 Thank you and thank you for the invitation. It's a pleasure to speak to international audience. So the my talk I found in my work in recent work that there's a common theme that keeps reappearing in a lot of things that I've been thinking about recently and from one perspective this idea of height functions and local height functions associated to close sub schemes keeps popping up in one manner or another. And so I thought I'd give a talk kind of talking about this theme and various ways that it appears and is being used in some recent work. Okay so I want to start off just going through kind of the classical theory before we get to how to generalize those two sub schemes. So to begin I recall that the height of a rational number written in reduced form so the GCD of the numerator and denominator is one is simply the max of the absolute values of the numerator and the denominator. And it turns out to be a little bit more convenient to work with the logarithm of this quantity. And so we'll work throughout with this logarithmic height and generalizations of this. And one way to think about the height is it's some kind of arithmetic notion of the size of the rational number or some I guess some physical notion if you look at the number of digits. On projective space we can define we can extend this quantity if you write your point with integer coordinates such that there's no common divisor among the coordinates then again the height is simply the log of the max of the absolute values of the coordinates. And we can think of the height of a rational number as related to this height if you think of the affine line in P1 and so it's related to this projective height. Alternatively another way to write the height is if you let mq be the set of places of q so we have a piatic absolute value for every prime and we have the infinite or Archimedean place with the usual absolute value. Then you can write the height in this form if you normalize the coordinates as above so that there's no common divisor then all of the sum is trivial except for the Archimedean the usual absolute value and so this is the same thing as the height at the top but the advantage of writing it in this form is that you don't have to normalize the coordinates this would work for any choice of homogeneous coordinates where the coordinates are integers and that follows from what's called the product formula that the product of the overall of the absolute values is one always easily from from saying okay so more generally this is all over the the rational numbers we have k as a number field and in case the set of places of k this is equivalence classes of absolute values and then for each place I'm going to choose an absolute value that's normalized in a convenient way it's not going to be so important but I'm going to stick into the normalization these factors like one over the degree of the number field so that when I use this everything works out to be the absolute height and with so with the right normalization of the absolute values then the height over number field is given by the similar formula where q is replaced by k more generally and it's a log of max of these absolute values on on projective space okay so more generally there's a machinery to associate a height to a divisor on a projective variety and the height satisfies certain nice properties so the height that we've been talking about in a projective space is just the height with respect to a hyperplane in fact for any hyperplane we have the same thing up to a downward function it satisfies some functoriality properties where the the if you take the height with respect to a divisor and have a morphism relating the varieties then the height with respect to the pullback is related to the height with respect to the divisor it satisfies a nice linearity property and in fact these first three properties already you can show determine the the whole theory moreover you also have a an important fact that it's preserved under linear equivalents of divisors or from another point of view you can attach a height to line bundles so we have all of these nice properties and the classical theory of this v height machine and the next thing we're going to do is talk about how this classical theory extends to higher codimensional objects okay so uh oh yeah sorry first let me talk about local heights so for you can also define a local height for each place of your number field and the intuition is roughly that the local height should be the negative log of the vatic distance from your the point to your divisor assuming say it's infected divisor and so with this negative log of the distance this means that this local height is large when your point is vatically close to the divisor d and these local heights also satisfy uh punctuality and additivity properties and of course the whole point of these local heights uh with in respect to global heights is that you can decompose your global height into a sum of of these local heights at least outside of the the support of your um let me just emphasize that the the global heights depend only on the linear equivalents of d but these local heights they really depend on on the divisor in a strong way you know there there's some kind of distance to the divisor and so they're definitely are not uh definitely depend exactly on the divisor matrix okay so what what do these look like so let me give the kind of easiest example on pn so if you have a hyper surface defined by some degree d homogeneous polynomial then a local height function with respect to this hyper surface and v is given by the following formula you take the log of the max of the coordinates to the d to power so things will be invariant under how you choose coordinates with the since f has degree d and you divide by the absolute value of f evaluated at your point and so you can see that since f defines d if your point is close to d that means that f of p should be small and so indeed this is going to be large when your point is is close to d okay so it is capturing some kind of uh negative log of the distance uh just to relate things back if you sum over all these local places then the the product formula for a number filled k tells you that the denominator there the the f of x not to x n is going to vanish when you take the sum of all places and you just left with the numerator and you get d times the usual project of height which agrees with the fact that the divisor d is d times a hyper plane and so if you believe the stuff I said about linear equivalents you should believe that this gives you d times the the project of height okay so anyway for for hyper surfaces which maybe are one of the more basic situations that we'll consider it's just given by this very explicit easy point okay so that's the classical theory in a very very quick nutshell and now we're we're going to go to uh setting of close up schemes so let me describe uh if close up schemes are a scary object let me make it a little bit less scary so divisors are our code dimension one objects and now we want to do this in in higher code dimensions they attach a height to just a point or or a curve on a higher dimensional variety and so a close close sub scheme of writing is roughly a close subset with some kind of scheme structure on it another way to think about it is that close up schemes are in bijection with quasi-coherent sheaves of ideals on x and that looks like a a mouthful but it's really just the kind of sheafification of ideals if you want to try to make sense of that it's way of giving an ideal on a writing uh in the simplest case where you're on affine space then close sub schemes are just in bijection with ideals in the polynomial ring and the way that you get a close sub scheme out of the ideal is you look at the vanishing set of the ideal like you typically do an algebraic geometry and then you put some scheme structure on it that reflects the ideal and it's speck of arm or the eye anyway depending on your algebraic geometry background it's a way to talk about the the set that the ideal defines in affine space with some scheme structure that that reflects the the ideal okay and on projective space there's a similar story where you can think about things in terms of ideals you have to there's a bijection with saturated homogeneous ideals I don't want to get into details but just that you can also think of these things as being objects that you get associated to ideals that then we want to do operations with close sub schemes and so the convenient way to do that is to talk about what happens on the ideal side and so this is with ideal sheaves but if you want to think about and or PN this is just operations with ideals and so the the close up scheme that we're going to define the sum to be the the close up scheme associated to the product of the ideal shoes and the intersection corresponds to the the sum of the the ideal shoes so we can construct we can begin to find these operations by looking at the ideals we say that the close up scheme y is contained in x if you get have a corresponding relation between the ideal shoes and the order reverses because you're looking at where the ideal vanishes we're also going to want to talk about punctuality and so you can also define the pullback of a close sub scheme and this is the the definition what happens on ideal sheaves if you don't know what this means it's just the natural scheme that you get by taking the pre image and putting the natural scheme structure on confusingly this is not the same ideal sheaf as the pullback of i sub x because that's not generally an ideal sheaf but anyway for people that know this okay so Silverman then defined heights associated to close up schemes generalized in the case for divisors and they satisfy up to bounded functions so everything is up to bounded functions unless you have some group action or something to normalize things and so i may accidentally omit an o of one in places but everything is basically up to o of one always so the the theory agrees with the classical case when the it's a divisor that you view as a as a closed sub scheme you have additivity just as you did with divisors you have punctuality in the same way that you do with with divisors if x is a closed sub scheme that's contained in y then you get a corresponding relationship between the the heights and in fact even if the supportive one is contained in the support of the other the support is just the the closed subset associated to the closed sub scheme then you get that one height is bounded by a constant times the other this is some kind of height version of the nulstons okay uh similarly you can also define local heights and in fact i'm doing a bit backwards here and in Silverman's paper he starts with local heights and then gives the global heights but anyway the you can define local heights and again the global height decomposes as a sum of the local heights and they they satisfy the same properties and additivity functorality and and so on additionally and this is going to be an important property up to a bounded function one way to think about these is that the intersection of two closed sub schemes the height the local height associated that corresponds to the minimum of the the local heights on on x and y this is only for this local height it doesn't work for the global heights and that's just because the the sum of minimums is not the same as the minimum of the sums you know the men maybe taking that ai or bi depending on i and so you only get an inequality if you look at the the global heights anyway the intersection corresponds to the to the minimums the quick question yes is this height a function on point algebraic points of the variety uh yes yes that's right and so generally speaking we'll uh we'll think about it as you know we'll always uh we'll be interested in primarily rational points and so uh if you want something that's over some other field uh we'll typically just take a an extension and and work over but yeah you can do everything over the over the algebra okay so i want to talk about ways to interpret these heights and there are at least four interpretations that all are useful and we'll we'll see use in various ways so the the first interpretation is as a distance function so it still holds that when x is a closed subscene that you can think of this as giving negative log of the the adic distance between a point in your closed subscene a another way to view it is as a minimum of local heights so if you want to reduce everything to divisors uh you can do this you can always write a closed subscene as an intersection of uh effective divisors and then you have the formula that the the closed subscene is just the the minimum of those uh heights attached to the divisors so in this way there there's nothing new everything is coming from divisors but we'll see i i think there's a there's a uh it gives a deeper intuition to to use this machinery okay so we we can think of things as um these heights as what keeping devices if you like to keep track of of minimums of classical heights so another way to think about it i yet another way is to relate things to greatest common divisors so let's look in detail what happens when you're working with hypersurfaces and for simplicity let me take them to be the same degree then if you look at this men construction or men property that i mentioned that will correspond to a max in the denominator and you get that the um height of the intersection of two hypersurfaces of the same degree is given by this formula where you where you have a max of the two defining polynomials in the denominator let me simplify this if we work over the integers and let me normalize things so that uh the coordinates are are co-prime then for any uh prime if i if i look at the piatic height the numerator up top is going to be uh one because i'm assuming there there's no common divisor and so i'm just left with the denominator and so we just get the simple formula that it's negative log max of f1 and f2 evaluated at your uh tuple and then in this case if you uh sum over all of the the non-archimedian places over all the the piatic absolute values if you think about this negative log max it's picking off the uh p part of the greatest common divisor when you do this and when you sum over all of these you just get exactly the logarithm of the gcd of f1 evaluate at the point and f2 evaluate and so if i look at the full height which also has an archimedian contribution uh it's some kind of generalization of the logarithm of the gcd of f1 evaluated at your point and f2 evaluated so this is going to be an important point of view is that there these heights of the higher codimensional objects they correspond to some kind of gcd okay and just to record now later um we'll talk about the uh generalized greatest common divisor and this is a formula for it that coincides with the the usual gcd when everything's an integer it's some extension to to number folks more okay a fourth way to think about things is that these are about heights associated to exceptional divisors so if i if i have a closed sub scheme then there's a you can blow up along your your closed sub scheme and you get a variety uh ax tilde and you have an associated exceptional divisor which is just the the pullback of your your closed sub scheme and uh that turns out to always be a cardiac divisor this kind of the uh universal property of blowing up is that it's it's minimal with with this property and then now if i look at the height with respect to this exceptional divisor by functoriality i see that it's the same thing as the height with respect to this closed sub scheme uh downstairs and so uh another way to view these heights for closed sub schemes is again they come from divisors you can always reduce to the theory of divisors by blowing up appropriately uh but again i think it gives some insight into uh on the one hand the heights with respect to these closed sub schemes but also what the heights with respect to exceptional divisors look uh and so this is another another nice connection that we'll see is useful okay so the uh first set of results we're going to talk about are related to this uh greatest common divisor point of view so let me first give give some motivation for things coming from voidus conjecture so uh we'll have a project to move projective variety x some data k capital k we'll always note the canonical divisor then a very special case of voidus conjecture is that you have this inequality for the height associated to the canonical divisor that it's less than epsilon times some ample height at least outside of some closed subset uh qualitatively what this inequality is saying one consequence of it is that if your varieties of general type that means that the canonical divisor is a big divisor then uh from north cuts there and for big divisors this tells you that on a variety of general type rational points should not be these risk limits in other words uh this is a quantitative version of the the boundary language okay now what happens if your canonical divisor is trivial so like a k3 surface or a new variety then this inequality doesn't really tell you anything because the height is just a bounded function and that's not really surprising in this case it's believed that rational points are potentially dense of course this is an open problem say for for k3 surfaces but the rational point should be dense and there's a more general set of conjectures due to campana that explains when this kind of thing should happen but in particular should happen when the canonical divisor should be okay the reason why I'm talking about this is there is a kind of a surprising way in this situation to uh I would say get something from nothing uh this because it's what he looks trivial in this case uh so how do you do this um let me take a a separate of your variety x that has a trivial canonical divisor and let me blow up along the closed subset then there's a well-known relation between the canonical divisors when you blow up in the situation uh it's given by the the pullback of the canonical divisor downstairs plus some uh multiplicity times the exceptional divisor and with our assumptions the canonical divisor is zero and so we just get some multiple of the exceptional divisors the canonical divisor upstairs but now if I apply voyage conjecture upstairs I get that the exceptional divisor is bounded by uh some kind of big height and then using functorality you get that the height with respect to this closed sub scheme should be bounded by an ample height outside of some closed subs okay and this is especially meaningful in this case for the canonical divisors divisors trivial because you should have a dense set of rational points and so this is really telling you something about uh heights with respect to these higher codomention closed sub schemes on on such a variety that some fact about gcd is on on on this right okay and it's a it's a bit magical that you can you can get something kind of extra from voyage conjecture by blowing up but that's that's what happens okay a simple case of this is let's take an elliptic curve and a point in say a five-shot form and a point x y on it and let me write the the point in as a fraction and then let me look on on the product of the elliptic curve with itself and look at the closed sub scheme being the origin then if you believe what I've been saying this should correspond to some kind of gcd and it's exactly says exactly the following that the the height of the with respect to y of this pair pq on this square is the log of the gcd of the denominators of the points p and q and that it should be bounded by the height of these points outside of some special curves in this abelian surface okay so you out of this conjecture you get these kinds of very interesting inequalities involving gcds okay this is all conjectural you know conjectural on voyage conjecture and so this particular problem for even for e cross for the self seems I've thought about it some it seems rather difficult there's some work of Dave McKinnon when e has rank one or rank one over the endomorphism ring there are also some results over function fields but I would say in general it's uh it seems like a hard problem even more so if you if you look at a an abelian surface that's not split now the the same argument from voyage conjecture which I won't go through it works also if you work with integral points on an open subset of a projected variety given by some complement of a divisor and again for simplicity I'll assume that the canonical divisor plus the boundary is trivial and there's some technical assumptions you need your close-up scheme why to intersect your divisor in some nice way more generally you get things out of voice conjecture you get some kind of refined set of inequalities and this was worked out by by silverman and a nice paper about GC is connecting all of this let me take the the case of a power of GM which we can think of in one model is Pn minus the coordinate hyperplanes so here there's a group structure where you multiply your n tuples coordinate wise and the log canonical divisor here is trivial the from this compactification the canonical divisor and Pn is negative n plus one times a hyperplane and so when we add the boundary we get a trivial canonical divisor and integral points here are just n tuples of units of s units so I haven't defined integral points but I'll just tell you what it is in the situation and so if you follow the previous setup from voyage conjecture you to expect that certain kinds of gcd inequalities hold when on this on this right so let me give the the story for for this setting where we can prove some things so I'm going a bit backwards this is a viewpoint from after these results were known but the starting point was back to result of the jorque corbeia and zanier and they show that if a and b are multiplicatively independent integers then you have the following inequality for the logarithmic gcd that the gc of a to the n minus one and b to the n minus one should be bounded by epsilon times n for all but finitely many positive integers and so it's a very nice elementary looking inequality that that has a lot of depth to to to the proof and to the the theory behind it how is this connected to the point of view of units well a to the n and b to the n are units when you put some primes into your set s and so this is really about the logarithm of the logarithmic gcd of u minus one and v minus one where where you and v are s units and and some and this corresponds to this height with respect to the point one one that's the minus one minus one up there on whoops I put gm to the end that should be gm squared this is the the two dimensional case in p2 and so you know you have this very elementary looking statement and on the geometric side it corresponds to this height with respect to a point on gm squared on gm squared okay then they proved a general result for polynomials in u and v which on on gm squared and I extend the result and and for n-gray equal to three they proved the n equal to case of the following theorem so it's a bit long but the instead of working with the units it's equivalent really to work with a finitely generated group and in gm and a few bar such a group will always be a group of s units over some number field the coordinates will be s units in some number field and we need some non vanishing condition at the origin for the the polynomials that's some kind of general position condition and then we get this inequality that says that the logarithm of the gcd of these polynomials evaluated at these unit points is bounded by epsilon times the times some height outside of some exceptional set which in this setting you can take to be translates of sub-torn okay so let's say but anyway if you if you can't follow that the statement it's a generalization of the a to the n minus one b to the n minus one on the previous slide but in more variables okay a height version of this theorem it's not quite equivalent but it's very closely related is that this is really saying something about heights with respect to closed sub-schemes of codimension at least two the connection is that if you're looking at the gcd of these polynomials f and g then the closed sub-scheme defined by f and g is where the the connection is and so another way to formulate this in terms of heights is that all of this is giving a bound for this height with respect to a closed sub-scheme on of codimension at least two on on on pn with respect to these integral points on on gene to the n which again you can think of as in tuples of the units okay so there's a height a height formulation of of the term okay a key ingredient in the proof is the the subspace and and I'll say more about that later let me just mention some more recent work in this direction so yasafuku and wang gave a different proof of this inequality using ruvoidus inequality and relaxed some of the hypotheses a bit proven in a more general setting my phd student jingxiao has some work that relaxes the unity condition so in fact you don't really need the points to be units they can just be close to a unit in a in a precise sense and so you can extend this inequality to rational points with some penalty if you choose a fixed epsilon then then you then you can specify how close they need to be to a unit for other work kping huang and i postdoc at msu also extended the yasafuku wang result to an even more general setting instead of going through the subspace machinery or the ruvoid inequality we use some geometry and the the result on gene to the n this is this is work in progress that hasn't hasn't okay so um let me go back to the subspace term and mention some connections with uh close subs schemes in the subspace so to start let me go back to the the basic theorem and all end results i want to consider which is ross theorem and ross theorem tells one how closely you can approximate some algebraic number by a rational number in terms of the denominator and so it says that with finite many exceptions you can't get closer than one over the denominator squared plus well times the denominator to the epsilon famous result and then ross theorem you can write in a more general form over number fields and including uh non-archimedian absolute values and this is a height formulation of the ross theorem due to write out in lane and the the form of the theorem which is going to be a model for some things we'll talk about later is that on the left hand side you have a sum of local heights and the right hand side you have some constant times of global height and the ross theorem is giving you some uh inequality between the two yeah so the sum of local heights is bounded by two plus epsilon times the height on on p and uh this is um just a reformulation in the height language okay then uh schmidt generalized this whole thing from the projective line to projective space and hyperplanes and chocovier proved the general form where again you allow non-archimedian absolute values and here's the statement if you're not familiar with it it's probably going to be too much to take in but let me just say the the left hand side is again it's some some of local heights attached to hyperplanes is bounded by some constant times of global height and it happens to be that uh n plus one which is related to the canonical divisor is is the right number um there are okay there are many important aspects of this but uh the high points have to be in general position you can change them with the place v and they're all very important but uh i don't want to get bogged down into explaining all the all the details about them another important aspect is that the exceptional set consists of hyperplanes hence the the subspace in okay and then uh the story is this has been was generalized to hypersurfaces or another way to think about it is more generally it can be generalized to linear linear equivalent divisors on a projective variety and so there are uh theorems of everett sofredi and corvaya zanier in in this direction and since i think the subspace theorem already is a lot to take in i don't want to restate the theorems but it's an analogous result for hypersurfaces or or more generally these linear equivalent divisors and what i really want to want to talk about is how you can use close-up schemes to understand these things so this will be a close-up scheme perspective on on such inequalities and so what's the key idea the key relation so suppose we have effective divisors d1 to dn and i'm looking at one of these local uh sums of local heights so i'm going to fix my point p and i'm looking at the sums of these local heights attached to these divisors df typically any hero would be the dimension of the okay so if i'm fixing my point after re-indexing i can assume my local heights are ordered and in this way this depends on p of course but i'm going to fix my point p and uh re-index thing so i have this ordering on the local heights and then now here's the the basic trick if i order them in this way then if i look at the men of these height functions at some point and you use the machinery of close-up schemes we get that the this local height attached to the i-th one is the men of the previous guys and from the close-up scheme machinery this is the same as the height associated to the intersection of the of these divisors and so if i set yi to be the intersection of these first i divisors then i see that my some of these local heights with respect to these divisors it's equivalent to a sum of heights with respect to close-up schemes where you have these nested close-up schemes given by these intersections so the the trick here is that we replace the sum of local heights of divisors by this equivalent sum of local heights of these nested close-up schemes and the subspace theorem what you're doing is you have all these hyperplanes and you get this like flag of hyperplanes you know that's replaced by these linear subspaces of where the dimension goes down each time generally if the if you assume the divisors take ample in general position then the these y sub i's will go down in dimension by by one each time okay and so it's it's a trade-off on one hand these close-up scheme machinery is maybe more complicated from a certain point of view on the other hand they have smaller dimension and so some things work better from this point of view one reason why i think it's important to look at this kind of thing is that when you're only at the level of divisors it's kind of hard to make use of information about how the divisors intersect and so for instance the a lot of these theorems like the Ebertzo-Fredi-Cordvia-Zanier theorems they don't really see the intersections they work just as long as you have some general position condition whereas here the close-up schemes capture information about how the divisors intersect in an important way and so the right-hand side when you rewrite things in terms of these close-up schemes it contains a lot of information about the intersections that you might hope to leverage in in arguments more generally i think because of this uh connection it makes sense to study diaphragmatic approximation in the setting where where you try to prove results more generally about close-up schemes okay so in this direction uh Gordon Hyer and I proved a version of Schmitt's theorem for close-up schemes and the set up is similar at each place you have some close-up schemes that are in general position with an appropriate notion and they appear with some coefficient on the left hand side that i'll describe it's it's a so-called sushadri constant if you know what that is then you can uh generalize the classical sushadri constant to close-up schemes and given the definition but if you don't know it's just you have to you have to put in some coefficient some natural coefficient for some quality to to hold i'll describe what it is in a in a simple setting in a moment but anyway this is some kind of analog of the subspace theorem for for close-up schemes on on the left hand side uh what are these sushadri constants about so if you're in the situation where your ample divisor on the right is a hyperplane and why is a hyper surface of degree d then this is simply uh the coefficient is one over the degree and if you're uh know the the these theorems about uh that hold for hyper surfaces they all have some kind of normalizing factor where you normalize by the degree and so it's nothing more than this natural coefficient in in those theorems all right in this case um I feel for completeness I need to say what the general position means the it means the following that the the code emission behaves in a code emission the intersections behaves in a natural way it's a bit counterintuitive in some ways so like if you have a point and look at this definition you can repeat it a bunch of times which doesn't seem very general position but uh is general position for for the prior to our work on the stem there was uh the case of points is essentially equivalent to a result of McKinnon and Roth and prior to our work there was work of ru and wang and there's more recent work of ru and wang and voida where they replace the sushadri constants by these so-called beta constants which I'm not going to define but if you know what they are there are some recent papers that improve our our result by replacing these sushadri constants by these these beta constants okay um then the the last thing I want to talk about is some current work or this section could probably be subtitled work that would have been finished a year ago if it weren't for a once in a century pandemic throwing life into chaos but uh right now it is current work uh one of the uh projects that I'm pursuing has to do with what's called the nacho ru wang theorem uh and this is a generalization of the subspace theorem to hyperplanes that are in uh m subjournal position so m subjournal position is a relaxation of the general position condition where you require only that um m of the hyperplanes intersect at a at a point at most m intersect at a point so this this is a important generalization of the subspace theorem where you weaken the general position hypothesis the proof of this theorem uses this uh combinatorial machinery that's named after nacho these nacho weights that you attach to your set of hyperplanes and the proof seems rather specific to hyperplanes so the machinery used seems to not generalize in a straightforward way to hypersurfaces and so uh proving a version of this theorem for hypersurfaces is is uh open uh in work in progress with with gordon hire we proved some kind of version of the subspace theorem for closed sub schemes in m subjournal position and I uh well first it's work in progress but also the statement is rather technical so uh I'll just say that it's it's some version of this for um m subjournal position and even though it's about closed sub schemes I you can apply it to the case of divisor so again if you're only interested in divisors it's still I think um crucial to to think about this machinery and uh we've made some some uh substantial progress towards proving an analog of the nacho wrong theorem for uh in a general setting which includes hypersurfaces on projective space um and and essential to these arguments is really this trick of replacing a sum of local heights by this equivalent sum of local heights of these nested closed sub schemes and so the this closed sub scheme perspective uh even for the stem that's just only about divisors uh seems to really give a lot of insight in um how to how to approach this this problem okay the other project that I want to mention is uh another work of provian zanier for most of my talks I could probably randomly label my slides for by zanier and it would make sense uh so here the the problem uh that they looked at was this equation f of a to the n y equals b to the n where uh y is an integer and a and b are fixed integers and the key thing is that they're not co-prime they they share some uh common divisor and as a uh amusing application they mentioned that you could use their study of this equation to solve the problem uh of when a square has three non-zero digits to to classify the the solutions that there's some infinite families um and finally many uh along with finally many solutions uh there is uh a lot of other work on this kind of problem where you're looking at uh squares or powers that have few digits and I'm not going to be able to uh mention them mention all of them but uh Mike Bennett has some some work on this and that it's very neat stuff uh the what I want to mention is um the the more general problem this f of a to the m y equals b to the n uh in joint work with uh Huang and Xiao we have a uh kind of a new perspective on on this equation uh including some high dimensional versions of this kind of um defanting equation so for instance if you look at squares with a fixed number of non-zero digits uh so more than three digits then uh when you apply these techniques you can say something about the uh two smallest exponents and the two largest exponents you don't get a fully uh uh satisfying theorem but it's at least put some constraints on the uh smallest and largest exponents and um let me just say that uh I find kind of funny that in this analysis the Archimedean GCD uh gives you information uh so the Archimedean GCD is typically it's large when you have some uh polynomials that take small values in the the Archimedean sense uh what does it all have to do with what I'm talking about but geometrically this kind of question is related to studying uh height bands for close subschains where uh your sub scheme y is on your divisor d so this is a situation that's not quite contained in the um previous results that I was discussing there your subscheme doesn't intersect your divisor in a nice way uh in this case your your subscheme is on the divisor or say uh is a point on the divisor at maybe a nasty point of uh intersection of the the components uh and uh again the the these arguments for studying this the close up scheme machinery is really crucial to uh constructing the arguments and and figuring out uh geometrically what what's going on um so yeah I um you know provides any I had a had a a different method and this is kind of trying to recast it in this close up scheme framework and get some arguments that are um from some point of view rely on some geometry of the the close up schemes that that appear uh in in this problem um okay so I that's the the last thing I want to say in the the new projects I didn't get very many details for but uh I guess my overarching point is just that I think the close up scheme machinery um it's more than just a a bookkeeping device it can really give you some insight into the geometry of things uh and help you come up with new arguments to uh tackle some of these problems even if you're only interested about uh uh divisors ultimately this translation into into this world is is useful um okay so I I think I'll stop stop there