C Hermite's 'Sur l\'extension du théorème de M Sturm' (part 2)

0:00 So the first one of a real function, and then what he proves is that you can relate his results with the Cauchy index. because simply you can express so it's expressed at the at the top of the page 410 the following page you can obtain the quadratic form associated with the coefficients in described by the f prime and f1 and so the the sign the various sign of the coefficients will be given by the the variation of the the sign of this relationship this ratio and so this is connected to the to the cauchy index that you know i think and in the sixth section he gave a bit more example of that and uh well a new formulation of this of this thing but perhaps i will not comment that where you combine this index in the situation where you look inside the curve okay so this is essentially the the paper and now we want all to hear about the detail of The proofs in section 2 and section 3. I have a question about the solution that appears here. It appears. That would be a claim, I guess, but because it seems to appear also interested. So how can you see it? What is the priority? No, no. You can't. it's well yes and no i mean it's very difficult uh all these papers are related together and so for each notion you have to check really carefully what it had been read at the same because they were written to each other they were exchanging papers it depends on the place of publication so i've not looked for the vision to say i've looked for you know the admission form look for the Sturm function, I mean, you know, you have a full list of such notions, for instance, some of the determinants are taken from Borchardt, you know, you have a lot of

2:30 little things like that, and, well, perhaps we can discuss that when we discuss the source book, but the impression you have with the Hermit text is that it's not a closed text, if I may say so, it's really a completely open text with bits of other text coming in, you know, it's really this sort of picture. we need a carton. It's really a sort of germ of a ship of text together, if I may say so. It's really this image which I have in mind. So, you know, you are asking me here what is the thinness of the thing, I don't know. And all these texts, so you have Sylvester, you have Borchardt, you have Jacobi, you have Sturm and Cauchy, and you have Hermes. And they are all exchanging things all the time and not quoting very precisely what they are taking from each other. So this is something about the Bezutian or about precisely? Because also, for example, in Artin, it is the same for my specialization. But that's Artin. Yes, Artin, so it means it was really part of what people knew about it. But much later. but also sometimes they don't you know you have this expression with what happened also it's that sometimes you find the expression without any uh hint about the importance of it in the middle of a computation and then somebody of the the bond would jump on this part and isolate it so i mean it's also you know it's So, yeah, we have some, okay, sorry. Okay, I tried to explain the proofs, but without changing the proof, but with a question for me. So, yeah, because for me, coefficients a, b, l, and roots,

5:00 a little e, for me, so I take a polynomial of this kind, I have numbered the roots, OK? This is our point. Two and then three. It is at the beginning of section two, page 309. What you call F0, I call it F bar, complex conjugate. And now introduce a form, something which is called phi, And we choose, in fact, this thing. E is the square root of minus 1. So indexes are always g of k. P is a general polynomial, v over t, of degree n minus 1. So the coefficients of the polynomial P becomes the variable of the form 5. and I have I have taken money polynomial but it is not very important because all formulas are homogeneous so this work also for non money polynomial so in order that this thing as a meaning you have to the roots of F have to be distinct there are these things and they are also to be distinct from the root of f-bar. So no root of no alpha, they are all distinct.

7:30 The conjugate of the roots are not, no, it is a polynomial without a real root, particular. I have said that I is not an index. I is square root of minus 1. So this is some strange thing, because the value is complex. So it is not a quadratic form at all. But the theorem says that, in fact, the value is real. So it becomes a real quadratic form. a strange thing which takes real numbers to complex numbers. So as says Hermit, it is much better to consider the sesquilinear associated form, which is So proofs are more easier with this sesquilinear form, which becomes the quadratic form when And so Q is a polynomial with the R. But now, now these are complex numbers, so if you consider the sesquilinear, this is the sesquilinear form, really. F-frame is a derivative, yeah. So it's very strange. Theta and Theta bar? Yeah, in Hermit, this is Theta and this is Theta bar. Theta zero, sorry. page 400. We have to prove two things.

10:00 We have to prove that, first, this is an Hermitian form, not only but Hermitian. And so this becomes a quadratic real form. And the number of positive elements in this quadratric real form, which is the same thing of the number of positive elements when they compose the sesquiline transform, is the same as the number of roots whose imaginary part is positive. And this is an easy computation. It's not the way to place exactly an extremely, OK? So this symbol, P, and we can consider the P of P, which is P of P, P. And the two symbols, P, are different from each other. P, P bar, not P, P is this. three things. Yeah, yeah, this, yeah, there are three. Three. Phi, this. The phi of PQ is not Nermit? It's capital phi, it's about phi. Yeah. The next paragraph. There is only a phi in Nermit. Big phi. I know that you were on the right is the same time as the one you were. Ah, excuse me. Excuse me. It is important to have the square root of my instrument here. And how this works? We have to know the following thing, which is well known, P of t over R of t. You see a rational function with a numerator which is a degree less than the denominator. So

12:30 So there is a decomposition in elements simple. And what is a decomposition in elements simple? This is given by... it is very well known. It is easy, in fact. t minus alpha k. Yeah, hello. Yeah, t. In fact, this is, excuse me, this is this. So, Why this happens? Ah, voila. In fact, we know that this is a combination of some terms with the roots. We know that this is a combination. And in order to know what is this, you take the limit when t tends to alpha k, to the root. And what is the limit when you multiply by t minus, yeah, in order to know this coefficient. It is a well-known, what, first year school, yeah. Non, ce n'est même pas ça, c'est, non, non, c'est... Integration. It's the first course calculus. Décomposition of the elements simple, you take the limit, so the limit of Pt, it's P k bar, and the limit of F bar divided by T minus alpha k bar, because it has to be multiplied by T minus alpha k bar. So, before we take the limit, it's F prime. So, yeah. Yo, okay. This is the first calculus. Yeah.

15:00 But your notation after that becomes complicated. My notations are more complicated than the one of... You have it? They are exactly the same, right? It's an inside joke. Sorry. Okay. And now, you take this and you replace T by alpha G, and the miracle appears. Okay. Not very miraculous, but it is true. So, and so, you get, and here, exactly what he calls the norm. And on va sceller les dérivés. Okay. So, in this form, this, this, we have a square, the square of the module of this complex number, and here the square of the module of this complex number, it is clearly Hermitian. Because if you permit G and K, you get the same thing because you get alpha K bar minus alpha G, but I has to be replaced by minus I and so forth. This is Hermitian. Ah, excuse me. This is the sum of our gk. Okay, and now you can see this as a linear form.

17:30 Excuse me. Excuse me. I have to copy exactly what is urgent. Because it is impossible to make such a computation on the black bar for me. So this is L of, this is Lk of P, Lg of P, Lg bar of P. LK is the valuation in alpha-K bar. So, now, in order to find the signature of this Hermitian form, you can take these new variables, these new linear forms, which are independent, as new variables. just to do this one There we go. Okay. Well, Zg and Zk are the new variables. And Zk, so it is this.

20:00 OK, in order to find the signature of this emission form, you have to compute the delta M, or something of this kind, And so for Hermit, the solution is clear, because he knows how to invert this matrix. So in fact, I think that all people at this time know how to invert the matrix. alpha j are roots of f, beta k are roots of j. You take this for people, it is not a matrix, but it is just a linear system. And they know how to invert this linear system the solutions, and the solution is given by a nice formula, which is given by Hermit without comment, because it is evident at the, I think, at this time, and in order to to compute the inverse, it is not difficult. You have to use the same trick of the composition of the elements simple. It is the same trick. And you find the inverse of this one, too. Cacavote. Ah-ha. You have to multiply by that.

22:30 He inverses the system, saying that it is clear. I think that this is an exercise in Matzpé 50 years ago, but not when I . So, we have to assume that the roots of F and G are distinct, in the contrary, yeah. And there are also the root of F, sorry, all this, alpha G and Bittaker are parallel distinct, all distinct. And so, yeah, you have the formula. OK. OK, and so you arrive at the final form, the end of section 2, where I compute, because Because we have the big matrix, but we want to consider the ratio of two consecutive determinants. And for us, it is exactly the ratio of two consecutive determinants. It is exactly what appears here when you invert this matrix. The ratio of two consecutive determinants is this coefficient in the inverse of the matter. And he says this and saying it is sufficient to compute mu over mu prime when mu m over mu prime m when we assume in the second, the unknown are all zero except the last one, not the unknown, the second member of the system, linear system, is made by zero, zero, zero, zero, and non-zero system. So when it says when the white-handled is a function?

25:00 Yeah, it is the, this is the word for saying that he diagonalizes. So, he, here, he applies the Hermitian-Jacobi theorem, Jacobi theorem for Hermitian forms. And this theorem, he gives this theorem without proof, but surely it is clear for, since it is the same proof as the proof of Jacobi theorem for quadratic, real quadratic forms. No, no, no. Yes, rectangles are not the square term, but not the symmetric term, yes. This is exactly the terminology. This is the first page. Yeah? This is actually the determinant that you have just reminded me of the exercise of aggregation. So Cauchy has introduced the inversion of this linear system. One of our competitive examinations. It's an examination. One of them, our favorite. So what you call the... What equals P, the pi, what equals pi is, in fact, I think, f bar. For me, I take pi equal F and pi bar equal G. So when you put this computation of this

27:30 inverse inside. You find the sign, we find that the sign is given by the square. Here is M. Alors c'est alpha, pour moi c'est, si je me suis pas trompé, c'est f de alpha m bar sur f prime de alpha m. Okay, and in this form, you have a, this is a journal, and you have the, we have a decomposition of the quadratic form in sum of squares with signed coefficients, and the signs are exactly what he announced, which is the sign of the real, the imaginary part of the root. No, now for Section 3. On Section 3, he comes back to the initial definition, so I recall that I introduced This is a file P, file X0XN, which is a P.

30:00 OK, now you define Zj that are not the same thing that I have called the Lj previous section. And so if you consider the polar form, this is a... Okay, and the ZJ are linearly independent, linearly independent since phi is non-degenerated, so this can be taken as new variable, but the inversion of the matrix is complicated And it's preferable to follow Hermits. So we introduce, he introduced. Excuse me. So I call it, I give a name to the coefficient. In fact, it is P, or the distinct of K. And for the sequel, I give a name to the coefficient.

32:30 OK, and you see that GK at alpha K is F prime, and GK at alpha G for G different is K. is zero. OK. And now, you take what becomes this when I replace Z to mu by Z index mu, and he called it FK, who is a gothic. And for him, it is F of alpha k. But for the moment, it's not so clear, so I prefer to call it. And then G is defined by this one. And he says that an extremely simple computation shows that, and so I'll show you the extremely Fk, you have to rewrite. We need a third index. Ah, I have no, excuse me. We need to compute ZL, which is 1 over 2, 5 for the xl. And And you saw, you see from the expression, it is exactly alpha, alpha core, alpha G, L, over, the same thing, f bar, actually, f . So you have to put this thing here.

35:00 In fact, when the computation is made, it's an extremistic rule. It is right. So, you permit the two sum index, you permit the two sum, here it is sum over, over, over Yeah. What do you have to say? The F-cab Zahoya, the gothic F of K in Amrita. So middle of page 430. It is what he calls F of Alpha K, but it is not clear if it is F of Alpha K. It appears on me at the end of the computation. And so, you get CKL alpha G LU, you find G, you find JK of alpha G here, here, here, here, f bar and j k. But since they are all 0 except 1, you find And Hermit doesn't give this result. It would be better for us, but it doesn't give. So the f prime disappears. So and in this form, it is effectively a linear form that depends on alpha k, a reasonable meaning. So fk is, in fact, f of alpha k and x is 0, x minus 1.

37:30 But it is OK. And from this, from this, you have the new expression here, which is given by Harmit. So it doesn't give the intermediate computation. Here, you write this in this form, F . So this is a new expression of the same polar form, and there is a little job in order to make appear the position. In order to understand what makes Hermit, I think it is better to introduce the polar is now minus sum over J so f prime is exactly the same thing replacing x 0 x n minus 1 by x prime 0 x prime minutes okay and since since this come from f of it this comes from

40:00 f of z prim. So you have to say, this form is this polynomial when I replace the exponent indexes. And the same thing for here. And now you make the formal computation. You make the formal computation and you have to understand why it works. But I think it is possible. So I take this for this, and I transform it. I make Fz, Fz train, the sum. I replace this by the difference. This is good. I have to put z minus z prime somewhere. And now you see the composition of the elements simple. You get... Yeah? I'm a little confused because I'm going to follow the... Yeah. And I see the next question, except that instead of having f prime over f bar, he has f0, which means f bar over f prime.

42:30 Yeah, and perhaps I have to make... Yeah, you are right. It's the coefficient and not the inside of the form. As usual, I mean, you remember it. I've made a mistake. The form you want to compute is the sum of f over pi, f bar over pi. Ah, so I did not understand anything. Je ne vois pas comment tu peux, si tu n'as pas... Comment tu fais ? Quand tu fais le changement, tu trouves bien F' sur F' ? Ah, c'est moi qui me suis trompé when j'ai réécrit la forme. Ah, okay. Here, I have this one, this. If you write, if you can follow the text, in fact, you... The previous computation gives this. Yeah, it's there. And this is, so I understood a little, so you get this one. And, yes, that means it's free. And finally, this is the position of F and S bar that appear here.

45:00 And now we have to see what happens really. And what happens really is that this is obtained from this, replacing exponents by indexes. And this is true. Because this formal computation works. I think it is difficult to make a clear reduction of this point. Here, in Hermit, it is strange because he has a square, which has two big exponents, so he replaced by f z of f z prime without speaking of the polar form, and without giving giving the polar form in this form. It seems that this proof is correct, but some details have to be maybe added in order to make the proof clearer. And so he has a direct way of computation of the, an equivalent form of the quadratic forms of the beginning, which is given by the matrix obtained from this polynomial by putting coefficients inside the, inside the attacks here and there, which is automatically our mission. So this is a proof of Section 3, which is hidden in Section 3, because he gives only some little steps, in order to understand what he makes here.

47:30 It's because he didn't use a mixed notation. You clear? Especially when he defines the new variable z0, z n minus 1, he doesn't derive, make the partial derivative with respect to x0, x n minus 1, but with respect to coefficients, which have no number. So there is a number on one side and no number on the other side. We need to find the gothic F. So, what we can say is that computations, like Hermit, work, do work. So perhaps you have comments or questions, you can dream of it on all this. what surprised the reader the paper is absolutely impossible to understand in the first dance after when you take line after line and you see what what he's saying and Finally, computations are reasonable, but after a big effort, for example, the inversion of the matrix for me was very strange, but since it is a well-known exercise of aggregation. It's in Cauchy, but probably also in Jacobi, so it's a...

50:00 In France this is called the Cauchy matrix, and in Germany it's called the Jacobi matrix. There was an industry of determinants in the fifties, fifties, I mean, they know very well determinants, I think. Very, very well. In the introduction, he stresses the fact that the whole is based upon the inertial law. So he has a clear view of what happens in this. But I think it's a very striking thing that we see if your general theory is about because of, say, wanting to down to the world of the president's system. It's a way for, because it appears to be a very basic problem they want to do. Yeah, but for me, this computation he has made, apparently, in the case of Naga-Thori. So I think of other things. Theriot-Korapi-Form was really Gauss, somehow, for all these people. And so in a very arithmetical situation. And then they were developing it in all directions, all possible directions. It appears also in mechanics. and it's really a main topic there are some reductions to sum of square of course they were looking for for I mean they were looking for reduction theory So, in fact, it's interesting because reduction theory in an arithmetical case, of course, is much more difficult. And there are some comments because they first study quadratic forms with integral coefficients.

52:30 It's a difficult case, very difficult case. And then they began, especially Hermit, to study with real quadratic forms. And they said things like, oh, it's absolutely marvelous. You have these results where you can really decompose into squares. It's incredible, these results, and so on. And the first time I saw this, I was a bit surprised. But in fact, it's true that it's so much more simple, much simpler than in the reduction theory for the arithmetical case. So they were looking for invariants. We were looking for things which happened in the arithmetical case. So various invariants, sort of reduction to sort of normalized form, simple form. Equivalence and the substitution. Of course, here, real substitution and not a substitution with integral coefficient, that sort of stuff. And it's really in the 50s that Hermit did this work, general work, on the quadratic form, from 1847 to 1850-something, and I'm not alone, as I said. I look at Hoya if she has something to add. Are there any other questions, comments, generalities, surprises? Is this a question, are these papers of their meat? Are they more of a source of real anthropology than the paper of Stern? It's a very good question which I would like to have asked. I don't, I don't, I mean...

55:00 Tarski doesn't mention it. In fact, it was in the Cours d'Ingèbes Supérieure de Serret and it was in Néanbourg d'Ingèbes, There is a full section at the end of the 19th century, turning of the century, sir. But that wasn't real after it was geometry. No, no, no, not at all. In the best books of literature. Would you argument on the inversion of Koshima, which is well known? Yeah, I think. I think. No, but sources for whom? I mean, for you, for you, you have to tell us, because, to tell me, I mean, perhaps Henry and Solenio, that I have not the least idea. I mean, I just, in some of the books I read in real algebraic geometry, I saw Hermit mentioned the method of Hermit to compute, I mean, storm functions or to, I mean, and so on. I think it's, well, I've learned the Hermes method without any mention to the Sturm sequence. It's possible to just look at the quadratic form and not look on the Sturm sequences to calculate the signatures, the signature of the quadratic form. So you can study both things separately if it's to calculate the same thing at the end. And it was for me, I discovered that it was really strongly linked, both methods were strongly linked. For me it wasn't the case. Since in a real algebraic geometry you are considering zeros of polynomials, and if you have a polynomial in one variable, this is, Hermite is concerned with this question. So it's on the basis.

57:30 Because at that time, I don't know that several people were asking how is it possible to generalize from the oil into the regular office. These people were starting to . And then it appears that it was already known, for example, . So it's not an informal game, right? And it's . There is also another thing, that the computation with the storm sequence is very often much more difficult than the computation with the Hermit form. And it was known at this time, Celler mentions a very ugly polynomial with respect to Sturm and Weber. And Weber is explicit on the fact that Hermit is better for the computations. But as we say, I mean, it's 1900, so what happens next, you have to tell us. Dr. Jacobson's textbook said he follows Weber's exposition, which I never read, but he was never mentioned. In Weber there are a full section on this method, and a small discussion on the fact that Hermite is better for computations, for instance. I also speak with Thomas Ressio a week ago, and he told me that for him, well, for him, Hermit's method is not Hermit's method, it's uncle's method or something like that. he didn't know of what I was talking about. He said it's the French, the French name of this method.

1:00:00 And he told me, he... Well, he... ...Kain Limarque, which is the reference paper on this kind of things, he cites as Hermit. Hermit, yeah. ...Kain Limarque, they are not French. But he said he discovered it in the 80s when he was concerned in problem of specialization of polynomials with parameters and he wanted to know if there exists one, well, the existence of a real root for a polynomial and with storm methods it was a bit it was complicated to do specialization and it was better with what we call Hermes methods that's what it's all about is that what's new now is that we can use them to make a different proof even when there are gaps in the debris and the roof and the same. And also the cushion. Here, for example, this is very important that the roots are simple because we have a friend at the elevator. And all these methods are more in general, they can really be used to exclude the children in general and excluded from the social attack. Brebert spoke a bit about that, but I don't remember what he said. Oh, in quotes, there was some work, I mean, because that's the beginning of a lot of works also, which tried to develop some aspects and I don't know who looked at that besides.

1:02:30 I'm taking care of the degenerate pieces of this paper they call it. So he gives a rule. So he's generalizing the rules, which say that you just have to leave as a sign of the minors. But when some minors are heroes, then you have a gap of even length and you have a rule to save out of the national system. And this rule is not ready for general It was a special case of . Thank you. The question, but maybe it was said that, so that's a reference to theorem W. Cauchy. So what is this theorem that is mentioning? You can compute the number of roots of imaginary polynomials inside, for instance, inside a a circle with a variant of the residue formula, in fact, with Cauchy, it's a variant of that, which, so the index appears in this circumstance. But he got Bezoutian Cauchy?

1:05:00 No, I don't think so, at least. But he has Cauchy? But he has the Cauchy index. Yeah, but the contour has not to be algebraic or... No, no, no, of course not. No, it's completely general, the contour in Cauchy. Because with some algebraic contours that... ...like hyperbola, circles and other curves, this becomes entirely algebraic. The imaginary part, yes, it's the, yes, it's exactly, I think, on page 410. The integral input, what Hermit called the integral index. Integral index, yes. But, so it's a Sturm commentary on Cauchy's theorem. Not surprising, surprising. So that's one of the things which, one of the stakes of the paper too, to go on with the question on sources. I don't know for sources, but this paper is normally taken as a first step in algebraization of the whole issue of computing roots. Also with Sylvester's paper, of course. Before, what we've been called the finding of roots, That's the problem. For Sturm, I mean, it's funny because, of course, as I said, what

1:07:30 is used in terms of continuity in Sturm theorem is a whole theorem or intermediate value theorem. So we don't consider that really now as very algebraic, as very continuous, somehow. We know how to... An algebraic action. As we have seen in the Artin Schreier paper, for instance, if I remember well, you can think of that as something really algebraisable. But at the time, of course, it was seen as analysis, really. So the fact that you can get rid of that was commented by a lot of persons and also commented in history. I refer to our Bible for the history of the field. So it is my main surprise, I would say, because knowing Hermit is not somebody with concern with the type of algebraization which, for instance, Artin would be concerned with. Not at all. It's not at all a question of reductionism or a question of arithmetization or a question of getting rid of analysis. For him, it's much more a question of if you can put continuity where there is none, it's very nice. If you can put algebra where there was only continuity, it's perfect. I mean, you know, when you change the method, it's good, whatever the direction. So he has a vision of the mathematics as a sort of big field, including analysis, algebra and arithmetic, and excluding geometry, with a very strong opinion about that. And he wants to travel as much as he can between this field. And you see really this travel in his papers, because you have this argument about arithmetics here, and you have nice things about continuity as well, I mean, it's really...

1:10:00 Thank you. Thank you.