H Whitney's 'Elementary structure of real algebraic varieties' (part 2)

0:00 We are ready. Do you want some water for you? Do you want to drink something? Oh, yes. I'm fine. Can I have my children? Thank you. Have a nice meal. So, I guess we're ready. Yeah. Yeah. So I think we should repeat briefly what we did last time. And the idea was to construct a stratification of a real variety, real or complex variety. And the technique used by Whitney is to use generating sets of ideals, of the vanishing ideals of varieties, and then look at the ranks of Jacobian matrices at the points of the variety. Then he splits up the variety successively by first taking a variety and looking at the set of points where the rank of this Tracovian matrix, the minors of the Tracovian matrix are maximal, or of the Tracovian matrix, I should say, is maximal. He looks at the rest where it's not maximum and that's a proper sub-variety and then repeats the process. So that's essentially what he does. Was this the first? First he doesn't use the word stratification at all. And what is understood now by stratification is something which is not only this partition

2:30 into these partial manifolds but also with some limit conditions that the closure of strata and this is work that was done later by Mignet himself in his papers in 1965. But apparently, this paper by Wittnes seems to be the first moment where there is this notion of having decomposition in pieces which are themselves manifolds with a decreasing dimension. It is decreasing in any event, depending on which construction you look at, decreasing dimension. Of various dimensions. Yes, yes, and, uh, yeah. Does this stratification have the properties of the later picture, really? No, no, no. You can't say anything about the closure. No, no, we are, you do not, uh... He doesn't discuss this question. No, no, but he doesn't discuss it. He doesn't have those properties yet. So it only has the properties that he claims it has. I cannot say for sure, to be honest. I did not look at this, if these conditions A and B are satisfied. In order to prove, I mean, when you do the conditions A and B, you have to add, you cannot just do it only with rank, you have to do something more. Yeah, but I'm not sure if this moor is not possibly available here. Personally, I'm not sure. I didn't check. so we showed you how the how the splitting process works in the complex case we essentially worked out his proof and then the real splitting process is done by using the complex complex vacation doing the splitting off of one piece and the complex case and then continuing so this is This is not a major step anymore. We did not really complete the proof of that, but there is no technical difficulty at all involved in that. So we can safely leave it here, I think, and just have indicated what the technique is that he uses without doing all the details, I think.

5:00 So maybe some remark that he had this invariant which is going down when he's making the construction, which was two to the sum of two to the power of the dimensions of the pieces which appear. So if these are the equidimensional pieces, I mean, these are the equidimensional pieces, i is equal to ri, then r1 is less than, and so on, less than alpha. And this invariant associated with v is the sum of powers of 2. This invariant here. And so in the real case, as you see page 9 of the . But it's in the splitting process real case. So it's in paragraph 11, it defines tau of V in the real case to be, in fact, tau of the complexification of V. So the invariant is really, the invariant is there only for the proof, to prove that the splitting process finishes, and finishes in a given number of steps. But he is associating the tau to the complexification of V. And as we've seen, there may be pieces, so there may be differences between the real dimension and the complex dimension. And so he deals with the complex dimension that appears in the composition of the complexification. So that's why he needs to do it already in a complex case. And in the real case, this means in the real case, the splitting can go faster than in the complex case. So then there's an example in which he shows that actually in general the number of spaces, the estimate for the number of steps he needs for the complete splitting of the variety is optimal.

7:30 So he says, in general, you can do it always with at most 2 to the n minus 1 steps if n is the dimension of the ambient space. And he shows an example in three-dimensional space which needs seven steps. That's in the complex case with the real case. Oh, no, that's real here. this is the the example is real and I mean the number of steps you need in the complex in the real case or the same if there is no no drop of dimension from the complex to the real and if he uses a construction which this does not occur then the number of steps is exactly the same the real complex places I can I can go essentially a picture here so he has one point let's see there's one point then there's a let's see and there is a curve like where's his curve it could be it could be something like this with a cusp it's a cusp yes and then dimensional. It's a cone. It's a cone. Okay. Something like this. No, that's not quite good enough. We need some more singularities. What does the surface look like? The surface example E. Yeah. What is the surface? Take this in your climate. Y to your climate. Square. Square. So what does this surface look like? So it's a cusp in the YZ plane. and then what example is this one? Example E the surface Y is something

10:00 if I want to draw it I should have a good picture of what it is of course but we could also use essentially we don't have to use this one but so it looks like yes you have a in the YZ plane and then it's the whole curve is singular in the surface. Can you draw that? No. Can you draw that? So you have discussed in the y-z plane, and then x cube is always positive, x cube equals the square of this quantity. It was the square of the square. So it means you have to take the cubic quarter. The square of the square. The square of the square. So it's only… So it ends this way. You know next constant

12:30 But to me it looks like, I think it's something like that. so maybe this is that it is because it is not a three a covering three times uh ramified along the casp the take the take a plane take a customer you you have this excuse equals a square so it means x is uh you have only one hook Okay. X3 is equal to a square. So this means that for every point of the plane, also the points where this is negative, there is almost one X. At least one X. Exactly. And in the ramification, the three points coincide. And the ramification is around the casp. The ramification is covering three times in the inside of the casp. The casp is the ramification. And outside the casp is only one. You have three sheets covering where... I don't see three. Where is the three? But in the real? You have three. In the real? There is only one sheet in the real. It's the same. So I didn't have the sign. So it's the same. So it's the same. It's the same. Why is it the same? Why is it the same? Why is it the complete sign in the real? Ah, yes. D'accord. The complete sign in the real. Now in the real case, you have only one, I'm not able to draw it, it's a surface where the whole curve, it's a surface which has a whole, where this curve is singular, and then on this singular curve there is a singular point. So I think that's basically what you need for the construction. You need that the singular locus is itself a singular curve.

15:00 So that you will create, in this process, when you take the singular curve, you will get again a void. So I'm not able to try this. Of course, it's a little bit difficult to draw this, but we have essentially four points. The point, this curve with the cusp, the cone, and then this curve here. And then if you look at the splitting, the splitting starts with the irreducible components of smallest dimension. That's this point. This here is number one in our splitting. Now that's gone, and then we have the union of these three varieties. And these are the irreducible components. and now the next step is on this curve here this has lowest dimension now on this curve everything except except the singularity so this would be number two and then we what's left after moving this we have this point and these two pieces here and now number three is this point here and now we have these two surfaces and then the next step would be to remove the non-singular points of the surfaces so it's the cone without the or without this this vertex so this would be and the surfaces and here the two-dimensional particles whatever it looks like so this would be number four and then we what's yeah the lowest dimensional component of the rest is this here this point which is goes as number five and then we are left with this singular curve which being split into two parts first six yes the curve the one-dimensional part of the curve so this would be number six without the cost number four is here

17:30 this the two-dimensional sheets here and the two-dimensional and the the the cone without the words there's a yeah if there is a line if there is a line then this line number four and then number seven the last point yeah number seven And then you'll be at this point. So this is the splitting process. And if you want to do this in higher dimensions, you take this, if you want to do it in four dimensions, you take this here, put it into a three-dimensional subspace and add something four-dimensional, or three-dimensional in four-space and then you can essentially repeat such a process. Why not take a square and a thesis to improve the equation to a square? I don't know exactly, but if you would take the same equations, but it certainly should not be too difficult for us to produce something three-dimensional and four-dimensional space with enough singularities so this this means that his splitting process in general needs two to the n minus one steps so I guess this is the splitting process now the now the fact that uh he has two more theorems about connected number of connected components he doesn't give any estimate he just says connect there are only finitely many connected components for real varieties and for complements of real varieties in other varieties so if you have a sub variety the complement has finally components and so these are the two things that we still need to discuss

20:00 and my preference is to look only at the first one of these two results the the second proof is in in spirit it's similar but technically more complicated and completely unnecessarily so yes yes because if you take a complement of a if you have a real variety and the sub variety then the complement of the sub variety is itself a variety you just move the sub variety to infinity and in a very simple way and so this complement of a variety is itself a variety so the he could have done that. I don't know why he didn't. No idea. Instead, he has an extremely complicated argument, and really, there's some topological details that I do not really know exactly how he wants to do it. I can work out my own proof. I can do that, but I'm not quite sure if it's exactly the same proof he was thinking of. It would be very interesting, actually, I think, to understand why you didn't think of such a similar argument. Doors and slang. I can't think. No, stop. So the simple argument would just be to take the sub-variety. You take the sum of squares, which is doing, in fact. And then so it means that the sub-variety is defined And now you just add a new variable and invert this equation, and this, so you write some new variable, say t times this sum of square equals 1, and it means that you add one algebraic equation, and it means that I know it's big. So the situation is, his sub-variety is defined by polynomials H1 prime to H gamma prime, and he actually looks at the sum of squares, he calls this H, the sum of squares of these, here, he uses this sum of squares, So he'll introduce an additional variable T and moves it to what he finds this here in addition to the equations he has in any event for V, and then he has a variety which is homeomorphic, certainly homeomorphic to the complement of the other, of the sub-variety.

22:30 and for such a variety he has already finiteness of the number of components and so all the ingredients are there really I have no idea no idea whatsoever maybe he just didn't think of it So essentially what's left to do now is look at the proof of theorem 3. Who else shall do that? Shall I do it? Okay, so the proof of theorem three is by contradiction. He suppose that the theorem is false and he will consider the smallest variety with an infinite number of topological components. Smallest in what sense? It was smallest in the sense of inclusion. So because otherwise, as I said, if there was no such smallest variety for inclusion, then it could make proper sub-varieties forever, and then it would contradict the Hilbert-Basys theorem. And now he's going to prove a contradiction by supposing that he's an infinite number of connected components and he considers the splitting. So again, he's using the same technique of splitting. And then he has decomposed this set into a variety V1 and a manifold, an algebraic variety V1, and manifold M1, but then of course M1 is strictly contained in V, that was what you were doing in theorem 1, we had a proper sub-variety.

25:00 And hence, it must be the M1 that has an infinite number of connected components. And all of them are of the same dimension. And then he first showed that this dimension must be strictly positive, because if it was equal to zero. As he said before, it should be a fact. He mentioned that zero-dimensional complex irreducible variety consists of a single point. So this, if it was the case that are equal to zero, then the connected component would be single points, and so... So using the fact that he knows that, ah, sorry, I'm lost now. The argument essentially is that then these points you irreducible components infinitely many irreducible components of your variety and there cannot be an algebraic variety with infinitely many irreducible components wait I don't see now why he's using lemma 8 and lemma 2 and lemma 3 What does Lama 8 say? Lama 8 says that the rank in the complex and real case is the same. Sweet soup.

27:30 So, you have points with rank, real rank n, and they also have complex rank n then. So if it was of dimension zero, it would lead immediately to contradiction, because the number of irreducible components of a complex algebraic variety of dimension zero is finite. Okay, now if, since now we know that R is strictly positive, then we know there are not single points, and since there are not single points, it's possible to choose a point which are not equal, which is not equidistant to all the points of, all the points of M'1. You have V1, you have N1, and on V1, you have only finitely many components, that's by induction. And in N1, you may have many, something like this, at least one-dimensional. And so, there's one of them, he chooses the point which is on none of them, and which is not equidistant to all the points of one of them. Since it's one-dimensional, you can always find a point which is not on this component and is not equidistant to all of these points. So take the point Q0 here, something like this, not equidistant to all points of that. And you add, so you add the equations which corresponds to, let's say, the equation of the sphere of... But I think it's important to reduce, so M1, it is not a clause, right? ensuring that you have a nearest point to this point, you need to have M1 plus, you have to replace this by M1 prime, that's why I think. No, it takes that close to political components. I think there is some detail here that the other points which are not part of the closed

30:00 political component are taken care of by... I mean, it's all inside V. yes yeah but being connected components or not our compact connected components are closed I think are they not know these connected components of m1 are not closed. Well, they are closed inside M1, but they are not closed. Yeah, but these are connected components of V, of the variety V. Yes, they are. We're talking about all the connected components of the variety V. Some of them are in V1, and some of them are not in V1, are disjoint from V1 because V1 can have only finitely many connected components. So there are infinitely many which are not connected to V1. And they must be closed inside V. And since V consists of V1 and M1, if they don't have anything in V1, then they belong to M1. Then they're inside M1. So you consider the splitting V into V1 and V1 as a theorem 1. Because it's splitting V1 and V1 is a variety, so that's closed. And the complement might be possibly not closed. Yes, it's not closed, but the connected components of V that are connected components, is if you look at any connected component of V, then it must be closed inside V. So it's closed in the ambient space, because V is closed. Now, if there is a connected component, which is contained in M1, then it's also closed. No, in the ambient space, in everything, here and everything we are looking at

32:30 now if you look at this here be what we want to do that yet the one would be the cusp and the rest here but this is the rest is not a connected component of the variety we're talking about connected components of the variety V yes V V is the original variety yes Since P1 has finally many connected components, it has finally finally many components on P3. We have also many ones which are three numbers, so this provides a three number of actually of connected components of M1 and which are called. Yeah. And hence, M1 has a good number which are closed cells. This is rather quick, but not all of them are closed. M1 has a good number of closed cells. Yeah. Those which have no connection with V1 must be closed. And there are infinitely many, because V1 has only finite many components. I think that's been clarified. right there.

35:00 So the next step is that he introduces this function g, which essentially gives the distance from this point q0. And now he looks at all these connected components and looks at the closest point on the connected component. Since these are manifolds, he can detect the closest point, looking at the Jacobian, or minus of the Jacobian, here. This is the closest point, this is the closest point, and so on, for each one of them. And now he constructs a sub-variety, a proper sub-variety of V, using the Jacobians, using the defining functions of our variety V, plus minors of the Jacobian, which single out essentially these closest points. And then what he gets is this proper sub-variety which contains here closest points on all of these sub-varieties here, plus the original variety V1 is also there, or not the original one, the sub-variety that he got by the splitting. But this must be a proper sub-variety because Q0 is not equidistant to all the points of one of these connected components. So singling out the closest point, or the closest points, there may be several of course, he loses some points from this connected component. And so his new sub-variety is a proper sub-variety of V. And therefore, because V was minimal with the property of having infinitely many common components, this must have finitely many components. The point V was chosen, so it was not equidistant from all points of one of those. Yeah, that's sufficient. But then it might pick up all the, oh, it just misses at least one point. Yeah, it's important to show that you get a proper subvariety. only important issue and uh and but you have infinitely many components because you get all these closest points on all these different different uh connected components so there

37:30 are infinitely many components and this contradicts the assumption about minimality And then the proof-of-theorem before, as we say, this is quite complicated, so we don't know. We decided not to present this here, because, I mean, the techniques are not much different from what we talked about now, a somewhat more complicated version of this. said there are some topological arguments which I really don't know what he actually exactly means and this may depend on on the fact that I do not know exactly what what the the notion about the the general notion about topologies of the topology of real varieties was at that time. He constructs functions that are locally constant and concludes that they must be constant. And of course, depending on your topological space, for some spaces this is true, for some it isn't. And he uses the fact that he can use such a, do this argument for subsets of varieties. I don't know exactly how he wants to do it. He's a little bit vague about this. So, I think that's been the paper. So, maybe you say a word about... About Pocahontas? Because we were talking of surprises, and... Yes, I... There were surprises for me. I provided a surprise to Niels. Okay. Yeah, the first thing that surprised me about this paper was that nothing like this finiteness of components or so had been done before.

40:00 And also the techniques he uses, because nowadays we would just say it's a semi-algebraic set he talks about, so the number of components is finite. And if you look back in history, of course, there were times when this was not known or things were not being identified as being semi-algebraic, that fifty years ago this was really new. And now Marie-Francoise showed me a dissertation which had been written in 1954, so predating this year, about semi-agric sets over arbitrary real closed fields and it deals with triangulations of semi-algebraic sets shows that semi-algebraic sets can be triangulated and of course this also includes all questions about finitely many components it's clear from this that there are finitely many components so at least as far as the question about the of the number of components is concerned this is after reading looking through Rakhage's dissertation, this is really something, this dissertation was never published and after the We received a mail from Royel telling us that he has a colleague named Rakage who studied semi-algebraic sense of this dissertation. And so now we included it as a reference in the English edition. And, but apparently, so he was working with Abisht, so Abisht is, of course, I mean, he's involved in a lot of interesting stuff in the rural world county and some resultants.

42:30 And apparently, from what you say, I thought it was under the direction of Habicht, but at least formally... Yeah, formally, this may have to do with hierarchies in German universities, because this was written in Heidelberg, written, submitted in Heidelberg, and it says, Referent Herr Professor Dr. F.K. Schmidt, and Co-Referent Herr Dr. W. Habicht. Maybe Habicht was a private docent at that time, and Schmitt ranked higher in the hierarchy, so he had to be named the referent. But probably, quite likely, Habicht was the true advisor. This is not unusual in German universities, even to this day. That was, pardon me, the date, 1954. And in that thesis did he prove the finiteness theorem for open semi-Eltraic sets? He proves that semi-Eltraic sets can be triangulated. And does that imply the finiteness theorem? No, but finiteness theorem, I think cheap means the fact that if you have an open semi-algebraic set, it can be described as a finite union of basic opens, which is the finiteness theorem in this. Right? That's what you were talking about? I think this should follow from this here. it comes from a good triangulation I'm not quite sure about that point but I think it should this is once again a source might have been a source it's even worth in the situation for Krivine where the source was a source that appears as being a source was not a source during several dozens of years. In this case, in a way, it's not a source at all. Yeah, I was also surprised because the triangulation, when I learned about triangulations, that was 25 years ago in Regensburg. And Delfs and Knebus did

45:00 triangulations then and showed that they have simultaneous triangulations and all kinds of things. They were certainly not aware of Dracargas. How is there? There's no publication of published. No, even if it's not been published, of course, you know a lot of things that have not been published. But apparently, Knebosch, definitely Delves and Knebosch didn't know about this here. It's really after our book was published. And well distributed, and it happened. So I met him twice, I guess, and I asked him if he went on this topic, and he went to numerical analysis, a kind of . Yes, it's nice too. It's important to mention that Whitney is saying that he doesn't enter, at the end of of his introduction, we do not enter into the more difficult and important question about local structure. Possibility of triangulation of V is a theorem in this direction. And then he's quoting this famous work of van der Waarden, which needs some . I want to answer to what you said about including these references to a source book. So I would like to say something about the kind of thing we, of course, we'll discuss much more tomorrow, but the kind of thing we had in mind when we, about this source book, that there may be two versions, I mean, one extended version of the source book, which is not even a book. It's kind of a website or a DVD or something where all the texts you would like to see are present and then the source book itself can be a book which is only part of the information so now we technically it's possible and then some other maybe approach also to what could be a source compared to existing source books i don't know for logic because when it used to be a book in a way you need to make choices but here you can be much more exhausted also so that's part

47:30 There was a paper in the 70s by Ephraimson and Dubois who got Ephraimson, he's proved the finiteness of the number of components of a semi-algiberic set by some cylindrical And I forgot if he claimed to have been dispersed or if he knew earlier or more, but I guess, didn't Boya Shavitz also prove finiteness of confluence for semi-analytic sets? I think it's improved again and again. It is not true. Take the joint disk, as many as you want. Locally. Locally, yes, of course. I suggest. so maybe with the right moment to say a few words about this paper on some semi-analytic which we were very unlucky with because two people were in charge of freedom here so one was pavlutski who was planning to come with his whole family and at the last moment There was a physics defense in Krakow that could not be moved, so he decided not to go. And then the second one was Norbert, Chapinchev, who couldn't make his flight. Well, not he couldn't make his flight. I mean, the flight was cancelled on Monday. So, of course, it's not possible to read it, especially it's quite a long paper, but maybe you should want to see a few words, but... Well... It's not only a source text, because it was never published. Because it was never published. Although it was never really published, but it's definitely a source of all the study of Samia Chabak, both Samia Chabak and Samia Chabak,

50:00 It's a reference which is unavoidable in the subject. So now, well, it's centered on semi-analytic geometry, but I don't remember that. I'm not sure that the terminology semi-algebraic appears in all this notes. I need to guess here. So then what does he say? I don't know how he says. He says elementarming. I have a note. Pardon me? No, the name semi-algebraic doesn't appear. His sets are called Elementarmengen, and the maps are Elementarabbildungen. The word semi-algebraic is replaced by Elementar, his dissertation. He sees connections with Klein's Erlanger Program and draws the name from there. But I never read Klein's Erlanger program, so I don't know exactly whether this is an analogy or more than that. So, for example, page 120, there is the terminology of semi-algebraic sets. I don't know where the definition is, but there is a theorem 1 saying every semi-algebraic set has only a finite number of connected components, each of them being semi-algebraic. Where is that? I think one hundred and twenty, where he is at Beech. There's a demonstration of the prophecy. There's a one at Tom in 1932. Sumi-a-Jibraic was used by Tom already. It was by Tom. Where does he say?

52:30 And then he's quoting the theorem by Seidenberg, interestingly, second theorem 2, on the same page, which said that the projection of a set of a semi-ergebraic set containing m cross n is semi-ergebraic. But Tarski disappeared. Oh, by the way, also there's a connection with Brackhage and Tarski. Brackhage has an appendix in this dissertation, which gives a, I didn't read this, but he claims he gives a proof of Tarski theorem. simplified proof which is based on an alternative discussion of Harbicht by Harbicht of Sturm sequences. It's simpler than what Tarski does. He gives a simpler proof of Tarski's theorem, he claims. As I said, I didn't read it. Yes. Well, I can guess. Of course, I can't read but probably I mean the so what so because it's something we have we'll mention in any case before the end of meeting this is what are called sub resultant polynomials that have been studied by a dish which are giving which which are giving a way to compute the number of real roots without all this branching that appears in Strom theorem because of the denominators that appear. So this method is, in fact, very close than what we do in cylindrical algebraic decomposition, which is to use these subresultants of . So it means that maybe it was already doing something similar to Colleen's method. It's just a very few pages. Pages 46 to 52.

55:00 I like the singer, Michael Jackson, he held the baby over the... Yeah. That's the only one, yeah. Someone takes a picture and you do it too. Okay. The question comes out of this paper. Should I consider it a source paper? It seems to me that there are two notions of source. One is in terms of the intellectual ancestry of current ideas, and the other is a source for current workers to look at how things have been done that might be useful. In other words, the fact that it was never, in other words, it had no color doesn't mean that it might not be worthwhile a particular paper to study. No, it's certainly interesting to study. Especially since, what better occasion to republish it than here? Yeah. Who else is going to republish it? But as I said, nowadays to publish may mean just to put it on a website. By the way, in the way I say which I also find on page 76, it says the family of connected components of the semi-analytic set is locally defined. Where is it? So it means that in this approach, the semi-algebraic case comes as kind of a byproduct of the semi-analytic study. And it's not really a particular case. What does it say, exactly? The theorem 4 is every semi-alibre set is a 25 set.

57:30 with me as a proof that every algebraic real set has a condition with the property of a real large incidence, results not get published. The other thing, yes, the first part of the center is this paper, and the first part of the . No, but there he . And it's well known also that Tom and Voyachevich were in contact, close contact. And even with it. Is there more to be said about, say Whitney and connections? I think that, well, concerning the result was more or less known It doesn't change, it doesn't change to be original result, to say that proofs are commonly very difficult to work in, so that we can see the originality of this result of a finite number of, but it's a part of the decomposition into, uh, into, uh, uh, so many, uh, pieces

1:00:00 which are, you know, this is quite the attribute of . But, and also concerning the, concerning the methods which are employed, I thought of, but also very influential. It's rather strange the method which is used to prove the finiteness of the number of the logical components, connected components. It's very, very strange. And one can say this is an example of estimating the number of connected components by two point meters. Thank you. Except that it proves it's not infinite. Yeah. I agree, I don't know if there was much influence on the development and the development of the I'm not sure, except it was certainly much less grandchildren than the north side, I don't know what I'm saying, right? Can you explain more one more support than two? Because for me, it sounds, I mean, it's like you're saying the 15-metre community, I love it, so probably now it's 15-metre city. For me, it looked like a typical proof in Algevois in the 13th, so I would ask a question like what were the contact with them or the topologies which I forgot to be written in I guess so, in terms of American magic, it's about...

1:02:30 Alexander. It's to be... Whitney. Yeah, exactly. Because it really looks like something that we have to speak about. But is it that... So, is it surprising for you, this aspect of surprising also, or other aspects? I guess it's like the better goal that you would say in the book of people have more. Well, this idea of considering the smallest It's not the topological components, it's, I don't know, it's a surprise, it's strange. Well, because maybe we are used to a A group of topological components would be, I think, for instance, the most natural, from some point of view, the most natural thing to do with, to proceed by projection If you have some in Rn, you can have a projection, say for instance, a projection of Rn-1 by choosing a good direction of projection and then considering the complement of the discriminant. some sort of thing that he was on that tool and maybe more answers in the spirit of the study of the next one that we are more interested in. But this part of considering the smallest variety with an infinite number of logical components, it's very strange. I mean, if he wants to use induction, I think this is one way to do it.

1:05:00 If he uses projection, of course, then he always has to deal with the question that projections of real varieties do not have to be real varieties. So he would have to use other techniques, too. Yeah. And, uh... Sure, but the... I think it's from the point of view of... from my point of view. Yeah, sure, of course. I wouldn't do it this way either, but I really, this, of course, if we want to understand it historically, we have to take into account what was the usual techniques at that time. I don't know. the connection with contingency plus the reference from the vector graph. But, for example, he's quoting also the Olenic paper and the Olenic paper is estimating directly the number of connected components by computing some number of critical points. So it's a direct proof which is not involving any reduction at absurdo I keep the bound on the number of connected components. And the Holenic papers? Yes, well apparently yes, I mean he's quoting a title in English, I think it was, well it has to be checked. And regarding the research I was very surprised to find this reference of the HHS concerning which at this time of development of ethics, as far as I know, they don't know.

1:07:30 And to find it here was, for me, a complete surprise. Because even in the tutorial of lectures, this book was completely obsolete very soon. And to find it there is quite surprising for me. He has notes, and then he has supplementary notes, seven supplementary notes. He says, let R be an m-dimensional region defined as the set of all points, the set of all points, x0 to xm-1, satisfying a finite system of inequalities, pi greater than or equal to zero, where the pi's are polynomials can be at most repeat. So it's a basic closed semi-algebraic set. That f, the irrational function, the denominator does not vanish on r. It's weakened. It's a basic closed semi-algebraic set. Then there is a positive integer q dependent exclusively on m, n, and p, such that it's set s of all function values of f. On r, it's an image of this rational function on that closed semi-algebraic set, is the sum of the most q closed intervals. and so the image will have a finite number of pieces right and so the inverse image does that mean the inverse image has finite dimension no but if you use enough rational functions to embed it into a finite dimensional space and of course doing it into just the line Yes, if you use several of them, then you can embed your set into it. I don't think that the citation means that it was translated at the time. I think that's because it wasn't translated at the time, and he's not given another place where the translation appears.

1:10:00 It may just be told about it, but it's just someone's translation of the reference. You can also think about the wrong viewer, because we don't know anything about that, but the wrong viewer has pointed out, pointed out that some work has been done, and then I don't know about that, I mean, probably it's not easy to know, we can think about this that came to the river in the river, so... ...at the beginning. In the first... Yes, that's the only place. No, no, there's no other mention of anything. anything. Had Whitney done anything in real Ashberg geometry before this? Maybe he was just a newcomer. Well, we've seen yesterday in the references of Nash's paper, reference to witness work on differential manifolds. That was generally approximating differential manifolds with analytic manifolds. But that was certainly not algebraic geometry. So he may have just been a newcomer and didn't know the current techniques and literature very And apparently, he didn't want to use much algebra, because he's involving no algebraic theory. But what he's using is this notoriety again.

1:12:30 Thank you. Shall I raise it again? Wait a second, this is here. nobody will know about this specific interest now so so I will first explain you quickly the structure of the paper this paper and then at one point in the second and the third part of the paper there is a note by Hermit saying that easy computation shows that, and this very easy computation took us some hours, defeated MAPL for a while and so on. So Henri will speak more carefully and at length about this easy computation in the middle of the paper. Okay, so what you have probably in mind, or hopefully, concerning the first paper, Remarques sur le théorème de Monsieur Sturm, Remarques some that Solène explained at the beginning of the week, is a property of Hermit's texts. And the property is that the main result is normally not very well proved. And there are a lot of mention of other results in various orders, which make the reading quite difficult at first sight. I mean, you have really to understand how the paper is structured before you can really enter in it. It's a general property, I think, of Hermit's text. And this new one, the second one, is exactly of the same type. That is, you have a lot of allusions to various other papers.

1:15:00 Sometimes very important to understand what is going on. And as I just explained, the details of the proof are not really given, or are given in such a way that you are not sure you know all the things you need to know to understand the paper. So here, for instance, if you look at the beginning of the paper, so the introduction, so the paper is a letter to Borchardt, which was written in 1854 and published only two years later. And so Hermits announced that he will prove by the same principle he has proved the Sturm theorem in the paper Solen described. He will prove this time the Cauchy theorem, that is, evaluate the number of the solution of a complex equation inside a curve, I mean, sort of a closed curve. uh and again uh the proof will uh will be uh depending on the theory of quadratic forms without continuity so at that point as you remember probably from solen's talk uh the the elimination of the continuity meant essentially the elimination of the the theorem of intermediate value, or a whole theorem in Sturm's demonstration. You remember that, or? Not really, partly. Okay. Okay. And then he said that he mentioned a very important memoir of Boshart, the reduction of quadratic form to a sum of square, which has been the subject of your memoir on the equation of secular inequalities is fundamental in my research. So what I wanted to tell you is two things. First of all, what is very briefly in this memoir of Borchardt, because that was also a requirement, and then we will go on to quickly for the description. So in the memoir of Borchardt of 1847, so you have the secular equations, so it was introduced in the 18th century in connection to the movement of planets. But essentially, as you see, it's simply you are looking for the elimination of the x in such equations, so linear equations, and this means that you are looking for the characteristic polynomial of a certain matrix, and because of the origin of the problem,

1:17:30 the matrix was in fact symmetric. So the proof, what has to be proved is that in such a case, the characteristic polynomial, of course, the vocabulary of characteristic polynomial is not at all, the terminology is not there at all. the equation when you have eliminated all the X in this equation. So what you have to prove is that this polynomial has only real roots, okay? And there have been a lot of different proofs of its results in the 18th and 19th century, but the point of the proof of Borchardt is to connect that with Sturm theorem. That was really his point. So what he did, I mean, I just summarized very briefly, but what he said is that, so So, what you need to know to have all the real roots is to evaluate the leading coefficient of all the Sturm function, and so, in fact, you can write this leading coefficient, and what appears up to a certain coefficient lambda, which are all positives, are the determinant built from the Newton's sums of the roots of the equation. Okay, so the S-car are the sum of the roots, I mean the Newton's sums of the roots with the right powers, and you evaluate all the determinants. And so the fact that this equation has real roots only is equivalent to the proof that all the Pi, all these determinants, are positive. And now the proof of Borchardt, which is incredibly complicated, is to prove in fact that all these This PI are sum of squares, and it involves very complicated matrices, which I won't have time to comment here. Do you have the roots here, or only the coefficients? You have only the coefficients at the end. At the end, but in the... But the matrices are built, but I don't know if you can limit them. built with everything uh uh recurrently on the on the what what is available so all the roots are there but but then at the end you have explicit expression explicit expression yes in fact he said that he has proved that by looking at the the proof of that for the case

1:20:00 m equals 3 by Kummer for cubic equations. So, and he has analyzed this proof and found the result. In any case, so these functions are used, these matrices are used by Hermit in other occasions too. Okay, so this is the allusion in the introduction. Now, if we go through the theorem, you have to remember, so for the paper, you have to remember what Solen explained about the structure of the proof, which was, in fact, hidden in these little four lines in the first papers, of the first paper, because it's exactly the same sort of structure. So on one side, so I explained that very briefly, but on one side, you have something connected with the law of inertia, if you will, of Sylvester, that is how to reduce a quadratic form in the sum of squares. So it's a very general result, which is completely implicit in the first paper of Hermit. But it works like that. I mean, you can find a new change of variables with only one variable at one point. I mean, you cancel one variable at each level. And with that sort of change of variable, you can put the quadratic form under this form, and the delta I are, well, the first delta I is the determinant of all the coefficients of the quadratic form, and then the other delta I, in fact, corresponds to when you eliminate subvariables, I mean, sort of minors, minors of the matrices, okay? So this is very general, nothing to do with equations or whatever. It's only a result on quadratic form. But now what you do, what Ehrlich did was to associate to an equation, a real equation with some roots, so such quadratic forms. And the quadratic forms are made with this sort of polynomial with the roots, the power of the roots, but also with coefficients which are chosen in such a way that you can decide, I mean, they can really show you when it's positive or when something is superior, when the root is greater or inferior to a certain value. So I say that because you will see how... The root is ABL.

1:22:30 I should say root, perhaps. And so when you look at this quadratic form, and you apply this theorem, the points of all this was that the various delta i, which appears are the Sturm functions essentially in the Sylvester expression. So the invariance of the signature for this quadratic form, specific quadratic form, gives because here you see really that, of course, it's connected with the number of roots greater or inferior to a certain value. And as you connect it with the Sturm function, you can see that it's the sign of the Sturm function, the sign of the data I involved in the process. I would like to ask a question because it seems that they always consider that all the roots are distinct. Not only all those are this thing, but even all the intermediate miners are not zero, so everything is generating this sense, and apparently, not so much later, I mean, there is this paper by Provenius, where he is in a room for the cases where there are some missing miners which are vanishing. If you push... When did it start, I mean? No, it will start. I mean, it was done. I mean, you have some remarks of Hermit saying, well, if some things vanishes, you can do a bit differently, but the same, you know. But you don't have anything in the papers. But because in Frobenius' paper, obviously, it's different from the place. Yeah, but Frobenius is a completely different guy. He writes it much later. For instance, in Serret, he did some things about that. Frobenius' paper is when? I mean, it's eight, six, seven. Yes, 70s to 80s, I would say something, but first of all, it's German, and that's very important for the question, I mean, if you look at the German papers, it's very different, I mean, it's really a question of how you write papers at the time, and in France, you write them like that, and you, okay, and even in the book by Serret, where he discussed the case where you have several, you have common roots and things like that, sometimes they say we can extend the result to, I mean, it doesn't prove it to me. So it's not only a question of time, but also a question of place, for instance.

1:25:00 Okay, so I say that because, as you see in the paper, so in the first paragraph, so the paragraph one, In fact, you will have the same delta-1 and so on, which appears in a very general framework, and at the end of the page, so it's page 398, you find the same sequence of relation ratios of of Delta I that we encounters in the in the real case but the new point is that it will be applied not to quadratic forms but to emission forms and that's the main novelty no no the first place as far as I know it's in a former paper about quadratic forms published in 53 by Hermit of course but all these things were elaborated at the same time so it's you know and he mentioned that because in the first paragraph it said 1C, so at the beginning of the page 399, 1C by this way that the forms like phi, so emission form, have the same behavior as quadratic forms with half the number of indeterminate, which is especially important for arithmetics as I proved it in the new theory, etc. This is the 53 papers I alluded to. So he will do exactly the same sort of thing he has done for the real case, that is the point is to construct also. So once you have this theorem, which he doesn't explain, but he's just in the paragraph one, he will try to construct that sort of specific functions, Hermitian form. I think he considers Hermesand forms not as forms with complex numbers, but forms with the real part and the imaginary part of the complex number. So there are two n real variables and not n complex variables.

1:27:30 So there is a duplication of all the results. of all the results and in particular it explained the the formulation of the the theorem page 398 the number of square will be the double of the number of positive terms in the in this sequence and that sort of stuff but okay so in the paragraph There are real substitutions on the real part. On the real part. Yes, imagine. There are more general than the Hermitian substitutions, the one we perform today with this kind of sesquilinar form. Okay, so in the second and third part of the paper, so in the second part, he will construct exactly what Solen described for the real case, and that will be explained by Henri in a few minutes. So I will not go on like that, so it's a specific Hermitian form adapted to the subject. And in the third part, he explained how to compute very concretely all these things, and it's where the Bézoutienne entered the picture. So that will also be explained by Henri. Try to explain. Well, I mean, at least exhibited by Henri, something like that, in a few minutes. So, okay, so at that point, you arrive to the theorem stated at the beginning of the section 2, where in fact you know how to compute or you will know how to compute the number of square the number of roots of the complex equation which have imaginary part positive and also the number of roots in this complex equation where the imaginary part is negative okay so this is the the theorem that he will explain so what what what I mean does in the sequel of the paper so in the fourth

1:30:00 section is to show the connection with Cauchy theorem and so to do that so at the beginning of part four so it takes in fact a function which is which in fact he will enter in the definition of complex function but which is real and and which he considered as a two-variable real equation for a curve. And he will explain, in fact, how you compute the number of roots inside the curve delimited by this function. So to do this, so you see in the middle of page 405, so he has, in fact, now these two equations, so the f is the original equation, so he wants to compute the roots of these equations. And the u equals phi of z, so phi of z, sorry, so phi of z is a new expression of which the imaginary part is related to the curve which is at the heart of the subject. So, in this case, you can write a new equation, say f tilde, there is no notation for it in Hermit, but f tilde, depending on mu, that is the phi of z, such that f to z could be written f tilde of u. Okay, so this F tilde has roots phi of A, phi of L, and so you have a, it's a quadratic form, an Hermitian form of the type you have dealt with before, and so you can compute the number of phi of A of roots of this equation with imaginary part positive. That's what you have learned to do. Of course, the number of this one is also a number of the A, K solution of this equation. And when you translate this condition in terms of the first equation, you see that you can obtain the roots inside a certain curve. give one example, which is, I give you the first example, but with phi of z equals, so that I can perhaps erase now, with phi of z equals yc minus 2 to the square, so if you

1:32:30 compute if you compute what is the imaginary part of that when you take the complex number you see that what you reach so here would be delimited by and eta is here, so I have the color here, so this is x equals xi and this is y equals eta and so you are dealing, in fact the imaginary part is something like x minus xi y minus eta, this is the imaginary part of that, so in order that you get the positive thing you obtain this part. And so as he explained, if you change this, you can obtain, for instance, the number of solutions inside a rectangle or something of the kind by choosing correctly the various phi and combining them. And he gave other examples when you obtain the thing inside a square, a circle or something of the kind. That's five, that's a four, and in five, you're a four.