 Nam, the university's FPT, and his name is Quy, and he talks about normal and tight Hilbert polynomials. And before Quy starts talking, I would like to remind you that, actually Vietnam produces a lot of very, very good commutative algebraists. And one of maybe the most famous one is with us, and is one of the organizers of this beautiful conference. And is Chung, and I would like everybody to applaud Chung because yesterday was his 70th birthday. And so also, I would like personally, and I assume from all of us and from the whole commutative algebra community, thank Chung for the enormous work that he has done in Vietnam to produce such a wonderful generation, multi-generation of very strong commutative algebraists, and for the connection that he has done on all developing countries for commutative algebra. So thank you so much for all your work. And I also would like to say that there is a wonderful conference in his honor at the end of June in Vietnam. So if you have the possibility to go, it's a fantastic country to visit as well. So now I'd like to give his talk, and let's go. I would like to thank the organization to give me a chance to present my talk. The talk of, the talk of, the talk is Norma and Thay, he was pulling the man. Did he join work with Li Chuan Ma to paper with Li Chuan Ma and paper with Sabri Dupe and Suga Pema? Yeah, in the talk, I was the first. I was the first to introduce Norma and Thay, he was pulling the man. The second part is Thay books about, and it's Thay, he was pulling the man. The last talk is about Norma and Thay, he was a coefficient. In this talk, I.M. is a compilate local domain of, local ring of dimension D. And at I.M.R. is local code. I is all the way, I'm primary idea and Q is x1 to xD is parameter. E, I.M. here is multi-policy and the co-lang of I. I bar is integral closer and I star is Thay closer of I. We know that the length, the length function of R mode I2 and third one becomes a HINBIRD polynomial of degree D. We have a Z formula here, where E0I is multi-policy of R which is affected to I and EII is HINBIRD coefficient, coefficient of R which is affected to I. So now what Norma, Norma HINBIRD polynomial is I bar is contained in own integral element X in I. And if I bar is ZD, so we have a theorem of Z which says that we have a number C such that I to N bar contained in I to N minus C for own I, our own N. So the length function, the co-lang function of I to N plus one bar become a HINBIRD, become a polynomial of degree D in Z form. Z is the same HINBIRD polynomial with the third coefficient is the same with HINBIRD polynomial. So Z is called Norma HINBIRD polynomial of R which is affected to I and E0 bar is Norma multi-policy and EI bar to N of N is Norma HINBIRD coefficient. So now what is Thai HINBIRD polynomial? Thai HINBIRD of characteristic P, so of what Z in the slide I will skip this slide because it define Norma. So the Thai collode I star is by Z formula yet we know about that. And I is called Thai collode if I equal to I star. So we have a Z. I content it in I star and I star content it in I bar. So if I is Z ring, so we have a Z the co-lang of I to N plus one star become a polynomial of degree D. Z polynomial in here with the third coefficient is the same with HINBIRD polynomial. So Z equals the Thai HINBIRD polynomial and E0 star is Thai multi-policy or Thai multi-policy is the same with multi-policy for Z ring. I star here is HINBIRD coefficient. So if the ring is Z deal, we have a Z the first coefficient owner of Z is the same. The same but we have a Z the length the length of co-lang of I plus one bar is less than or equal to I to N plus one star and of course it's ordinary function. So Z it means the second the first coefficient of Z is I one bar of I always bigger than or equal to I one star of I and it is bigger than I one bar I one of I. Z will always have a Z. So for the normal idea we should have said about the work of C. O. Voto in Z-1, Z-D. Q is parameter idea. So suppose Q is integral clout. So Z ring is regular and the mill of M over Q is bigger than or equal to one. So Z is the next theorem which is easier is the work with Moralette and Milan Marzop Perceptor Chung in here he says that RM is zidiu equidimension local ring suppose the I one Q bar is equal to I one Q. So the ring is regular and Z-1. Z is the same here we talk. So now we talk more than Z we always have that the clout of Q is always bigger than the multiplicity of Q. Q is parameter idea. So Z will talk about the theorem of Hayasaka and Higiri in that we have that the clout of Q to n plus one is always bigger than Z-1. So if we have the equality for some n then the ring is clout. So apply Z inequality here we easy to say that the first number first coefficient of Z is non-positive Z is theorem of Mada Singh and Bama. If Q is parameter idea of R then the I one Q of Q is always less than or equal to zero. So what happen if we have an equality in here. So Z is van Gogh's vanishing conjecture for I one. Z conjecture was set by Reiji, Goto, Hong, Ozeke, Phuong and Van Gogh's Salute in 2010. The theorem is that if Q is parameter idea of an unbeat local ring. So if I one Q is zero then ring is cohenmokole. Ring is cohenmokole ZD. So what about the similar statement for integral collow or tag collow. So we have that the theorem of Goto, Hong and Mada is that if ring is ZD and equidimensional so we have that the I one bar of Q is always bigger than or equal to zero. So we have something in here we have that I one of Q is always bigger than zero less than or equal to zero. And if I one of Q is zero we have that the ring is cohenmokole. We need to admit in here. So now what is here Goto, Hong and Mada so the I one bar of Q be always bigger than or equal to zero. So what about ZD one the vanishing conjecture in here and I've got for tag collow of Q what happened, collow of Q is zero what happened. So now here we just note that for any for integral collow integral collow is not collical for parameter idea because for any primitive idea I which I Q is minimal reduction of I so therefore we have that the I one bar of I is the same with I one bar of Q so it's always bigger than zero in here. Okay so now the theorem of Vema and Mukuda in 2012 he says that if I ZD cohenmokole this cohenmokole of positive characteristic P then if I is epsilon epsilon then I I one star of Q is zero for some Q for some Q we not to not work Z theorem work only for cohenmokole condition. So the question is here of Percepts Hooniki for work with maybe we are Percepts Vema in here and here we public in paper with his student in ZD if I am a ZD and equidimensional locaring of positive characteristic so if I am a ZD if and only I star I one star of Q is zero for some Q ZD question. Okay so now we we talk about for Z1 in here Z question for cohenmokole so we have to consider Z question for more general case so the first case is for books power we talk Z member that the length, the co-length of Q is only bigger than or equal to the multiplicity of Q so defy the stucco and vulgar says that I is co-books power if and only Z different is a constant for every for all Q for all parameter idea Q okay so Z is some defy definition of books power it's the same almost the same of cohenmokole but here is regular sequence we don't have a M in here okay if it is books power so we have a Z different of the length the co-length and the multiplicity can be equidimensional by Z formula by Z formula in term of local cohenmokole okay so here we talk about the defy definition of bitter sense is that BR is intersection of Z model for every one to XD is a system of parameter so if ring is book power ring we have a Z if ring is book power ring we can see that we can see that in here so the M is content in this idea so if the ring is book power ring if and only if the BR is all the way bigger than or equal to the maximum idea M okay or what if BR is equal to M R the ring is cohenmokole so let's talk a bit about epsilon so we go to try to take P epsilon if every idea the fundamental idea is the type of ring for F rational ring a theorem of Hochtl and Hoenecky is that Z is normal and cohenmokole and a theorem of Karen Smith is that if rational if and only if ring is cohenmokole and the type of ring submodule of top local co-length is panis so here we we link two problem with Z by a conjecture of Watanabe and Josida R M is unmet local ring of dimension D of characteristic P so it choose that for every parameter idea the idea Q we have is that the multiplicity all the way bigger than the co-length of type A low of Q and if the multiplicity of Q equal to the arm mode Q star for some parameter idea Q then R is epsilon Z is a conjecture of Watanabe and Josida I've got fortunately Z conjecture is true Z conjecture is true Z by Koto and Naka Mura in 2001 so we you can see that in here the cohenmokole imply books about and a person implies cohenmokole so what we want to find in here is that Z is we want to set in here in here so it imply books about and Z1 ok so there are answer for Z problem by Bach, Ma and Sweeter and myself both in the public in the 2018 ok so now but ZD is a partial answer so here the R we define by parameter that idea parameter that idea is by Z1 Z formula parameter that idea if R is F regular if and only if the parameter that idea of R is equal to R R is F regular on function spectrum if and only the parameter that idea of R is an employee my idea ok so a question of Watanabe Z do you question of Watanabe is that suppose the content in the parameter that idea in here so it choose up the different of multiplicity and the column of title of Q is a constant for own parameter idea Q ok so fortunately in Z we confirm Z question of Watanabe ZD a theorem with Li-Chuan Ma public in 2022 if R is un-missed local ring of characteristic P R is type of small here we define the new F singularity is type of small if of the multiplicity and the column of Q star is independent of Q ZD we call Z is type of small if and only the parameter that idea a content in bigger than the maximum idea so for type book powering we have a that different of the multiplicity of the length of R mode Q star is by Z formula the same with book power we have Z1 of what here we we found Z diagram is a type book power type book power and coordinate and book power here so here is example of type book powering book powering with 0 type of low of 0 sub module of Z1 is 0 is type book power so for Z string is type book power Z string is type book power and Z is book power and here but here if R string is Z1 here we have that R is cohenmuculate and the 0 sub module type of low of 0 sub module of top-level concord is K so Z is type so we go to the ok now we go to the next part we study the Hingbert function of type book powering so Hingbert function of book powering we have a theorem of sensor we have Z formula in here so for the type book powering we have the same Hingbert function for Z1 so we have a theorem of dober we have dober and for type book power powering so we have that the of Q and plus 1 star by Z formula and Z formula we we can we have Z coefficient of Z is here so what about E1 E1 of Q star is 0 what happen so you can see in here we have that E1 of Q E1 star of Q is 0 so you have that for Z formula E1 of Q we have that so the I2M of R is 0 and the local quality Z is one so of course Z of Q maybe have some we send because we don't have a Z E1 of M of God so we have that if RM is type book power of F2 so suppose I1 star of Q is 0 for some parameter idea so the ring is here ok so we talk about the here we go to Thai for any ring we generalize work of Hayasaka and here if RM here we have that of Q N plus 1 star is let go down or equal to Z formula Z formula in here so we have that Z1 E1 star of Q E o low 8 bigger than or equal to 0 so for Z1 we have so Z1 a o low 8 bigger than 0 in here here for any parameter idea Q is minimal reduction of I so we have that E1 star of I o low 8 bigger than E1 star of Q and o low 8 bigger than or equal to 0 for any idea for any idea ok so now we talk a bit about the similar work of Moralette Chung and Villa Mahjong Villa Mahjong so the work of Moralette Chung and Villa Mahjong E1 star of Q equal to E1 star of Q E1 star of Q so ring E F Z formula so our Z if Rm is Z deal Equanimational of Positivity P and suppose E1 star of Q equal to E1 of Q so Z ring is Epracenal we have the same here and so with Z can extend for not only positive characteristic but also for mixed characteristic of what we I don't talk about Z because I'm not familiar with Z Z is for any balancing big so we can define the same the same way by Z1 so we have Z E1 B of Q all the way bigger than 0 mixed characteristic and of course here we get I star of Q is all the way bigger than 0 because ok so the last theorem I need something sorry the last theorem of Z talk is we answer we answer the question in here Z1 we have that E1 of Q is 0 and if you leave I regular me of M of Q let's say 0 Z so the only question is about Z1 so Z is a question if I Z deal and add 2 so suppose I 1 star of Q equal to 0 for some parameter IDQ so Z is Eprasina ok so thank you let's thank Kui and are there any question for Kui yes you were explaining this relationship between Eprasino and type Buxbaum and Buxbaum Eprasino yes that picture the diagram so how do FN important singularity fit in that picture FN important is a FN important but FN important is not implied of what here but of what FN important is very close with Eprasino I mean of what here we can for some important element and FN so of what but no direct relation means that a second follow up question would be would you try to relate the Milpoten property with something similar as the this type polynomial but you use Frobenius powers instead of type closure yes I don't understand what do you say I was wondering if you could try to mimic the construction of the type Hilbert polynomial by replacing type closure with the Frobenius closure and if that would tell you maybe anything about Milpoten you mean Frobenius Frobenius closure I'm not sure we can define Frobenius close I'm not sure because Frobenius close may be very different okay thank you any other questions okay then I would yes