 IIT from Garagapo, and he will speak about he has several multiplicities of fabulous powers of an ideal. Am I, is it okay now? Okay, so it was reminding me my days in Kansas, Virginia, and Purdue had really nice and stimulating time with the extended Hunikeh-Hockster family, I mean, Alessandro, Luis, Ilya, Julio, Long. So, although, I mean, Craig was my advisor, although I never got a chance to interact with Mel directly, but like, interacted with many in Mel family, so two of them are sitting right in front of me right now, Long and Luis, and I just want to thank all of you for being excellent friends and colleagues. So, I'll talk about Hilbert Samuel multiplicities of Frobenius powers of an ideal. This is a joint work with Jacobi and Krithi. So, let me just set it up a bit. So, we have a D-dimensional Noetherian local ring and also characteristic P. We take Q to be P to the power E, and we I to be an M primary ideal, Q with Frobenius power of I, as many people have defined, just let me just for once define it. So, the Hilbert Kun's function is defined to be this. So, the length of the quotient by Frobenius powers, and this is, as we have already seen several times, this is well defined because this is M primary and Monsky proved in 83. This is basically, and this is a positive, and this number we call it Hilbert Kun's multiplicity. Some simple terminologies by HK of R, we denote by Hilbert Kun's of the maximal ideal for other M primary, it was proved by Hens in 2003. This is, so I mean this Frobenius power commutes with usual power, there was this connection with the usual Hilbert-Samuel coefficients or the usual multiplicity coefficients were established. E zero I is just a Hilbert-Samuel multiplicity. I is so basically, you define this as this length for N large enough, this is, one knows that this is this polynomial. Of course, all these are very well known, but I just wanted to. So, Trivedi in, I think before that I should mention the relation, so how Hilbert-Samuel multiplicity and Hilbert Kun's multiplicity are related. This is a theme, several people have explored and before saying this, I just want to mention one result by Watanabe and Roshida in 2001 proved that when Macaulay local and dimension is greater than equal to two, I stable ideal, I'll just define in a moment what this table means. Then for all k greater than equal to one, the Hilbert Kun's multiplicity is actually the 0th E zero k plus D minus one, Q's D minus E zero I minus H k, minus H k, this whole thing times k plus D minus two, D minus one, where, so for stable, let me just quickly define what is stable. So, J subset of, sub ideal of I called reduction of I, if for N large enough, J I power N is N plus one. So, we defined, so this is the reduction, J is called minimal reduction, J is minimal with respect to inclusion, all reductions. I is called stable, if for any minimal reduction, J I is I squared, it sort of starts from the first step itself. So, that result of Watanabe-Roshida established this interaction between Hilbert Kun's and Hilbert Samuel, the result I was trying to mention earlier, it was a result by Trivedi in 2017. So, R standard graded over a perfect field, I homogeneous M primary generated same then limit k going to infinity, E H k divided by D factorial, D minus one, minus one factorial. So, this is what Trivedi demonstrated in 2017, then in 2019, earlier to the following result, if you have taken weatherian local and like the very general setup and I M primary, then you can actually write down Hilbert Kun's multiplicity of power in terms of this Hilbert Samuel coefficients of Frobenius powers in general. So, you have E one, this times k plus, even for an abstract in this area like me, this look like a really beautiful result because this kind of connects Hilbert Kun's, 0th, first and also Frobenius power, usual power and the first power. So, this kind of gives a grand kind of brings everything in the same picture. In the same paper, Ilya posed couple of questions, same paper actually. So, he asked does limit q going to infinity, E I of this for all and also for large enough k is it true 0 to D? Now, of course, the second question to some extent, I use the affirmative answer to first question because it uses this limiting expression. One can also explore if this doesn't exist whether that automatically say this doesn't exist or not but like, so, but this was, this is how it was posed in his paper. So, that was the time I, so the simultaneously similar time I was working with Professor Verma and Krithi on another paper on related topic where we are exploring Hilbert Kun's multiplicities. This problem came to our attention and we kind of started talking about it. Show the following. So, if you take a Bushbomb ring, which is slightly more general than Cohen Macaulay ring, I mean, if you remember, Koei talked about Bushbomb ring in his talk of dimension D, force characteristic P, that's our setup and I an ideal generated by system of parameters. Then for all I limit Q going to infinity. I guess I can, it is the definition of reduction now. For all K greater than equal to one could also show that this Hilbert Kun's of I power K E0, choose D. Now, the proof of this result basically uses a result by Chu in 18, 1983, which sort of gives a formula for this higher Hilbert Samuel coefficient EIs of Frobenius powers in terms of length of local cohomology. So, that's, we use that heavily to prove this and we could also show again same setup. This is Cohen Macaulay, well, not same setup. This is now more restrictive Cohen Macaulay local dimension greater than equal to one, characteristic P, M primary ideal. Suppose the depth of this associated graded object of I power Q plus one, for all Q large enough, limit Q going to infinity, E to the power I, I to the power K over Q to the power D exists. In fact, we give it, you know, close the expression for that and that's keeping it for the sake of time. Now, this condition was something we sort of tried to remove but we really heavily needed this condition and this was required because we were using a paper by Tom Marlin, Huckabur from 1993 and where this condition was needed. So, we tried but we couldn't do anything about it. So, this, at least in these two cases, we could answer Smirnov's question affirmatively. So, next, just like this, you know, so I have five more minutes. Just like he asked these two questions, Ilya also made a conjecture, in the same paper, Smirnov made this conjecture. Cohen Macaulay local I in primary, then I is table equivalent to this limit of this first hill-backed salmon equal to this. So, this is the time when we saw this conjecture. We were working on the multiplicities in Stanley-Riesner setup and we had several examples in our hand for some other reason, handy. So, in fact, so essentially that time in a different work, what we were showing is a following, passing mention that that was the time we were working on this, proving this. So, if R is a d-dimensional phase ring of a simplicial complex, j equal to x1 power nu1 xR power nuR, where R number of variables and each mu i is strictly greater than zero. Then this length, so the basically so-called generalized Hilbert function, Hilbert-Gunz function is a polynomial q and k both. So, this is what we were the time working on, I mean. And so we sort of were dealing with lots of examples involving phase rings and that was the time when we kind of started working on this and we found the following counter example. You just take the phase ring of this thing. So, your ring will be R x4. So, x1, x4, that R is x1, x4, intersection x1, x2, intersection x1, x3, x3, x4. And take n to be extension of the, you know, the maximal ideal basically. Now, here one can show that both the first Hilbert-Samuel, the Hilbert-Samuel multiplicity and Hilbert-Gunz, both are four and also the limit, this is zero, one can show. So, the right hand side matches. This showing stable involves a little bit of Hilbert-Samuel function and reduction number. Okay, thank you very much for the talk. Kind of, is there some questions or remarks? Is the construction known in dimension one? Is it known in dimension one? I mean, I... Okay, the first theorem in here is for book borrowing. Yeah. Yeah, but of course, I think it leads to for Jean Lacroix and Maculee. That's what we were discussing yesterday. That's what we think so too. I mean, it just occurred to me, occurred to us yesterday. I mean, like, yeah, I mean, probably, yes. Yes, we need to sit down and make sure that all the... We hope it's true for any ring. Anything? Well, I mean, that's the question, you know? I mean, I don't know. I mean, I don't have any counter-example for that. But that's the question, yeah, for any ring. But at least it's true for Jean Lacroix and Maculee. Yeah, hopefully, that's what we're hoping. You too, you too. So, thank you again.