 opportunity to speak here. So, earlier Prof. Trivedi introduced some compactly supported real valued continuous function called the Hilbert-Kunz density function to study Hilbert-Kunz multiplicity. So, in this joint project with Shuprajo and Prof. Trivedi, we are trying to look for a density function for another invariant that is the epsilon multiplicity and in the next talk Shuprajo will set more light on that, but to do that we realize that we first need to introduce some kind of density function for field rations and which has its own importance. So, throughout my talk I will stick to these notations. So, a will be standard graded finitely generated algebra over the field K and r is a pi graded K algebra with bi degrees like this. So, if we take all these dj's are 0 and dj's are 1, this is nothing but standard graded phi graded algebra and in that case it is well known that this length of these components of r mn they are basically given by some polynomials for large values of m and n, but in the non-standard graded setting this is not quite true. So, in different cases in the non-standard graded setting several authors including Huang and Thru they studied and they gave several descriptions and then Brucefield and Nien they showed that there are finitely many chambers and in each chamber there are some quasi-pollinomials, so that gives us this length of the components, but they built in the generality and they did not give an explicit description for the chambers neither they talk about the top degree terms of the quasi-pollinomials, but for our purpose we need those descriptions and that is why for this restricted setup we had to study more about the Hilbert functions of this bi graded components. So, this bi graded algebra can be seen as a quotient of a polynomial ring and we can assign appropriate degrees to the variable to make this map a bi homogeneous map and then since r is finitely generated S module we have a finite bi graded minimal resolution of r and the bi graded Hilbert series of S is given in this manner and due to the finite free resolution of r we have the that Hilbert series of r that is represented in this way where this polynomial is nothing but this captures the degree shifting that occurs in the bi graded minimal free resolution. And so the length of this function this is basically alternate sum of the length of shifted copies components of this S. So, here note that this length of this components of this polynomial ring they are basically given by such a vector partition function and here this matrix this column of this matrix is basically the degrees of the generators of the base ring r or as well as of S. So, it gives us some hint that we need to then study more about this vector partition function, but before going into details I would like to give some quick examples of such ring r. So, an ingraded filtration that is a collection of ideals of decreasing filtration of ideals which satisfies this property and the associated result is given in this manner it has a bi graded structure if all these ideals are homogeneous ideals in A. So, with this if we further assume that the result debris is noetherian then this filtration is called noetherian filtration and then adic filtration, integral closure filtration, tight closure filtration and this generalized symbolic power filtration and also there are many more symbolic power filtrations which are noetherian under certain conditions. So, for all these rings we have that rational Hilbert series. So, here I would like to point out that since we are going to use this vector partition function we just need the Hilbert series to be a rational function if any and there are examples where this associated result debris is not noetherian, but its Hilbert series is a rational function. So, our results are also valid for those kind of filtrations. So, now I want to discuss about this vector partition function we are mainly going to use a structure theorem due to Sturmfeld and for that I need couple of notations. So, here this matrix M is given by this where this v is basically the columns of the matrices and the polyhedral cone that is defined by these vectors of these matrices that is defined in this manner and for any subset sigma of this set 1 to n 1 we denote this sub matrix that is generated by this respective columns by M sigma. So, there are few more notations, but I am going to skip it and describe it in terms of pictures. So, I am restricting to this matrix. So, if we notice carefully, so if we take some results associated to some ideal that is generated in degree 2, 4 and 7 then one can see that the migrated results has these degrees. So, for this case the polyhedral cone associated to this matrix M that is the whole thing and that has this polyhedral subdivision given by this. So, notice that this is the third column and fourth column of this matrix and they basically this is the cone that is associated here and all these matrices here that are listed here. So, they are basically the maximal linearly independent vectors that occurs in the matrix. But there are some other combinations also for instance if we take 1, 2 and 1, 7 they also generate I mean they are also linearly independent, but there is a partition due to this line 1 and 4. So, we are ignoring that. So, in some sense we are taking the common refinement of all these partitions and this kind of representation is the chamber complex is called the chamber complex of this polyhedral cone. And then so just now as I told that if we take this 1, 7 and 1, 2 and the polyhedral cone associated to them. So, they are they contains this polyhedral cone containing 1, 2 and 1, 4 and so then we will say that this third column and first fifth column this sigma this sits here. So, this is basically collection of all these number of columns which contains this polyhedral cone. And this is the quotient of this is a finite group and this basically the z linear combination of the columns of the vectors. So, here this m 3 4 these are the these two columns and we are taking the z linear combination of these two vectors and it is known that this is finite and if we take that first column and the fifth column then one can easily see that the z linear combination is basically z 2. So, in that case this group is basically 0 and such kind of sigma that 1 5 this is called the trivial group. So, then now we are ready to state Sturmfield's results. So, what it says it says that so in each sorry I do not know how to make it a full screen, but if it is ok can I continue. So, it just says that so if we are in each chamber the pointer is not working it is ok. So, if we are in each chamber then this is given by this vector partition function can be represented as some of these two polynomials, but I mean I have marked in red. So, that is the main part that says that it will be a quasi polynomial and that denotes the image of U in the quotient group G and as I mentioned that in the top chamber. So, that is defined by 1 7 and 1 0. So, in that chamber since the group is trivial. So, it is basically a polynomial. So, based on this observation we studied I mean like we studied for any Noetherian filtration, but here the degrees are associated to just Noetherian filtrations of ideals since I mean one can actually follow the same argument for Noetherian filtrations and one can say see that these are the maximal possible cells. And so in this setup one can say that for A's the quasi polynomial has same top degree terms, but as we saw that I mean for the length of the components of R this is basically alternate some of the component components of A's. So, it does not ensure that the for R also the top degree term of R that quasi polynomials have I mean they are constant they are not quasi I mean they are not periodic. So, that is not ensured. So, we had to work more on that. So, this is the initial chamber complex then we are taking this shifted chamber complex. So, if we take some point here one can see that that point actually sits in the second column with respect to the blue chamber and in the first chamber with respect to the red one, but if we extend this line. So, if we take n further along the slope then it falls in the same second chamber. So, based on this observation we can define some restricted number the restricted chamber that is the intersection of all these shifted chambers. And then we short that. So, if we take the depth of the ring is positive and the dimension of A is D. So, here I would like to mention that we can remove the depth assumption, but then the description of D will be changed and in that case if the ideal is generated in these distinct degrees and Cj's are the corresponding cones then we short that this Hilbert function associated to the Ries algebra of I t l that is given by some of these two polynomials. So, first one is the polynomial and second one is the quasi polynomial and here we also showed that the quasi polynomial has degree strictly less than the degree of P. So, it ensures that the top degree term in the quasi polynomial that is constant. So, this helps us to define. So, basically it gives us like this we know that. So, D 1 is the lowest degree element in I below that it is 0 above B s it is largely studied by several authors including Huang and Thru and several people also studied diagonal subalgebra in the yellow region, but we also found that in the intermediate region also they have quasi polynomial type behavior and their slopes are exactly the degrees of the generators of I and they are also the top degree terms they are constant. So, this helps us to define a function in this manner. So, we are basically taking the slope x and we are seeing the growth along that slope and this is we show that this is a well defined continuous function. And we further one can see verify that these are basically given I mean piecewise polynomials and in the previous chambers in the. So, these non trivial chambers it is are they are polynomial of I mean at most degree D minus 1, but P s this exactly equals to D minus 1 because in that case there is no quasi polynomial term and we can show more on that. We also explore when I mean we also gave some sufficient condition when these are non zero and when they are 0. And besides we define the limit function which is again a piecewise quasi polynomial function and which can have possible discontinuities at exactly the degrees of the generators of the ideal. Although we have computed several examples and more or less it says that it is continuous except at D 1 because before D 1 it is 0 from D 1 it is becoming non zero. So, it is highly expected that at before at the slope D 1 there is certain discontinuity, but all our examples that ensure that it should be continuous we have some proof, but not verified yet. So, that is all I wanted to say.