Investigation of Hessian Matrices
Turner Blick
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08/03/2021
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I am sorry that this video might seem rushed, there is just a lot of information! The relationship between the rank of Hessians and waring rank are investigated in this video. (Slide 5 is meant to say "...same total degree among its terms", not "...same total degree between its variables".
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- [00:00:00.750]Hello, my name is Turner Blick, and I'm here to share about
- [00:00:03.180]UCARE project titled Hessian matrices.
- [00:00:06.750]So my outline for this video, let me give you a little bit of introduction,
- [00:00:10.140]about what you'll need to know, then I'll tell you my research question,
- [00:00:13.110]and finally we'll walk through the results. So for the sake of the video,
- [00:00:16.980]I just don't have time to explain every single concept I'll be dealing with and
- [00:00:20.540]every buzzword I'll be using. So you'll need to know about number theory,
- [00:00:24.450]partial differentiation and linear algebra, uh,
- [00:00:28.410]as well as three more subjects, uh, lined outlined there.
- [00:00:32.820]But we'll talk about those briefly right now. So here,
- [00:00:36.120]just think the field is basically just the number of times need to add one to
- [00:00:39.960]itself in order to get zero. And if there isn't a finite number of times,
- [00:00:44.910]you can do that to get zero. And we just called the characteristic zero.
- [00:00:49.500]Uh, again, if you need to have more explanation,
- [00:00:51.840]you can read it and next we
- [00:00:56.160]have, uh, definitions about rank.
- [00:01:00.390]So homogeneous just means that each term in a polynomial is constant.
- [00:01:05.280]Uh, in terms of degree,
- [00:01:07.350]linear form just requires that degree is one in a homogeneous binomial
- [00:01:12.930]and warring rank is just the smallest, uh,
- [00:01:16.260]number of linear forms that you can sum together to get F
- [00:01:21.240]uh, when every linear form is raised to the same power deep,
- [00:01:25.380]which is the degree of the polynomial F lastly,
- [00:01:30.360]uh, contraction contraction is just like differentiation,
- [00:01:34.440]except you're not multiplying by the previous power,
- [00:01:37.410]which is very useful because then the derivative won't vanish.
- [00:01:41.940]If the previous power was the characteristic like saying this example,
- [00:01:46.240]the characteristic is three,
- [00:01:48.210]then three X squared would vanish because three would be equal to
- [00:01:52.950]zero because three is the characteristic,
- [00:01:55.320]but X squared would be preserved because you're not multiplying by three.
- [00:01:59.910]And this is a very important fact because a Haitian major
- [00:02:04.470]sees as, uh, detailed,
- [00:02:06.720]there are calculated by partial derivatives.
- [00:02:09.720]And if any of them were to vanish,
- [00:02:13.290]then the rank of the Haitian would be lower than probably what it should be.
- [00:02:17.280]And so contraction helps us define a
- [00:02:22.080]hessian major CS, which behave a little bit better.
- [00:02:26.460]And if a Haitian is defined to be with respect to contraction,
- [00:02:31.230]then we normally add a prime symbol here. So H prime, uh,
- [00:02:34.950]sub a and B of F would be, um, the hessian with respect to contraction of F
- [00:02:41.280]and then my research question. So roughly,
- [00:02:42.900]we're just trying to find a relationship between the column rank of a Haitian
- [00:02:46.410]matrix of a polynomial and the warring rank of a polynomial.
- [00:02:51.420]And we do this in fields with characteristic zero positive characteristic and
- [00:02:56.430]a special positive characteristic. So characteristic zero,
- [00:03:00.790]we'll start there first recall from the slide with characteristic
- [00:03:05.680]that if characteristic is zero,
- [00:03:07.840]that means that there's no finite number of times, you can have one,
- [00:03:10.510]two itself to get zero, uh,
- [00:03:12.880]proposition 1.1 lays out that, uh,
- [00:03:17.170]there are infinitely many polynomials in our ring, uh,
- [00:03:21.940]and each of these polynomials have a finite warring rank, uh,
- [00:03:25.480]as long as the field has a characteristic zero.
- [00:03:27.970]And we'll see how that's contrasted later, uh,
- [00:03:31.660]because there are an infinite amount I had to pick some of them obviously.
- [00:03:35.680]And so I took, uh,
- [00:03:37.000]31 forms of polynomials in the ring that I call
- [00:03:42.040]F, which is just the rationales and joined with variables,
- [00:03:46.030]excellent X for, uh,
- [00:03:48.850]these polynomials come from the paper written by a cycle
- [00:03:54.130]that I referenced at the very end. So you can see that, however,
- [00:03:58.430]conjecture 1.2 kind of lays out some observations and some facts that
- [00:04:03.160]I've, uh, investigated through the table that you've seen on the right.
- [00:04:08.320]And conjecture 1.3 extends that to more general cases,
- [00:04:12.610]but really there isn't much of a relationship to speak of in characteristic zero
- [00:04:17.440]between the rank of Hessians and the warrant rank.
- [00:04:21.910]So moving on to positive characteristic,
- [00:04:23.890]this is where it gets a little more interesting.
- [00:04:26.320]So remember this means that there is a finite number of times you can add one to
- [00:04:30.010]itself to get zero.
- [00:04:32.080]So this means that there are finite many polynomials in
- [00:04:36.580]rings with positive characteristic. Um,
- [00:04:41.020]specifically there are finally many polynomials with a finite warring, right?
- [00:04:45.580]And this is contrasted before because with characteristic zero,
- [00:04:49.420]there are infinite in many.
- [00:04:52.090]So in order to investigate the behavior here, I constructed, uh,
- [00:04:56.770]the two polynomial rings, uh,
- [00:05:00.250][inaudible] three each of join two variables,
- [00:05:04.240]X one through X four,
- [00:05:06.250]and know that the two tables show that they follow the consequences of
- [00:05:10.630]conjecture 1.3 and proposition 2.1,
- [00:05:14.380]as well as some remnants of conjecture 1.2.
- [00:05:17.080]It's just that I didn't feel like those needed to be restated again.
- [00:05:20.920]And a note, the remark that also explains some of the behavior as well.
- [00:05:26.230]There isn't, again,
- [00:05:28.150]there isn't a lot to speak of in terms of a relationship here
- [00:05:32.860]either,
- [00:05:33.430]except that there is something very interesting going on with the number
- [00:05:38.590]of these polynomials with the finite warring, right?
- [00:05:41.050]Because remember that's a finite number.
- [00:05:44.200]So theorem 2.2 just says that if the characteristic is prime
- [00:05:48.940]and the number of variables is finite, uh,
- [00:05:53.320]then all of the polynomials with a finite warring rank actually
- [00:05:58.220]have a warring rank of one, which is a very useful fact to note,
- [00:06:01.820]you saw that in the previous slide corollary 2.3
- [00:06:07.250]falls very closely in terms of logic to theorem 2.2.
- [00:06:10.640]And it just basically says that we can actually put a number to this finite
- [00:06:15.200]number of polynomials.
- [00:06:16.400]And it's just the characteristic raise to the power of the number of variables
- [00:06:20.890]minus one.
- [00:06:23.020]So that's the investigation of positive characteristic. And finally,
- [00:06:26.740]we get to the special positive characteristic,
- [00:06:28.810]which is actually where we start to see relationships that we can
- [00:06:33.490]equate two equations. So in this one,
- [00:06:36.910]we just call the field are, I mean, not the field [inaudible] arm,
- [00:06:41.320]and it's just, uh, S uh, joined with the variables X, Y, and Z,
- [00:06:46.960]where S is the quotient ring of the integers
- [00:06:51.970]adjoined with w and the quantity w squared plus w plus one,
- [00:06:57.260]in order to investigate in this, uh,
- [00:07:00.010]ring who wanted to find two different polynomials. Uh,
- [00:07:03.970]one is Pete in the deep,
- [00:07:05.710]and the other is Q to the deep where D is the, uh,
- [00:07:10.120]degree of the polynomial.
- [00:07:13.420]And each of them have a very clear warring rank,
- [00:07:16.870]which is why we choose to define them this way.
- [00:07:21.040]So the warring ranks of P to the DNQ,
- [00:07:23.800]to the D respectively are 12 and nine, which by definition makes sense.
- [00:07:28.840]But what about the relationships of the ranks of their Hessians
- [00:07:33.520]both with respect to differentiation and contraction and their warnings.
- [00:07:39.250]So investigating this showed that you can
- [00:07:43.990]actually make a pretty nice, uh,
- [00:07:46.360]formula depending on how well behaved, uh, the Hessians are.
- [00:07:51.910]So for the rank of the hessian of Peter,
- [00:07:54.850]the di with respect to differentiation is relatively well behaved.
- [00:07:59.380]It has a pretty nice equation there.
- [00:08:01.510]That's pretty easy to use a cue to the D has an admittedly
- [00:08:06.190]less beautiful and eloquent looking, uh,
- [00:08:11.680]equation, but it, it still works.
- [00:08:14.080]And it still works for any applicable values of a,
- [00:08:18.310]B and D as outlined in the conjecture and the ones that are
- [00:08:22.750]most well-behaved are the Hessians with respect to contraction,
- [00:08:26.320]which is just a very elegant, um, equation there,
- [00:08:31.270]and very easy to use.
- [00:08:32.740]And as the examples in both this slide and the previous slide show,
- [00:08:37.480]these equations can be used, uh,
- [00:08:40.480]to calculate ranks of very nontrivial,
- [00:08:45.790]uh, Hessians uh, relatively easily
- [00:08:50.410]and V and lastly, we will take W2 equal to,
- [00:08:55.260]and with the same notions of Peter, the deem Q to the D R an S.
- [00:09:00.030]So when you plug in w equals two into S S just turns out to
- [00:09:04.980]be the integers mod seven,
- [00:09:07.050]and that means that R is just the integers mod seven, and joined with X, Y,
- [00:09:11.700]and Z. And when w equals two, that also changes the notions of Peter,
- [00:09:15.900]the DNQ to the Dean,
- [00:09:17.430]but I investigated these relationships as well,
- [00:09:22.230]which yielded a very analogous, uh,
- [00:09:26.490]set of equations,
- [00:09:27.900]but counter-intuitively the ranks of
- [00:09:32.940]the Hessians with respect to differentiation are much more well behaved
- [00:09:38.340]than the ranks of the Hessians with respect to contraction.
- [00:09:42.180]These are the equations of the ranks of the
- [00:09:47.010]Hessians with respect to differentiation. However,
- [00:09:50.130]I couldn't find any sort of like meaningful, uh,
- [00:09:54.930]elegant relationship with the ranks of the Hessians with respect
- [00:09:59.700]to contraction.
- [00:10:02.220]And also just note when you're using these equations,
- [00:10:05.250]that a just means, uh, the values of little,
- [00:10:09.240]a little B and D that equal to two and four respectively,
- [00:10:13.740]and not a is satisfied if any of these aren't the
- [00:10:18.420]case and same thing with beat.
- [00:10:20.880]And so not a would be three to four,
- [00:10:25.740]uh, 2, 3, 4, et cetera,
- [00:10:28.020]like anything that doesn't satisfy those exact conditions and same thing B
- [00:10:33.300]uh, yeah. And so that's the relationship there. And lastly,
- [00:10:37.950]here's just the references to the, I guess,
- [00:10:42.570]not even just references,
- [00:10:43.710]just one reference to the paper I use to get some normal
- [00:10:48.570]forms of polynomials to investigate characteristic zero.
- [00:10:53.340]And I want to thank you for tuning in this video. I know it was a lot,
- [00:10:57.120]a lot information getting thrown at you very quickly.
- [00:10:59.280]I hope you paused throughout to read the slides and try to get
- [00:11:04.230]something out of this video. So thank you so much.
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