 Thanks a lot, Philip. Very happy to be here, virtually at least. So I want to explain, talk a little bit about the system of all the polynomial equations. So my, this is the subject I want to talk about. So we have a number N, which is the number of variables, and then a number R, which is the expected dimension of the zero set. So then I take, let me say, this is like the context, a family of polynomials FI over a field. Indeed, I will take Lorentz polynomials. I will admit negative exponents. I will take N minus R of them, right? And I will denote by Z, they are zero set. So the zero set, I will take it naturally in the multiplicative group, in the n-dimensional multiplicative group, right? Which is what I will also call a torus, like an algebraic torus, because the set of complex points of it, for instance, if you're feeling some characteristics zero, is what's typically called the algebraic torus. So the basic question, is the big question is, how large can it be this zero set? How large can it be Z? Or in a slightly more precise sense, whether I can read the size of my zero set with, in terms of my input polynomials without actually computing it, just in term of easy data that I can read from my polynomial. So of course, this is a very general question, not well-defined, for instance, if your field is any field, an arbitrary field, then you might think about the cardinality if you're satisfied, right? And otherwise, you can think of the degree of this set with respect to a family of like one. And if you are dealing with a field that is defined time, like the field of rational, the field of functions of some normal variety, then you might talk about the high. So this is what interests us. Can I read the height of the zero set in terms of some height of the polynomials? So let me go to an example. Let me think about a simple example, the polynomial F in two variables. This is an affine polynomials in two variables, one plus six plus X plus Y, okay? And then let me consider an arbitrary point in the torus, an arbitrary point omega in a q bar star to the square. And I will use that point to twist my given polynomial. So the twist that will be written with omega star of F is indeed the inverse image of that polynomial with respect to the multiplication map of the torus. And in elementary terms, it is the evaluation of F when you multiply the variables by the coordinates of that point. In this case, it is also the affine polynomial one plus omega one X plus omega two Y. So I will consider specifically twist by roots of unity, root twist by torsion points of the algebraic torus. So if this point omega is then a point which is a pair of roots of unity, then for instance, the coefficients of this twist have the same absolute value as the coefficients of my original polynomial as the coefficient of F. And then not only for the intermediate absolute value but also for all the periodic points, okay? So nice. So all these twists have the same somehow, the given polynomials of the same height principle. But of course, the height of the zero set is not independent of that. The height of the zero set will depend on omega. So if I am considering the intersection of F with this twist, that is the common solution of one plus X plus Y and one plus omega one X plus omega two Y, the bell height does depend on omega. For instance, if you take this torsion point, the third root of unity and it's a square, then the common solution will be also torsion point and the bell height will be zero, of course, because it is a torsion point. And if you take this other one, minus one and I, then you get a point that is not torsion and the height is not zero, okay? So, well, of course, the question was too naive and the answer was simply no, you cannot read directly in terms, at least in terms of quantities that are independent for each of the given polynomials. Of course, maybe some produce something that takes into account the relative position of the polynomials, you might obtain something. I don't know the answer for that. But if you look for quantities that depend on the individual Fives, you cannot. So, I modify my question. So, I asked instead if I can pick some typical value, some representative value. So, for instance, I consider the mean, that is I will look at all rows of unity, okay, of increasing degrees and then increasing order, sorry. And then I will compute the mean of the given heights and try to see if the limit exists. So, the answer in this case is yes. So, this is the thing in the right, in the left-hand side is the mean that is, I am taking some D, which is the order. Then, this is the group of unity to the square, right? I should take here the cardinality, there is a typo here. The cardinality of this group, for the square indeed. And then I'm taking the sum of the heights, the real height of this solution and this limit exists. And indeed, it is this number, which can be written as a question of two special values of a set-up function, that is two set-up three over three times set-up two, okay? And here you have some sage experiments that we did when we didn't know that the exact expression, okay? These were all experiments we did with Roberto Waldi, by the way, forgot to say, I'm explaining ongoing work with him, Roberto Waldi from Regasburg, okay? So you see with D equal to 36, you have this plot of roots, and finally with D equal to a hundred, you see that the values are concentrated, they are concentrated around this particular value. Because you don't only have this phenomenon that the average height is converging to something, but indeed you have a concentration phenomenon. Most heights are going to this value, right? Okay, so this is a bit of motivation to explain my problem. So let me go to the more technical context. So my context will be the three varieties, right? I will have studied this problem within the Arakeloge geometry of three varieties. So let me explain first what's the degrees of cyclos and product varieties. This is classical material, of course. So on the one hand, I will take as before, I will take as before, low-end polynomials in n variables, I will take n minus r of them. So I will extract some information from these low-end polynomials. On the one hand, I will consider the zero set. So the zero set, I will modify it slightly to fit it better with the intersection theory. It is rather the r-cycle. So associated to this family of polynomials, to the polynomial fn minus r up to, yes, up to fn, right? So it is the r-dimensional part of the scheme defined by these polynomials and with the corresponding multiplicities, of course. So this is on the one hand. And then for each individual polynomial, I have here the Newton polytope, right? Newton polytope is defined as the convex hole of the exponents that appear in my Lorentz polynomial. So this is a lattice polytope. It's a polytope in Rn with whose exponents are integral, right? So this is my information from my Lorentz polynomials. Then I will take x, the toric variety with this torus, with the torus, the multiplicative group, right? So toric variety, it is just a compactification of the torus that is equivariable, okay? That has the property that the action of the torus extends to the whole variety. So you can think of the finite space, the project... Well, indeed, sorry, there is something missing here. I should ask complete. I want the complete torus variety. So you can think of the projective space or the product of projective spaces, or you can think of hills of resurfaces or toric vandals, et cetera, et cetera. So to do intersection theory, I will need a device or a SCARTY device so these DIs are SCARTY devices on my toric varieties that will be NAF, that is the associate degrees will be non-negative and will be toric in the sense that they are invariant under the torus action. For instance, if you are in the projective space, this SCARTY device can be identified with associated by-devices, with hypersurfaces. They can be, they will be some of the hyperplanes at infinity, the different coordinate hyperplanes of my projective space. I will take in any case, R of M. So then, from these devices, I can obtain some combinatorial information which is a lattice polyp. There is a construction, a bit more involved, but classical too, that to a device associates a lattice polyp. And well, and indeed, it's a dictionary. I mean, you can get back your NAF toric device or from that lattice polyp. And then you come to the classical, very well-known Versten theorem from the 70s, from the beginnings of the theory of toric varieties. If you ask some mild genericity conditions to your system of Lorentz polynomials, which is essentially that they define a proper intersection and the zero set has no components outside the torus, then you can compute the degree. Then you have a yes answer to my question in the first slide, that is the degree, the size, in some sense of the zero set with respect to this family of divisors, is given by a combinatorial formula, which is what you have in the right-hand side. Here you have the polytops of the divisors and here you have the Newton polytops of the polynomials. And everything is measured in terms of the so-called mixed volume function. Mixed volume is a function that generalizes the usual volume. And it is adapted to take as input N convex bodies of the hidden space. Okay, so this is the model for us. So in any case, I want to recall a famous quotation by Christophe Soulet. Soulet is one of the founding fathers of Arachel of Geometry, which is the branch where I am treating this problem. So he said, I will read it. So the reader and the reader is likely to discover a new and interesting question by just asking for the arithmetic analog of her favorite statement in classical algebraic geometry. So Arachel of Geometry is a branch of arithmetic geometry that treats schemes over the integers with tools from biotic analysis, and complex analysis among others to get analogs of arithmetic analogs of theorem of classical algebraic geometry. And since one of my favorite statement in classical algebraic geometry is that Bernstein theorem. Well, okay, this is what I want to get. I want to get an arithmetic analog of that. So let me go with some precision, okay? So I want to take a Geophantane field like the fields of rational for simplicity, but of course, all that extends to bigger generality. So I will take as an ambient a toric variety x with this torus multiplicative group over the rational, okay? And the fundamental object is this d bar, okay? d bar is what's called a semi-positive toric divisor on my variety, okay? So this is a pair where on the one hand, you have a Cartier divisor. And on the other hand, you have a family of things. These things are indexed by the places of the field, okay? The places of the rational, that is the Archimena place and all the biotic ones. And for each of them, you have a metric, okay? This is the metric that has to be semi-positive, the technical notion has to be rotation invariant. This means some invariance, which is the toric analog of, well, the analog of being toric in this setting. And these are metrics on a line bundle. This is the biotic analytification of the line bundle associated to this. So the key here is that this object has also, I mean, through a theory that I developed with Bourgouis and Philippon, we could associate to this object an analog of the polyp. So the divisor itself has associated a polyp, okay? And to the Metrici divisor, we can associate a family, an Adelic family of functions. Each of the function is called a roof function. It's roof function because it will be like the roof of a small house, because we're talking about functions that are concave on the polyp. So concave functions are very much used for constructing roof of houses. And this is why we call it like that. So you have this Adelic function. It's also a bunch of an infinite family of functions like that. Indeed, you have only a finite number of them, which are non-trivial. There is a number of places where they are arbitrary and for the rest of the places, they are just the zero function. And so with this family is what we propose as an analog of the polytope and to support that claim, we prove this theorem that tells you that if you have a family of Toric Metrici divisor, VI, you have N plus one from zero up to M, then using that you can compute this arithmetic intersection number. So in this theory by Arakelo Falda, Shile, Sule, Anzan, among others, this H is the arithmetic analog of the arithmetic appearance of the degree, arithmetic intersection number. So it can be computed in terms of so-called combinatorial data. It is an Adelic sum. So there is one term for each place of the field of rationales and each term consists of something that depends on the root functions of the given divisors. And then they are measured through the mixed integral function. A mixed integral is the extension or the analog of the mixed volume for concave functions. Instead of looking just to concave, to convex bodies, you're looking at functions on them. And in that case, when you have only one functions, it's natural to consider the integral of the function over the polytope or over the convex body. And when you take N plus one concave functions, you can consider the mixed integral. Here in this picture, taking from one of my papers, you see the situation. Here I'm considering in the first diagram the place at infinity. For this place at infinity, there are two functions. This one and this one, they are concave. Indeed, they are linear in this example. And then the full line consists of the so-called subconvolution of this concave function, which is a kind of Minkowski sum of functions. And then the mixed integral is the integral of this function minus the integral of the component one. Has nice properties, it is symmetric, it is multilinear. And in some cases like this one, it can be computed. So this picture, which correspond to a curve two matrices of divisors that have non-trivial contributions for the place at infinity, the place two and the place three give explicit functions and you can actually compute using this formula, the height. Good, so we have not finished yet with the problem, I mean. In the sense that in the geometric case when you compute the degree of your atomic variety, the degree of the ambient space, this is, you have like automatically as a consequence, you have automatically the degree of the cycles, by using just the Bessou formula. It is not the case in the arithmetic setting because the arithmetic Bessou has also some correction term. So I will come back to that later. So in any case, we cannot readily deduce from this what is going to happen to the cycle, for instance, for hypersurfaces or zero dimensional cycles. So let me advance. So this was done indeed by Walden in his thesis for hypersurfaces. So he considers, let me say, consider one Lorentz polynomial f in n variables as before. He introduces a function. This function is called the Ronkin function. That is one of these functions for each place. This is a function defined over a nuclear space and the way of defining it is going through the torus and considering the function logarithm of absolute value and then computing the mean of this function in the fiber. The mean with respect to a natural measure there. So for instance, when you take the place at infinity, then the Ronkin function is a matter measure. It's indeed the matter measure of a twist of your given polynomial by some number, by the exponential of your point. So in the Archimedean case, the Ronkin function, which was named by that, by Pasi and Roulgar much earlier than Walden, is indeed a function for which each of the values, each of its values is a matter measure. This computer is a matter measure. Since we'll define it, it is not very easy to compute. But it is well-defined. And when the place is not Archimedean, then it is explicit. Then your Ronkin function is a piecewise linear function is the minimum between all these affine functions. This is an affine function. The scalar power of exponent m with u is an affine minus this constant. So you get what's called the tropical polynomial. This is indeed the tropicalization the via the tropicalization of your given Laurent polynomial in the language of tropical geometry. Okay, so given a polynomial, I know they also are somehow insist, you compute the family of functions of this space. The next step is considering the Legendre-Françel dual. This duality of Legendre-Françel comes from convex analysis and we'll give you through some definition which is even in terms of some optimization problem gives you a function on the polytop. I mean, the fact that the domain is the polytop is a result, it's not obvious, but given the definition of the Ronkin function, you find a concrete function on the polytop. So the statement somehow, just to advance a little bit, is that this family of functions that is the Legendre-Françel dual of the Ronkin functions of your Laurent polynomials are the kind of arithmetic analog of the Newton polynomial. That is, you want to, you want to go to, when you go to the arithmetic problem, you need to produce this analog because it is appearing in Berserk theorem, right? And so the statement is this, these are the, this is this analog. To support this, there is the theorem that tells you that if you have a family of semi-positive toric matrices, divisors on the toric variety, you have N of them, you start with zero and you end up with N minus one. Then the height, the arithmetic intersection number, the height of your hyper-surface, can be written exactly in combinatorial terms as an idyllic sum, exactly as before where you have the root functions of your given divisors and also the Legendre-Françel dual of your Ronkin function. This is a formula for the height of this hyper-surface. Even if this hyper-surface is not toric, of course, which is, this is the interesting thing of this theorem. Okay. I don't know if there are questions, if this is the case, please interrupt me. Okay. So now I want to go to the higher co-dimension case. So, so now I am joining what I was, the example and the considerations at the beginning. Okay. My example at the beginning was an example in co-dimension two. Okay. We have seen there that you cannot hope for a formula like this because of the fact that the height is not, you cannot read the height from the Ronkin functions directly because, in particular, because the Ronkin functions are independent of the twist. If you twist your polynomial by a torsion point, you will get the same Ronkin function. Okay. So it's impossible to get something that depends only on them because the height does depend on the torsion point. So, so for that, we have, first of all, we have conjecture. Okay. So it's a, well, this conjecture is quite natural because once you are in the, inside the problem, we have not proven it yet in this, in this generality, but at least it's, we can propose it. So, now I will take a family of polynomial, not only one, but N minus R of them, as before, of Laurent polynomials. I will take them. I will tell them the, the semi-positive divisors. And then additionally, I will take a, I will take a family, a torsion point of this product of tori. So I have here N minus R tori, that is one torus, one torus for each Laurent polynomial, torsion points, that is points with whose coordinates are roots of unity. And I suppose that this sequence, because I have a sequence indeed, is a strict in the sense of equidistribution theory. A strict means that it eventually escapes any proper right subgroup of the given torus. So our statement is that not just the average, but indeed the limit in this case, the limit of this intersection number should converge to, to a value. And this value is given by an expression that generalizes, generalizes the results, my results with Bulwos-Filipon and the result of Wadi. It's very natural. You have a delix sum of mixed integrals. And then root functions corresponding to matrices divisors and do some working functions. This is the concept. Okay. So, I'm sorry I had to be a little bit technical. Of course I am skipping many, many details, but at least this conjecture, this statement can be in some instances can be, it's like more elementary. So let me just mention an elementary consequence, elementary particular case that can be understood without any intersection theory. This is, I think this is much more elementary. That is, you can specialize to the case when your variety is the projective space and dimensional projective space. And then you can take a symmetric divisor, the hyperplane and infinity equipped with a canonical metric. And you can take R equal to zero. This is a typo. This should be a zero. Okay. So in any case, you are looking at, in this case, you have bail height. Okay. You are looking at the bail height of points in a projective space. And this cycle is a zero dimensional. R is the dimension. That's why it should be a zero. Another one. So this conjecture is saying that the limit of the bail height of the zero sets of the twist of, of the given family of Lorentz polynomials is going to be given by this combinatorial formula where here you put the, the function that is the zero function, the zero function on the standard simplex of RM standard simplex is the, the convex, convex hole of the standard basis on the zero, you took the zero function there. And here you put the rocking, the dual support in function. Of course, I could have avoided all the previous theory, but on the other hand, I just present this case, but I really find that the previous theories is, is, is important in, in order to be able to understand the meaning of this formula and the analogy with the burst and theory. So by the way, the previous conjecture is what we, we propose that need come back. This is what we propose as, as the arithmetic analog of burst and theorem, because on the one hand you have some, like typical value of the height. No, you might think in a very, very approximative form. You might think this is the value of the height for generic sense. If you think that this genericity, this genericity within polynomials with the, with the same rocking function. And on the right-hand side is, this is more clearly an analog of the, of the mixed volume of the, of the polytops I showed before. Okay. So now I want to go ahead. So what we could prove. Well, we could, we could prove for the moment. This is really, really small compared with the generality, but nevertheless it's something. Well, of course the, the, the, the, we're looking at the case of co-dimension two, where the things is really non-trivial in, in that, so in dimension two, in any case, and co-dimension two or dimension zero, we could prove the case where, where the two polynomials are a fine. That is they are two polynomials, not more than polynomials, polynomials of degree one. This is the case we have for the moment. So in particular, this has as a corollary, has as a corollary, the statement at the beginning from the beginning, the real height of the zero set of this polynomial, this should be one plus X plus Y. There is also a type of here. This is Y and also here. And this, and this twisted version of it are converging to the mixed integral to the alchemy part. Because all the other, all the other, all the other contributions are zero. And then this can be computed in this case, to this number. So, now I want to enter a little bit into the, into the techniques for proving that. So first of all, I will compute, I will prove this corollary. Okay. Because it's, it's, it's smaller. There is this first part is died, the second part is not that died. So this is the, this is the proof of the corollary from the theorem. So on the one hand, you can show that for the places, the periodic places, the non-alchemy and places, the formula that tells you that the, which is the ranking function in that case also tells you that the dual is the zero function. So in this case, the periodic contributions are mixed integrals of the functions, the zero function on a political, and by the multilinerity properties of mixed integral, it is, it is easy to see this case that, that is the value zero. So the, the, the periodic parts do not contribute in this case. So all the difficulty is in the Archimedean part. So this, the function that appears here inside without the check is the ranking function of the polynomial one plus X plus Y. I have explained that this ranking function indeed is contains in this case, well, it's contains malar measures or many twists, many twists of this affine form, affine polynomial. And in this case, you are, you need to compute it, you need to know the malar measure of an arbitrary affine polynomial, of an arbitrary polynomial of degree one. Okay, luckily this is known. All these malar measures have been computed by Mayo some 20 years ago or a little bit more, because this is, I think, and then you have to look at the amoeba, this is the amoeba, this picture is the amoeba, the Archimedean amoeba of this polynomial. That is the image of this curve, of the curve that it defines in the torus through the evaluation map. Has a shape. So outside the amoeba, the function, the ranking function is very easy. It's just a linear function. Here in this part is zero, in this part is the function Y and in this part is the function X essentially. But inside it has a complicated formula that depends on the block beginner dialog item, right? So the definition, I mean, this actual computation is not easy and computing the dual is even more complicated. So just to be sure, we don't have an idea what this function is. I mean, we don't have an explicit formula for this. But the mixed integral, as the integral, for instance, can be computed in terms of the border. You can use a formula similar to stocks. You can go to the border and you can profit from the asymptotics of the ranking functions, which are easy to compute on the one hand. And also the fact that you know what appears here is the Monchampere, the real Monchampere measure of these functions, which were studied by Passat and Roulgar and tells you that the Monchampere measure of this function is the Lebesgue measure on the amoeba, restricted to the amoeba. That makes the magic in this case. Divided by some constant, which is this pi to the square. Pi to the square is the Lebesgue measure of the whole amoeba. Now you compute it. So you are bound to compute a piecewise linear function on an amoeba. And indeed this can be done and you obtain from this the value of, you get the series, the usual series expansion of set of three. You get the expression here that you can rearrange to this, the value I explained. Okay. So this is how we, how we made this computation and now coming back to the theorem and to the general case. So how can we prove the theorem for, that is the particular case of dimension two for the ambient, dimension zero for the cycle. Two affine polynomials. It's quite representative of the general case, but of course easier and this is why we could, we could do it. So then you have to take an additional metrizate device. And then you can think that this metrizate device is again a hyperplane device or with a canonical matrix so you get very high. And then you take your, your strict sequence of torsion points. And you use what you want to use the arithmetic bassoon because you go, you want to go from the zero dimensional case to the one dimensional case, which is hyper surface. The case we know thanks to the result of what. So here, this tells you that the very high, the height with respect to the bar of the cycle defined by F and the twist of G, right? Because indeed one of the twist, you can save it because if you twist the whole family, the zero set, I mean, you might obtain that zero set by just twisting one of them. The diagonal action of the torus is not, I mean, has no effect on the zero set. In any case, the height of this zero cycle, the height of this point in this case, this point in picture is by the arithmetic bassoon theorem equal to the height of the curve defined by F with respect to some arbitrary matrices divisor that you can choose this E bar is any other device or you can choose matrices divisor you can choose plus some correction term that is this long thing. Let me just forget about this for the moment and concentrate on this. This one observation, first observation is that if you choose this matrices divisor wisely, indeed it is the so-called Ronkin matrices divisor of G. It's called Ronkin because it's a torus matrices divisor whose associate family of concave functions are exactly these two also from King function. It's a construction that you can do when you start from a given polynomial, you can construct a natural matrices divisor on your product variety on your picture. In this case, if you choose it like that, then this part will be equal, what the theorem will be equal to the formula you want. This will be equal to the sum of the mixed integrals you want. So finally, if you make this choice, what you need is to show that all this quantity is going to zero when you're when you're L, when your total points get more and more involved, more and more generic. Okay, so let me skip the exact terms here. I will maybe explain it better in the next slide. In any case, I am making some jump. Of course, I cannot do all the details, but when you enter into the formula that was there, the second part, then you see the formula thing. Then, so let me define for each v, each place v a function on the multiplicative group of the completion of the ish of right closure of the R or sorry, you understand, of QRV. So it is a function which might take minus infinite values, okay, and it is defined through a sum into that thing. In a point x, you use that point to twist your given polynomial g, and you consider the log of the absolute values of that twist, and you integrate that with respect to the to the motion per measure. This is the measure associated, when considered in Arachel geometry that is associated to my D bar. D bar is the matrices divisor, which dictates the height. I am considering. Why this function can be minus infinity? Well, I am doing the integral over a curve, over the curve of f, if by chance g, by a good chance or bad chance, I don't know if g is a twist of f, for instance, if g is equal to f, then for some point, for the point one, in that case, you will have, you are integrating an identical minus infinity, over your curve. So there, in this case, in the case where f and g differ by a twist, which is the example I showed at the beginning, you get a function that has a singularity, that has value minus infinity. So in any case, this function is, we can show it is an analytic function, except from some places where it has, may have logarithmic singularities. When you look at the second term, in the previous formula, you realize that this term, 10 to zero, is equivalent to the fact that the integrals of this function along Galois orbits of your torsion points and the sum over all places converges to some expression. So this is, somehow, it is an equidistribution problem. You have to show that, that for this so-called test function, for this particular function that appears in high theory, your, this, the integrals over the Galois orbits and the sum over all places converges to what it should. So the difficulty is exactly here. The technical difficulty of the problem is exactly here. First of all, you are integrating functions with logarithmic singularity, as in the type of here, right? Whereas when you want to use the standard, the standard equidistribu- equidistribution of theorems, like the theorem of Ulmo, Zang and Spiro, et cetera, Belou, all the functions considered there are continuous with complex support, right? And moreover, you want to consider all places at the same time. You want to consider the sum over all places. Okay. So in this particular case, which is, I mean, it has the advantage of being very explicit. Of course, when you have to affine, to affine polynomials, the solution is explicit that you can work with it and some things becomes easier. So we could do it. And so to solve it, we had to use these tools. I mean, first of all, we have to study the periodic distribution of these torsion points. And then this, this periodic distribution of torsion point indeed is what ensures us that you can get rid of, of all the places except a finite number of them. That is this, this study of reductions of torsion points is what allows us to go from Adelic to local. Secondly, in the Archimedean place, you have to use bounds for linear forms in logarithm. That is classical Baker theorem, Alan Baker theorem on that. Whereas for the, the, the, the periodic places, they are easier to treat with the table of theorem on linear forms in, periodic linear forms in roots of. Okay. This is indeed, this is indeed technically, this problem is very similar to the so-called e conjecture that was studied for instance, by Baker, Rumeli and also by, by Dimitrov and Abigail because you, you dealt exactly with the same, the same kind of technical problem. I think my time is almost over. I will stop here. Thanks a lot.