have their evidence below on my slides.  Also in the full generality I, in higher dimension as well, I outline, I have a  short preprint on archive with this criterion. It just follows although Daniel considered the  language of linear current sequences looking at the coefficients of the power series. But on the  level of power series it reads we have f of x, a formal power series over a global field k.  And for every place v at which we can reduce mod v, let's consider the degree of the reduction  if that is, if the reduction is a rational function. And if the reduction does not exist  or is not a rational function I'll just take the convention that dv is just infinity.  And then we have the criterion that this formal power series is in fact a rational power series  just as soon as these inequalities fulfilled. And now for this talk the quantity on the right  will be ubiquitous. This is a standard notion of height for formal power series. I'll call it the  height of the power series. I use this subscript k just to keep the general case in mind because  but maybe you want, maybe it is good to restrict to k to the rational field q then I will omit the  index just include the global function field case as well where we have no ground field.  And then the condition is this inequality on the residual characterities of the finite fields,  finite residue fields. It roughly says that if we have enough mod p rationality,  rationality in a quantitative sense against the growth of the coefficients of the function  ultimately all comes down to the product formula in the global field then the function is rational.  In fact this is obviously a characterization the converse is trivial. So this is completely  characterizes the rational functions in an arithmetic way. The proof is a typical use of  Ziegels lemma and I'll just indicate quickly because we will, the theme will occur in the more  in the more recent criteria. We have, we can use m equations in m variables for these choices  and there was this parameter kappa in this in the criterion that will come from exactly Ziegels lemma  as the Dirichlet exponent. And then we construct this is typical transcendence extrapolation proof  with auxiliary construction. We start with an auxiliary construction which is the approximate  relation. We solve a linear system of equations for a polynomial q with integer coefficients and  small height. We have degrees of p and q bounded by this and height bounded by that. And then  extrapolate the coefficients using the multi-information. This is like the trivial Bezou bound  in the projective space over the residue fields for OV. Just the bound on the intersection  multiplicity at zero by the total degree. And then we use the product formula and the details are  quite easy. Again here is the reference to the original paper actually it's 1982. And I also  mentioned that this is in fact can the optimal version is not what I wrote. The following  the generalization of the conjecture of Roussa on pseudo polynomials. The shark version should  delete somehow the parameter kappa that I will not discuss but that came artificially in Ziegels  lemma. In the in the applications of the arithmetic columnistic theorem this will be the key point  however in the analogous setting that I will turn to. And this is used I think I sketched  how this should imply a lower bound on the orbit goals for iterations of our fine space. Because  the iteration has a model as well as the lambda over the a-syntegers for a finite set of places  as the full height history is just controlled by by just the fine veil height of the individual  coordinates. The whole orbit is S integral we have the prime number theorem that tells us exactly how  much mod piationality we have. You see the dimension occurring here because of as one over the index  exponent. Because I know the reduction mod V is a rational function bounded of degree bounded by  that. So this is ultimately how how it enters. And finally we get the following the rationality  criterion does much better essentially improves the previous the trivial counting bound by north  code than exponential. It turns out it follows that either our orbit recorded by the coordinate  lambda I mean by the observable by the function lambda is either so this either piecewise and  a polynomial is wise in that the large enough n partitioning to finitely many arithmetic  progressions and on each of them these coordinates is a polynomial in n or else we have this  where now notice that my the height function was logarithmic and I've taken the log of the  logarithmic height on and also I have included the individual lambda here that is still not trivial  from the north code bounded for the full orbit growth. So this is the exponential improvement  in particular and all right I think I repeated the last display and also I'll just mention that  one can use a specialization argument to the global field to also get an implication from  this arithmetic statement this arithmetic method to a purely algebraic statement about the growth  of degrees but in restricted situation again we move and then lambda is just a polynomial map  then we know that the sequence of these degrees again I think what makes it non-trivial is that  I have first taken lambda and then composed with the nth iterate is either eventually periodic or  unbounded and if it's unbounded there is some explicit but horrible lower bound on the growth.  So before I turn to the holonomicity versions of such criteria I will I want to point out some  related work by different methods that there is the theorem of Ulle that contains exactly this  kind of bound on the degree but for the full for the full degree there is no lambda involved  and it is I think more general that it applies not only to morphisms of affine space but any  at the morphism of an affine variety and also even for the lemins of the height there is this work  of Kantat and Xie giving divergence of lemins if the sequence of degrees is not eventually periodic  and in the setting of rational meromorphic algebraic dynamics where we have any dominant  rational self-map of a quasi-projective variety then again we have the same observation starting  from Northcott on the limb soup of this kind of growth but this is for the full height and it does  not include the lambda the orbit is not sampled on the fixed function you cannot look at just one  coordinate and I would like to point out that it seems this same bound if you put lambda of  fn of p should be also fulfilled in the following with exactly this gap of one over the dimension of  x just as soon as the orbit is not piecewise polynomial in n and sorry and that is I apologize  that this is how it should read this not quite right I should have the orbit to be not piecewise  polynomial or arithmetic progressions and however there is recent work that I'll indicate on the  next slide that does show the positivity of this incomplete generality and these are the references  the I think please correct me if I get it wrong but I think there is no so there is some explicit  bound but it's worse than one over the dimension of the variety I think this should be expected  and there was the paper by Belkioka and Satyano in the case where f is an endomorphism by a  periodic analytic method as opposed to this one which is spot p but where p varies I think in  full generality and even more recent paper oh I apologize I do not okay I think on the slides  I sent you there I have updated the publication the full reference for this paper which is in  the case of a rational self map now I turn to two more general criteria and I will introduce the  height a bit of notation that I already mentioned this is for any go over any global field  and there is also another standard notion that I will call the tau tau of f is  a what remains of the height after you take account of all the convergence radii so that means  that the full height will decompose into an analytic piece so to speak which will involve  which will be just the log plus of one over the convergence radius of the power series  and so this would be the case if the coefficients were as in to go for a set of places but in general  think of the think of the logarithm series for example the convergence radii are all one and  the height is one by the prime number theorem so in general there remains another part that is the  true denominators part true arithmetic part if you if you that makes sense of the height  so technically it's defined by this formula that you first take a finite set of places then you  consider the same growth rate of the height but after it's depleted you remove the contributions  from the finite set of places and finally you see how small can you make that if by increasing the  finite set of places so the the moral of this is that tau is zero if the coefficients are a  syntagere for a finite set of places and the condition tau equal to zero should be read as a  slightly weaker quantitative version of a syntagere coefficient. Assyntagere coefficients appear in the  most celebrated classical rationality criterion the one of Emil Borrell that work revised in his  work on the congruent zeta function he used it to give the to prove the rationality of the zeta  function of any variety over a finite field and we will include this so the first result on this  will be refinement of the theorem of Borrell and work that also will be the sharp strength  strengthening of theorem from Andres book g functions and geometry the broad work criterion  will be the case but tau is equal to zero essentially a syntagere coefficients and  the improvement will be essentially by completely deleting the parameter kappa that was artificial  and we saw earlier but in this in this context I will I will not include mod p conditions at all  I'll just just limit the discussion to the attic the attic analytic conditions so as in the broad  work criterion the vatic conditions are that well we consider the largest radius of meromorphia of  the formal power series f we can do this for every place v of the global field and that means that  we can express f of x as a quotient of two convergent power series on the vatic  disc of given radius rv so I guess I should write sorry I should write mv of f or mv is the largest  radius and then the criterion for this the first new theorem on this on the subject but it is a  sharp rationality criterion it states that the f is rational just as soon as this whole that the  sum of all the logarithms of all the meromorphic adi is greater than this tau invariant that I  introduced so when the tau when the coefficients were a-symmetric the right hand side is zero  and that's just the familiar condition of braille and vortice you have meromorphic on disks or poly disks  whose product of radii strictly exceeds one but so what improves so this this will be just a small  improvement after all I will use Andrei's own method but show how a multivariable use of his  methods allows to bypass an artificial constant that he had so his theorem proved had a weaker  condition certain absolute constant times the tau invariant and this is sharp because conversely now  for any collection of such radii of real numbers there is a continuum not just uncountably many  but in fact the cardinality is the continuum of a formal power series that are even convergent on  those disks and where this this turns out on equality so as soon as we have a strict inequality  we have a countably many and in fact all of them are just rational functions this is a complete  characterization of rational functions for the so sorry thank you for the question so in  I should say yes I I want to consider the radius of meromorph so the lag is  the rv is what what we defined the mv of f and the remainder now I turn to holomicity and  the results that follow are joint promote work in progress with Frank Kalegai and Yunqing Tang  all possible errors in these examples I hope everything is fine but everything that that  is not fine is entirely my own responsibility we have also filtered invariants that  so I defined the invariant tau of f but for any every integer we can consider a sequence  of decreasing numbers by just looking at the string of not not the entire initial string  of coefficients of the power series but all coefficients in degrees between these two  two multiples of n and then we will have an improvement of the criterion that that is  sometimes useful here are some examples the polylogarithmic series have a tau invariant equal  to k but indeed this sequence of filtered invariants is strictly decreasing to zero the  tau r of the kth polylogarithmic series is just the asymptotic density of the primes up to n  that divide some number in the range that we use that range comes up in in the Zigo lemma  proof in a similar similar way to the profile sketch of the first criterion and however this  modification will have tau equal to k for all k and r so the sequence will not decrease  so we we work in a more general setting that this follows andres approach  that we can call simultaneous simultaneous neomorphic uniformization  we don't instead of asking for f of x to be neomorphic on a disk of a given radius we will  ask for both x of z and f of x of z to be to be neomorphic on on a disk in the z plane  and I will use notation for the polydisk of radius rv in this sense that the it is an  equalized polydisk we take the same radius for all the coordinates and the obvious extension  of my notation with these invariants in the multivagant setting using the total dp  and as we look at the growth rate of the vector of coefficients of tot of a bounded total dp  I apologize I should follow the chart okay there is nothing new and here is the setting that  we all the remaining criteria are placed we look at the radius for each place of v and  a holomorphic mapping that's just a convergent power series  for every place into the into cv to the d it's from the d dimensional polydisk  with this normalization that is all I ask of the map this is like a substitution but  it should be normalized to have a unit derivative at the origin otherwise it has just transcendental  coefficients and also we ask that it should be this is very important for for the criterion it  should be trivial for all but finally many v x v of z is just the identity for all but  finally many v I will call this an adelic template because I will be interested to  to fill the solutions of this template and by that I mean the solutions will be the formal  power series in the variables with this property that when you do the substitution f of x of z  at every place and then look at the z plane or in the dimensional polydisk then we get a germ of a  meromorph function on that polydisk but I also again stress that I allow this f of x of z to  be meromorphic in z but I insist and that is will be a crucial restriction because Andres  method is more general I insist x of z to be holomorphic in z and then the basic problem will  be to to describe all the solutions of this for given data this is the starting datum and then  to find all the formal power series with these properties subject to denominator constraint  for example on the invagans tau of f of the more precise denominator information for f  in this language the rationality theorem is expressed like like this if we take let's  this limit to dimension one here so and then we take x of z to be z of every place  with some radius rv and then all the formal solutions of this inequality  time variance more than the sum of the logs of all the radii precisely the rational power series  we complement this by dropping this restriction that the substitutions are all the identity  and now we are we can make any substitution which is holomorphic such that f of x of z is  meromorphic on at every place v and now if this and then we also relax the condition a bit to the  sum of the logs of all the radii the total arithmetic degree that we have to be  strictly larger than the tau infinity filtered invariant so this is the limit of all the tau r  if this is confusing then just put tau of f here that would be a special case  and so the simplest version of the theorem states that such formal power series are necessarily  defined power series they are holonomic functions and they are solutions of linear  differential equations over the function field k of x  I state a special case of the main delfantin criterion in Andrei's book  with a mild invariant to the filter invariant that I point out and in our notation I must stress  it is very important to additionally assume the set of places in the criterion must be finite  otherwise actually the proof breaks down and it treats this way  suppose we have these substitutions additionally I will have a second bit of notation to denote the  maximum sv of the absolute value of x on the disc on the polydisk so we suppose we have  holomorphic mappings that send the polydisk radius rv to the polydisk radius sv of course  s must be at least r just by Schwarz's lemma and if x equals z then we have the equality  and then Andrei's delfantin criterion gives linear relationship over the function field  it states that if there is a parameter kappa that arises similarly in serial lemma  the slides actually the extended slides indicate the full proof of this criterion taken from Andrei's  book but I will just indicate the choices of the parameters involved if this holds for some kappa  then the template problem simultaneous uniformization problem has at most  n minus 1 independent solutions over the function field again it is a similar kind of inequality  if you like it's also possible to include mod p conditions reducing also the power series that  some of the primes like we had in the first criterion the parameter sorry yes I missed what  this tau k when you have m functions for the tau oh yes it is defined just by taking the  totality of all the coefficients of the entire vector of the mod x to the n truncations for  all the power series that's a point in some huge projective space and then we take the height  so defined in the same way  and I just want to convey the intuition for the parameters involved this is how Andrei's  in a condition looks like we will be interested to remove the dependence in kappa that is completely  obstructs applications but the essential condition really is this the absolute  potential positivity condition without it we have just uncountably many formal solutions  with it we have very easily just countably many formal solutions and we would like to see some  structure about them the proof is by transcendence methods we just like before we use Ziko's lemma  to construct an auxiliary linear form in the functions where the qi will be polynomials  and the combination we just want to give an approximate linear relation the combination  will vanish highly at the origin and the polynomials will have heights bounded proportionally to kappa  times h kappa will be the deflex exponent in the Ziko lemma and the height of I mean the height of  the vector of formal functions fi and then of course there will be a parameter of a parameter  that will go to infinity I mean the proportionality parameter the degrees of the polynomials will  be bounded by this proportional to this and that is where this term will come from from the parameter  count but we can't really get rid of that kappa it comes there is an optimal choice that it's  a mess that to equalize the two terms with kappa and finally this term will come from  estimating the polynomials on on the polydisk so we need a growth information both on the x of z  as well as on the functions f of x and then I have essentially the full indications of the proof  on the on the next few slides I think I should just leave okay just just display Ziko's lemma  the the full statement works like this we have a parameter count we have to solve that many linear  equations in that many coefficients and as soon as n exceeds the number of equations we will have  a solution then we make an additional make this choice to to get a bound on the height and this  is where kappa will come from but and the rest will be actually that is taken from Andres book  so let me from now  but move to the move on to the proofs of the new parts the rationality criterion  is a multivariate application of Andres criterion again here we take the basic case where x of z  is trivial we introduce the auxiliary new variables like in Roth's method and that will improve the  parameter count in Ziko's lemma so I will denote the block of variables by by this x and out of the  original univariate power series we build a bunch actually quite a few of them super exponentially  many of them of new power series the crucial point is to take the split variable disjoint  disjoint variable product of f evaluated on products of pairs of polynomials so I arrange  this ranges over subset of pairwise disjoint pairs ij of indices their number is clearly  exponential in this square and the crucial point is that is super exponential in d  because we if we take all right if we take tau of a power of f it just in just one variable this  increases and can get as large as the original tau plus term like like this but  unless tau is equal to zero  but for disjoint variables we always have this if you take a new power series a power series in  x1 multiplied by power series in x2 and so forth the tau never increases so additionally when we do  change f of x to f of x times y the height as well as the radii will scale by the same factor  completely proportionally by factor of one half so we have this as well and then andre's condition  becomes the following and we can now firstly take kappa to zero that will kill this term  and then we will take d to infinity and it will because m was super exponentially large in d and  will also kill this term and what survives is just the essential condition that we have proved already  supplies rationality but this was in the case where all the substitutions were just trivial  x of z equal to z the proof also immediately shows the first part the qualitative part of  the holonomistic theorem actually that already follows from andre's criterion because we can  take enough sufficiently many derivatives in andre's criterion that never changes the  invariant style or the height of the power series and we can just kill the the additional terms we  can take kappa to zero and then sufficiently many derivatives will make the third term in andre's  criterion also go to zero however we have the following quantitative estimate on the order  of the differential equation this is how it reads not only are the solutions of our problem  the formal power series f satisfying these substitutions on the relativistic conditions  not only are they holonomic but also but in fact they satisfy a linear differential equation  I take the inhomogeneous form there exists a differential operator l of order r minus 1 such  that l applied to f is just a polynomial where and the bound on r can be taken it is explicitly  in terms of all the invariants that we have as soon as r satisfies this inequality then  then the rank of holonomicity will be bounded by will be in fact smaller than r  and a variant will be if we restrict the denominator types more precisely let's  consider rational coefficients for the power series and suppose we have a real number s and  an integer t such that all the coefficients when multiplied by the least common multiple of the  integers up to sn and then raised to the t power are as integral where the finite set of places can  can be allowed to depend on f we do not fix it in advance so this is the situation where tau is  bounded by s times t but we have a specific behavior for the denominators then the version  of the criterion states that the same condition entails a finite dimension okay a dimension  bound by this quantity on the linear span of all the solutions so they just range from not  only they are holonomic but the totality of them has a finite dimensional span there are just  essentially finitely many operate differential equations involved  the proof is works similarly by additional variables where we replace the original variety that was  the affine line by the dth power and we will take the asymptotic the number of variables to vote  infinity this time we cannot use a super exponential number of new power series because  that no longer applies if x of z is not the trivial substitution this will no longer be analytic on  the radius of square root of r the best we can do is to just take the disjoint variable products just  of the power series and their derivatives and that way we can generate exponentially many  exponentially many auxiliary solutions in d variables out of the original univariate polynomial  the key input is that when we take disjoint like split variables product like this f of x times g  of y and the variables are disjoint these two invariants the height and tau will not increase  and the the proof then works very similarly suppose for contradiction we have these derivatives are  linearly independent over the function field k of x and then for some are satisfying  um andre's condition then we will consider the r to the d new power series out of the original ones  by just taking derivatives that does not change and does not change the infargence  and multiplying in split in disjoint variable products and then and then this choice of the  substitution just calling just the Cartesian power of the univariate one in this procedure  the term from the ego's lemma the kappa will disappear if we let d to infinity and then  uh the resulting inequality is exactly the contradiction is exactly the opposite that is  I will uh I wanted to stress that the proof involves  uh crucially the holomorphi of x and z we can uh andre's criterion is nonetheless more general  it applies to uh meromorphic substitutions as well so this improvement is not always possible to just  remove the parameter kappa incomplete generality there was a not obviously related paper by Zudelin  in a journal basically in a different field apparently constructive approximation theory  a couple of years ago and the title of the paper is a determinant approach to irrationality  where he essentially gave rationality criterion in of the of a similar kind  and this is a particular case of our setting his criterion looks like this  uh we have a formal power series with rational coefficients uh we suppose it is um analytic  say holomorphic on a slit plane c minus this branch curve from minus infinity to minus r but  because of the Riemann map uh for for that simply connected domain that is explicit here that condition  is exactly equivalent to convergence when you make this substitution so that that places his uh  this in our setting uh and also to uh suppose f has unit uh piadika radii i mean convergent on the  piadic unit disc for all points p and finally that the total arithmetic degree the sum of logs of all  those radii is bigger than three halves times the tau invariant then Zudelin proves that uh f is a  rational function um and there are examples like the logarithm series for example log of one plus x  uh is um uh not rational it has to be holonomic because because of the  holonomist the theorem uh because um uh and because tau of the power series f is equal to one  and that is smaller than log four uh and uh this shows that uh there must we cannot have an algebraic  uh conclusion in general we we can uh uh at best have holonomicity uh and it remains an open problem  to find the best constant in Zudelin's criterion it is somewhere between 1.5 and log four  um there is another uh case uh where uh we can solve the uh problem precisely and that will be  similar criterion to Zudelin's um to this to precisely describe uh which holonomic functions  arise in in a special case of of this setting uh let me directly just take uh point out uh  this case take the prime two and uh this hypergeometric function that it just reduces to  essentially the arc sine function uh that has um a two-adic radius uh as as high as four um this  this combination although the arc sine uh although um if we uh so this is equivalent to taking this  function uh and this is a holonomic function it satisfies uh linear differential equation of order  one in homogeneous uh and with singularities only at zero one and infinity the two individual factors  this and this have two-adic radii as small as a quarter their product has a two-adic radius as high  as four sixteen times as big um the Taylor series of the function has denominated has  denominator type is we take the least common multiple of the first n denominators it divides  this it is essentially this uh and so does this function uh modifying h by the substitution x over  x minus one um so we are in a position to fully determine all functions with with these properties  so i will just state another case uh where we have a precise characterization of the solution space uh  it states like the following we take a power series uh with rational coefficients such that  it converges on the two-adic disc of radius four uh and on the p-adic disc of unit radius second uh  the if we take the nth coefficient multiplied by the least common multiple of the first two n plus  one integers uh we get uh an s integer uh where uh s can depend on the function f and third uh  they're up to multiplying the power series by some polynomial non-zero uh there is a linear  differential equation l of f equal to zero for some linear differential operator with singularities  uh close enough to either zero one or infinity uh which were the singularities for this function  then uh any such function is just a rational linear combination with  q of x combination of those three we have a three-dimensional space of solutions with the basis  uh given given like this uh there are not many cases where we can get an exact characterization  and uh i think i should uh all right so my slides also include the proof uh outline of this  as well as the application to rationality of the of the two-adic zeta value that i uh did not have  time to discuss um so it is what it was with uh such applications in mind that this was uh  these criteria are developed um i think uh i i get to the uh i'm out of time at this point so i should  um i want to thank everybody for for your attention on the wonderful conference  okay uh feel free to unmute yourself and and applaud  some people are putting little hands in whatever that is yeah that's cool uh are there any questions  can you say how your um how your thing results would relate to uh Sonye's results and strength  for for ruch's conjecture where um oh the one uh yeah if i go back i think i can do it just  just like that on the first slide basically the two slides are exactly of the same strength  actually i would say uh if you recall this paper he had um constant like uh e to the  i think three minus two root two something like this he proved that if we have um a linear recurrent  uh if sorry if we have a sequence of integers uh such that the reduction mod p is uh satisfied  a linear recurrence delay equation mod p of uh a length less than p uh and the growth rate  uh was smaller than this then uh the sequence satisfies a linear relation so the and if you  optimize kappa in this display it gives exactly the same constant i would say this is the same  result uh as i stated the one variable proof uh optimizes to Sonye's quantitative result exactly  all improvements come from for more variables and similarly in Andrei's book there is a related  constant for uh i i stated um his uh sorry um his rationality theorem is stated under the conditions  this some of the law talks of mermorphic radii exceed basically the same kind of constant i  wrote 12 but in fact he he the optimal constant is i think x of two times that the same three minus  two root two something like this times the tau invariant but uh again it is possible to mix  the two types of criteria if that is useful i would be very happy to know if there are more  applications if they come up useful for other things uh you can add mod p conditions and  um uh the attic analytic conditions at the same time can i ask a question sure hi i i'm uh  i'm you probably i don't know if this is a bad question or not but i would be interested in  actually trying to use this criteria or trying to apply it so it can it be is it practical to try  to make it effective you gave an example on a function which was somehow already known and  it was hyper geometric but what what kind of format of function do i need to be actually try to make  it effective if i actually like i would really love to try to use this yeah so i think you have in  mind the the holonomistic criteria with oh yeah the whole yeah sorry sorry i should jump ahead sorry  i had it in my head in the about the teaching five minutes so i uh right right no that's a great  question that's exactly what we uh we we try to do in some cases there are potential applications  where if you can describe completely the solution space we just give um semi semi effective result  with some dimension bound but we really have a finite dimensional space and we can't describe  it effectively this is somehow uh i apologize for the comparison but i think it's a little  similar to to isiga rope where you have a bound on the number of solutions but finding them  effective is hard by the method that you need some anchors some extremely good examples like in  bombilla effective three methods to start with uh and this was just an example where uh the arc sign  example or this hyper geometric case where it was possible yeah because we uh we have an excellent  two-adic radius uh in this case and we started with three solutions uh that are real solutions and  they they somehow are enough um to to bootstrap to to know solutions you you can find some  indications on the slides that are posted on the workshop but the short answer is we would very  much love to do it in more more interesting cases okay thank you but can you see it in some sense  it's just being a very general like paulia kirlson type theorem or something like this is it  you mean again um are you interested uh referring to the holonomist twist not just uh  yeah i mean you don't yeah you're getting logs log one plus x and things like this  mm-hmm yes already in zutilin i think zutilin's work was the first extension of this kind  because if you yeah we have seen that holonomicity holds even without this coefficient of 1.5 but then  it we really have holonomic solutions not rational and we have a criterion for holonomicity up to  given given order that is the what is used in in the application to transcendence we don't know the  best constant here in this gap but i don't i don't know if we can apply the holonomistic criterion to  uh orbit growth lower bounds in a similar way like irrationality the irrationality criterion  gives uh for example a height lower bound on holonomic functions uh for example the height  must be positive if uh the function is not rational of the special form some polynomial over one minus  x to the power uh i guess uh cyclotomics product like that i don't know if the holonomistic criterion  has similar strength stronger consequence are there other questions for vessel  if not let's thank vessel in again i've got to figure out how to stop recording somehow i uh let's see  it's still recording right now um there's supposed to be a button um  oh wait a minute maybe i see the button now here we go i was on the wrong page stop um  um so let me go back into
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