 Good evening everyone. I would like to thank the organizers and ICTP for giving me the opportunity to speak here and also for organizing this school and workshop. I'll be talking on Tite-Hilbert polynomial and effrational local rings. This is based on joint work with Professor Kui and Professor Verma. Let us begin with some notation and definitions. Let Rm be a netherial local ring and I be an m-primary ideal. A sequence of ideals i sub n is called an i-filtration if for all m and n and z the following holds. Firstly, i sub n is the entire ring r for all n less than equal to 0. Secondly, it is a descending sequence of ideals. Thirdly, it satisfies the graded property and i to the power n is contained in i sub n. Further, this filtration is said to be i admissible if there exists natural number r such that i sub n is contained in i to the power n minus r for all n and z. One of the trivial examples of i admissible filtration is i-addict filtration wherein we consider powers of ideal i. The first non-trivial example of i admissible filtration was given by Ries via integral closure filtration. The integral closure of an ideal i is denoted by i bar and is given by the collection of all elements x in r such that x satisfies the following equation wherein the coefficients come from the corresponding powers of the ideal i. These characterized analytically un-ramified rings as follows. Let me recall here that a ring is said to be local ring Rm is said to be analytically un-ramified if the completion of r is reduced. Ries proved that a Noetherian local ring Rm is said to be analytically un-ramified if and only if the normal filtration that is we consider the integral closure of powers of i of any primary ideal i is admissible. I admissible filtration possess a very interesting feature called as Hilbert-Sammel polynomial that FBN i admissible filtration and Rb Noetherian local ring with dimension D. The Hilbert-Sammel function of F is defined as the function which takes n to the length of r mod i sub n. And it turns out that this is a polynomial function that is there exists a polynomial with rational coefficients of degree D called the Hilbert-Sammel polynomial of F such that the Hilbert-Sammel function coincides with this polynomial for large values of n. The coefficients are referred to as the Hilbert coefficient and E0F is called the multiplicity of F. In this talk I would like to target a particular class of filtrations that is tight closure filtrations which is also talked about a lot in these two days. Let us set up some notations for a Noetherian ring R of prime characteristic P and an ideal i in R we have the following notations R0 denotes the complement of union of minimal primes of R. The letter Q is reserved for the Eth power of prime P. Eth for being its power of i is denoted by i square bracket Q and is given by the ideal generated by the Qth powers of the elements of i. Tight closure of i is defined as collection of all elements x in R such that there exists the in R0 the property that c times x to the Q is in i square bracket Q for large values of Q. An ideal is set to be tightly closed if i equals its tight closure. Ideal is contained in its tight closure which is further contained in its integral closure. Next we have Brankon's Coda theorem which states that for Noetherian ring R of prime characteristic P and an ideal i of R generated by n elements. The integral closure of i to the bar n plus R is contained in the tight closure of i to the bar R plus 1 for all natural numbers R. As an easy consequence to this we obtain that when i is the principal ideal when the tight closure coincides with the integral closure. Now we have the notion of test elements. As the name suggests this test element helps us detect whether an element belongs to the tight closure or not. An element c in R0 is set to be test element for R if for all ideals i and for all x in i star c times x to the Q belongs to the Qth power for all values of Q. The ideal generated by all the test elements is called as test ideal of R. The parameter test ideal of R is noted by tau bar R and is the ideal generated by all the elements c in R0 such that c times i star is in i for all parameter ideals i of R. An element is said to be parameter test element if it is in the parameter test ideal as well as in R0. Due to this result of Hoxton-Hunike we have test elements, existence of test elements in certain classes of rings. Where in the rings are reduced algebra of finite type over an excellent local ring of time characteristic. And let's see R0 be such that R localized at c is regular then some power of c turns out to be test element for R. Well Mukundan and Varma recently introduced the notion of tight Hilbert polynomial, they proved the following. Let R be a d dimensional analytically unramified local ring with prime characteristic P and I be an M primary ideal of i. Let tau be the tight closure filtration that is we consider the tight closure of powers of i. And tau is an admissible filtration and therefore corresponding to this we would have a polynomial which is called as tight Hilbert polynomial. And it is given by the following expression here is it a star of i is same as the multiplicity of i and the, and all the other coefficients are called as the tight Hilbert coefficients of i. The Hilbert coefficients are very useful in determining certain invariance of ideals and things. Here are some of the results from the past literature in the same direction. I showed that if R is a Cohen Macaulay local ring and I is an M primary ideal, then even if i is equal to zero if and only if I is a complete intersection. Agata showed in 1961 that no they didn't ring RM is regular if and only if I is unmixed and the multiplicity with respect to the maximum ideal is equal to one. And Cello's in 2008 conducted that for any ideal Q generated by a system of parameters, even if Q is strictly less than zero if and only if R is not Cohen Macaulay. Mandel Singh and Verma proved the following in 2011 that RM be a local ring, then even if I is less than equal to zero for any parameter ideal. And Z et al settled the conjecture in 2010 by proving that in a formally unmixed local ring if Cohen Macaulay if and only if even if Q is equal to zero for some parameter ideal. It is natural to expect that even the title but coefficients can be used to characterize certain properties of the ring. And it has been recently proved that the vanishing of first title but coefficients gives us a freshality of the rings under certain hypothesis. So let's look at the definition of a rational rings. H elements x1 so on till xh are called as parameters at the height of the ideal generated by them is exactly equal to edge, and the corresponding ideal is called as a parameter ideal. A ring is said to be a rational if all the parameter ideals are tightly closed. Let's look at some examples and non example, that K be a field of characteristic P and SP the polynomial ring K x y z. I'll be the quotient ring as smart ideal generated by x square minus y cube minus that seven, and I be the ideal generated by y comma z here throughout the small case letters denote the image of the corresponding uppercase letters in the quotient ring. So I is the ideal generated by system parameters and excellent I star. Therefore, we have that the ring is not a rational. On the other hand, in the second case, wherein we have the quotient as smart ideal generated by x square minus y cube minus that is a seven, and I is the ideal generated by y comma set. The next is not belong to I star if I don't leave these strictly greater than seven and therefore it is a rational if I don't leave these strictly greater than seven. So, I'm looking at Burma proved that in analytically un-ramified when my colleague local ring with prime characteristic P, the following characterization holes, the ring is a rational if and only even star of I is equal to zero for some I generated by system of parameters. Okay, as the question whether the same characterization holes if going Macaulay assumption is replaced by the unmixed. He asked the following let RMB and analytically un-ramified unmixed local new day rendering and QB and ideal generated by system of parameters. Is it true that even star of I is equal to zero if only R is a rational. In the case of dimension one we have positive answer in fact it turns out to be regular. But R be a ring of dimension one and I be principal ideal which is m primary. Since R is cohen Macaulay because R is reduced. And even star of I is equal to zero it forces the ring to be a rational by the. Now consider a minimal reduction B of M. And from bank on Skoda theorem we have that the tight closure coincides with the integral closure. As the ring is a rational in this case the tight closure is itself and therefore maximum ideal is generated by single element which says that the ring is regular local ring. In the case of higher dimensions and dimension to itself we have a counter example just given as follows let K be a field of prime characteristic be greater than equal to three. And I'll be the policy we bring K x4 x cube by x y cube by four that Q be any M primary parameter ideal of our then it turns out that even star of Q is equal to zero but the ring is not a crash. Now we have the definition of the tight closure of zero in the top most local homology that RMB a D dimensional no they're in local ring of characteristic P. The tight closure of zero in HDMR is given by collection of all elements in HDMR such that there exists C in RO with the property that C times the years for being as action on it at vanishes for large values of B. So compute the title but coefficients in this case let RMB an excellent reduced equidimensional local ring of prime characteristic P and dimension at least two that X1 so until XG be a parameter test element the parameter test elements and Q be the ideal generated by them. Then we have the following expressions for the title but coefficients in terms of the usual hill but coefficients. And we also have an expression of the title but coefficients in terms of the lens of local cumulogy modules and the length of tight closure of zero in HDMR. Now as Chang and Willemere prove the following let RMB an analytically un-ramified excellent local domain and IBM M primary parameter ideal. If even bar of i is equal to even of i then the ring is regular and i to the power the integral closure of i to the power n is equal to i to the power n for all n. By even bar of i we mean the first hill but coefficient corresponding to the integral closure filtration. We have an analog of this theorem for the tight closure that RMB an excellent reduced equidimensional local ring of prime characteristic P and dimension at least two X1 so until XG be parameter test elements and Q be the ideal generated by them. Then the ring is a fractional if and only if even star of Q is equal to zero and the depth of the ring is at least two and if and only if even star of Q is equal to even of Q. Here is an example to show that the assumption on the ring is not superfluous. Let S be the power series ring FB XYZW and R be the ring S mod i into ZJ where i is ideal generated by XY and J is ideal generated by ZW. A be the element X plus Z and B be the element Y plus Z and Q be the ideal generated by A comma B. Then even star of Q is equal to zero but the ring is not a fractional. What goes wrong in this case is the depth of the ring is equal to one so it does not satisfy this assumption. And this characterization also partially answers honey case question in certain classes of rings. With this I would like to conclude my talk here are some of the references used. Thank you very much.