 Okay, so I'm going to talk about, hi everyone, I'm going to talk about a random additive polynomial in the setting, and I will focus about the Galois theory. So what I want to tell you something is weird because I got this message. Okay, never mind. So what I want to cover is first the general Galois theory of additive polynomials, some model of randomness and, and in each model to tell you what, what is the typical Galois group and if time allows I will give you maybe a glimpse on some of the methods which, which are I find very interesting. So, maybe I'll start with some motivation or some introduction. What happens for a regular polynomial so not not not not non additive polynomials. So, two or three years ago I gave you a talk so I will use the same notation, I hope you remember. And so I have a polynomial and I take a polynomial over the integer which is uniform from the set of all polynomials of degree n and height. At most L, this man means that I choose a sequence of a random variables, which are independent and they take the values, the integer values between minus and L and L uniformly. So this is a my random polynomial. Okay, and I have a three models that I mean three natural models that are more but a to consider but before that, if the polynomial is a separate band, namely doesn't have a double roots I can view its Galois group as a subgroup of the symmetry group by action on the on the roots. And the three models are the three natural models is the large box model, well, I take the coefficient from a box. And the size of the box is growing and the degree, the dimension is fixed, so the degree is fixed and the height goes to infinity. And I have the restricted coefficient model where the dimension of the box is going but the size is fixed. And I said three but I meant two so. So these are two natural models for random polynomials over the integers. And some results that we have about this polynomial in the large box model, the Galois group of the polynomial is the full symmetric group asymptotically almost surely this is going back to Aletasy einer court, Vermeer Donder modern quantity versions, etc. And in the large box model, the big challenge is, is a is to understand what, how the Galois group that behaves on the on the complement when when the Galois group לא אסן, ובמקום הזה, אנחנו רואים הרבה פרשת פרקסטים, אולי שיש שם פרקסטים, ובמקום הרבה פרקסטים בגבולים הרבה יעדים שבארגיבה, שהגבולים לא אסן הם מבחינים בגבולים של פרקסטים. סמטוטיקל. Okay, and from this one can, and also it showed that if you are not ascended, most probably you are either S and minus one, which means the polynomial as a root, or you are A n, I will not get into this in more details, but I will just say that a numeric evidence showed that A n is very rare, but this is the best we can prove now. And in the restricted coefficient models, the situation is more difficult, and even irreducibility is a challenge, and here maybe the recent progress is by Kuniagin, myself and Cosma, בויאד ההמביאויום, וכהלבן הוא שם דמńskו כולופלוס ואז כוזמא. So let me not cover all the history I just give you two exemplary results, the first says that the probability that the polynomial is irreducible is always positive for every box that you take and it becomes probability one when the box sizes. אבל כשרונות בווקסה ה-35, כך אלי 17, וכך עד כדי pedestrians, זאת אומרת, אם אתם שערם בווקסים בווקסים בגלווה, אז עד כדי דבר היא שהגלווה קרופי הוא גריא, זאת אומרת, זאת אומרת, תגידה על איזה פרוביטי, שזה נמצא על פרוביטי גריא כדי איזה פרוביטי גריא. זה רק שם שם. אז זה לא מה שאני רוצה להסתכל על זה כי זו לא פלנומיאלים פלנומיאלים עכשיו אני רוצה להסתכל על פלנומיאלים פלנומיאלים. אז אולי נשאר איזה פלנומיאלים sorry, so what is an additive polynomial or slightly more general q additive polynomial? q additive polynomial is a polynomial that has only terms of the form x to the q to the i. So I assume that the base field where I choose the coefficient form contains the finite field fq. And I only allow terms that are of degree a power of q. And it's called q additive because it has some additive property as I wrote. And the point is that the zero set of f in the algebraic closure is a vector space, it's an fq vector space. Okay, because this is obvious right. Therefore, if f is separable, which in this case it means that the a0, the smallest coefficient is nonzero, then if f is separable, the Galois group, which acts on the zeros, but now the zeros are fq vector space, so the Galois group naturally embeds into the, into GLN q, so into invertible matrices over fq, because it acts on a vector space of dimension n. Okay, so now the group is not a subgroup of s and not every symmetry is allowed, because we have a structure on ZF. And if you randomize f in some way, then you get a random subgroup of GF, so the main question is, how this Galois group distribute for example, is GF typically irreducible subgroup of GLN, is it large and whatever interest you? Okay, maybe I should say that q additive polynomials is a very classical object and they come in, in the study of a Drainful module, etc., but let me not get into this too much. Okay, so the bottom line is that if we have q additive polynomial, then the Galois group is a group of matrices and not group of permutations. So, let me immediately restrict to the case where the base field, the field that I choose coefficients from is the rational function field in one variable over a finite field fq, and then I have, this is analog of the rational, so I have analog way to choose polynomial, random polynomial, so I take f to be a uniform polynomial of a degree, let's say monic of degree q to the n, and of height at most q to the d. More precisely, what do I mean by this? I just mean that the coefficients are independent random variables taking the values, polynomials of degree at most d, a uniform. Okay, so this is my uniform polynomial, this is my random polynomial. Okay. And here we have three models, the large box model where I fixed the degree of the polynomial in X, I fixed the finite field and I let the height, which is the degree in T goes to infinity. I have the large finite field model in which I fixed the height and the degree and let the finite field go to infinity. And maybe the most challenging case is when I call it the restricted coefficient model where I fixed the degree of the coefficients and I fixed the finite field and I let the degree of the polynomial go to infinity. Okay, so these are the three natural models for randomness. By the way, feel free to ask questions. For example, now if you have a good time. Okay, but don't worry to stop me, it's fine. And I order them in this order, not randomly, because maybe you can think that it's alphabetically but it's not, it's by the level of difficulty of each model. So the first model is very easy to handle. The second model is slightly more challenging, and we use deep result from finite group theory. The first model is what I will focus about in this talk. Okay, so let's start with the first two models. So, So this is the large box modern and the large finite field model. So here in these two models we have a very nice and clean result that asymptotically almost surely the Galois group is GLNQ. But the proofs. So this is a theorem with, I plan to say it at the beginning, but I forgot. So this is a joint, all of this project is joined with Alexantine from, with Alexantine and Ailey McKemmy. And look on her eyes to see if I said it correctly. Yeah, but an Ailey McKemmy. So, in these two models, the Galois group is asymptotically almost surely the full matrix group. And in the large box model it's, it's pretty easy. We just, we compute the generic Galois group so when the coefficients are actually variables. And this, this, this is easy to do, for example, Dixon did it in 1911 when he studied the invariance of this group. And then we just apply a function field of Healbert's irreducibility theorem. So it's very similar to the proof of Healbert of that irreducible, that the polynomial over the integers in the large box model has a full symmetry group is probability one it's, it's exactly the same proof, but modify to the, to the second. The large finite field is, is more challenging I don't want to, to get into too much details because I want to focus on the third limit, but the main thing that we use is that there are a few characteristic polynomial of element in certain characteristic classes in maximal subgroup of GLM. So, such as I just gave one example which is the, the read product this is the group that, that acts on, on a factorization of the vector space into a direct sum of L copies. And, and this is a deeply, and this is based on, on a Fulman-Gorannik classification of maximal subgroup of GLM Q. It's a series of papers of them, by them, that, that is a key ingredient in the proof. So, so it uses a very deep group theory. And we don't know how to prove it without. Okay. So, let me go now to, to the large degree limit. And I want to start because it's slightly more complicated in the large degree limit so let me start with what are the obstructions for the group to be GLM. There are some obstructions that, that occur with a positive probability. So first of all, we have a separability obstruction, so I remind you that a polynomial additive polynomial is separable if and only if it's, it's, if the coefficient of X is non zero. So if we restricted the coefficient model, we choose, I mean the probability that a zero is zero is positive. It's one over Q to the D plus one. Because it's a random polynomial degree at most D so it can be zero. So first of all, we have to take care of, of this issue that occurs with positive probability. Another obstruction is obstruction for the irreducibility, when I say irreducibility I think of the Gallup group as a subgroup of GLM Q and I think about irreducibility in the sense that there is no invariant subspace. And it's possible and it happens with a positive or symptotic probability that if I view F, this F that I wrote it as a polynomial in X. So now I view it, I look at it as a polynomial in T, and I take GCD of the coefficient. It's possible that, of course, the GCD of the coefficient of F as a polynomial in T, always divisible by X because X devised everything, but it's possible that it has a higher degree. GCD, I call it the content of, in short, CTF, this GCD is a Q additive polynomial with coefficient in a finite field in FQ. And this, and the zero set of this polynomial is an invariant subspace of the Gallup group. So it's possible with positive probability that the group is even not irreducible. So it has invariant space. But this invariant space is very special. It's something over a finite field. Okay, but this happens with positive probability. And we also have a determinant obstruction. So I just gave one example. The probability that the determinant of all elements in the group is one is to say, I mean, if I meant bigger or equal, I wrote equal, but I meant to write bigger or equal. So think that maybe the frequent a zero is some power of Q, some b zero to the Q minus one. So no matter what I will do, the determinant of all elements will be one. This also happens with positive probability. So the first theorem that I want to tell you about is what happened if all of this obstruction do not occur. Okay, so this is the simplest theorem that I can provide to you in this limit. And in this case, if I condition on the content of the polynomial to be X, so I don't have the irreducibility obstruction, and I don't care about this determinant, so I just want to check whether SLN is contained in the group. So the probability that SLN is contained contained in the Galois group, given that condition on the fact that the content of the polynomial is X goes to one as the degree goes to infinity. Okay, so this is a kind of avoiding all the obstructions. But we, in fact, we also analyzed what happened with the obstruction, but I wanted before to say one. So this theorem use a very deep group theory to prove it, and we get a corollary that myself, I don't know how to prove without deep results in group theory. But it seems kind of, how to say, something that, yeah, something maybe that is of interest to some people, so if I take this additive polynomial, of course it cannot be irreducible as a polynomial, because X divide all the coefficients, because X divide a polynomial, but if you divide F of X by X, and this polynomial when N goes to infinity is irreducible as a polynomial asymptotically almost sure, and this is something that I do not know how to prove without going through our theorem, and in particular using deep, when I say deep group theory for me as a number theory, it's, the minimal is that I use the classification of find simple group, but it's much, much further. Okay. So I don't know how to prove this corollary without the classification. Yeah. Okay. So let me try and explain what is the general result that we get, because we get a full result in this thing, so I need to analyze what happens if the content is not one, and I need to, so. So let's take H to be a polynomial with coefficient in the finite field FQ. And, and I assume that this polynomial H divides the polynomial F. Okay. So the zero set of H is invariant GF subspace, and because H has a coefficient in a finite field and it's very easy to understand the Gallo group because it's only the four binaries and then if you do some computation you can show that the Gallo group is generated by the by the companion matrix, the matrix that you put once and the coefficients of the polynomial. Right. So, so automatically we get that the Gallo group is a subgroup of upper triangular block matrices, wherein, in, I'm not sure if it transform me or not. So, in the upper left side, it's a group, the group generated by D, this is the action of ZH. There is something in the upper right corner, and, and in the lower right corner, again, it's, I don't know anything, so at the beginning it's contained in GLN minus eta, just so the dimension will be n. Okay, so automatically I know that the group factors into this is contained into this. Another thing is that if I take some non zero C and some K, the divides q minus one, then, then one can show that, and I assume that that a zero is I have something on the screen that doesn't allow me to read and I don't remember the notation so I will try to use my memory and if it's nonsense tell me and I assume that a zero is C times u to the K so, so, so this is the determinant obstruction so if, if a zero is, is some constant time time some polynomial to the K, where K is the maximum possible, then, then I know that the, and I write and I write the group like this, an element in the group like this, then, then I know that the determinant of B will be some C over determinant of D to the I times the K power and so I know that this is an obstruction on the determinant of the lower right block and, and, and I call it a gamma NHCK, probably this is the thing that, that I was missing I don't know how to delete the, the mating is being recorded by those participants, whatever so, so I let gamma to be all matrices of this form that they, that they are in the upper left side I have some power of D, D is the companion matrix of H I have something A in the corner, in the right top corner and in the bottom corner I have some matrix B and the determinant of B is, is, is given by something like, satisfies this condition that I wrote, this is something technical but it's exact and then what we can prove is that the probability that GF is this group a condition on the content being H and A0 being a K power times the constant it goes to 1 as the end goes to infinity. Okay, so, so we, we, we managed to analyze based on the content which is, which we denote by H and the maximum power that we can represent A0 is due to the K time constant to, to analyze exactly what will be the group I should, I should say that this, this, this thing that we condition on happens with positive, with asymptotic positive probability so we really have to, to, to worry about each, each one of these and also I should mention that this exhausts all asymptotically it exhausts all cases in the sense that if I take the union of all these events on, on which condition I get a asymptotic probability once so, so this is really a full result but it's a bit technical so, so let me go back to to the, to the easy result where we ignore all the obstructions so, so let me tell you the methods it's just a glimpse about the method because there are some more points there but, but I think those are maybe the most interesting parts so, so I restrict to the case the first theorem that I told you that I just want to check whether SLN is inside my group and I condition on the fact that the only polynomial that divides the only polynomial that divides the only polynomial in X that divides F is X so that the content is X okay and yeah so GF is irreducible, is reducible it means that there exists invariant subspace of some dimension between 1 to N so between 0 to N and, and we can write it as, as the fact that there exist some additive Q additive polynomial of degree compatible Q to D, to DM that divides my polynomial indeed if I have the vector space V I can construct G by taking the product of X minus the elements in V and vice versa if I have G I can construct V as the zero set of G and something that happens in function field that does not happen in number field and this makes the number field case much more complicated is that in function field the height function is, is multiplicative so in some sense in the number field this is illustrated by the fact that the height function is multiplicative once you, but is almost multiplicative and the constant that you have to pay is exponentially in N something like this which cause a lot of problem when you want to take N to infinity so, but here it's multiplicative so, so irreducibility is slightly easier and I wrote some, some lemma I mean we can, if, if I have such a polynomial that divides F then I can get bounds on the degree in T and therefore I can bound the, I can bound the probability that such G exist so, so irreducibility is significantly, significantly easier than what is happens in the number field case and I want to emphasize this because in the number field case when I multiply two polynomials the height is not multiplicative it's almost multiplicative but, but the constant that I, so the height of F times the height of G will, will be comparable to the height of F times G but the constant will be, I don't remember now if it's exponential or even larger in N this is coming from going through the model measure which is multiplicative and, and when you take N to infinity exponential constant in N is slightly problematic because it makes all arguments fail so you need to come up with something different but, but here irreducibility we, we managed to do it I mean I skip some details but, but this is the heart of it and the, the, so this is one, one, one thing that we have height bounds and then we can prove irreducibility in this method and the second part is specialization method we want to specialize for example T equals zero and then and then I can lift the Frobenius so what does it mean to specialize F is a polynomial in two variables and after I specialize I mean I'm in polynomial I have a polynomial in one variable over a finite field so, so I have the Frobenius that acts on the roots and, and I can lift this Frobenius to an element in the, in the Galois group of the original polynomial and, and the point is that at least if F zero X is separable and this means that the corresponding point is unremifed in some extension never mind the details so if F zero X is separable then I get an element in GLN Q in the Galois group that have characteristic polynomial which is the same polynomial as my original polynomial but now every time that I have X to the Q I write just X so this is the characteristic polynomial of the corresponding matrix and, and we can show that this is in some sense almost uniform that I, I don't want to get into the details of that and then we get a matrix whose characteristic polynomial is, is almost uniform in some very strong sense and so again I want to prove that GF contains SLM so now that I know that I have almost uniform characteristic polynomial I apply finite group theory and here this is the part that I, that, that I learned I had to learn for this project that I learned from ALEE is that we have a we have a, we have the nine Ashbacher, Ashbacher classes of maximal subgroup of GLN and for example C2 is, is this groups like the Ritz product that I wrote before it's the groups that stabilize such kind of factorization we, we want direct sum with VT which have the same dimension so, so I have the symmetry group acting on the component I can transform them and then for each component they have GL of the correct dimension probably N over T that acts on each component so, so this is one, one type so, so we have eight, nine classes like this one of them is of the reducible polynomial that we already took care of of them and the point is that this is a deep theorem in group theory that although there might be a lot of maximal subgroups there are not so many characteristic polynomials as N goes to infinity so, so I remind you I have a matrix I know how to generate a metric with high probability a matrix whose characteristic polynomial is close to uniform so I have some constant time Q to the N of these with high probability but only a very small portion of them is a, is in one of these classes of maximal subgroups that do not contain SLM so, so the conclusion is that, that I have irreducible group that contains that is not containing any maximal subgroup so it must contain SLM okay and this proof use deep results of Fulman and Goranlik about the number of, okay all kinds of results and you also need to, how to say to enhance it with a analytic number theory in the sense of an atom of integers and this was done by Eberhard and Gertzoni okay so what I told you, I told you what is Galois theory of additive polynomials about three models of them I explained the results that we have in each of the models we have in two of the models, the large finite field and the large box we just get GLN with probability one but in the restricted coefficient model we get we get all kinds of variables restricted coefficient model we get we get all kinds of abstraction and each of these abstraction of course with positive probability but we can handle it and I gave you some hints on what is hidden behind the proofs and I think that's it thank you