Re: Limit and Root Objects
- To: mathgroup at smc.vnet.net
- Subject: [mg72321] Re: Limit and Root Objects
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Tue, 26 Dec 2006 05:42:32 -0500 (EST)
- References: <em606c$2o1$1@smc.vnet.net> <4586C045.2020805@metrohm.ch> <DA28578F-4E16-49A8-A447-3FDE236F303F@akikoz.net> <C322A62D-E35E-4395-A264-AFCA59DE9608@mimuw.edu.pl>
*This message was transferred with a trial version of CommuniGate(tm) Pro* I forgot to add that the argument below shows that continuous "root objects" can't be constructed over the space of all complex polynomials of a fixed degree. If we consider only real polynomials, e.g. only real cubics (which is what I was doing in my original post in this thread) then the situation is quite different: indeed it is quite obvious that a continuous section exists and can be defined by using the function Piecwise to combine different root objects. However, it is impossible to extend this section continuously to all cubics with complex coefficients. Andrzej Kozlowski On 20 Dec 2006, at 09:46, Andrzej Kozlowski wrote: > I have thought that it might be of some interest to justify the > remarks below about using "elementary topology of configuration > spaces" to prove that one can't define continuously objects of the > form Root[f,1], Root[f,2] ... etc, where f is some polynomial. The > proof is actually not quite as "elementary" as I at first thought. > What it needs is the concept of "the Schwarz genus". Given two > normal Hausdorff spaces X and Y and a continuous map, f:X ->Y , the > Schwarz genus of the map f is the minimal cardinality of an open > cover of Y such that there exists a continuous section of the map f > over each set (a section over a subset U of Y is a continuous map > p:U->X such that f(p(u))=u). > Let's choose a degree of our polynomial d, say d=3 for cubics. The > set of all monic polynomials of degree d (monic means that the > highest coefficient is 1) is just C^d. Consider the space Y = C^d - > S, where S is the so caled discriminant, that is the space of all > monic polynomials of degree d with a multiple root. Let X be the > space of al pairs (m,x) where m is a monic polynomial of degree d > and x is a root of f. Then we have a map: > f: X -> Y, taking (m,x) to the point x. This map is actually a d- > fold covering map. > Now we can prove the following theorem: if d is a power of a prime > than the Schwarz genus of f is d. The proof requires quite a lot of > topology and obviously I am not going to include it here (it > appears in a book by V.A. Vassiliev). . But what it means is that > in the case of cubics you cannot define a continuous map Root[f,i], > where i is 1,2, or 3, over the space of cubics, since that would > provide exactly eh kind of global continuous section that can't > exist. Unless I am confused about something here, all attempts to > construct a "continuous real root" for a cubic over the parameter > space are clearly doomed. Of course all this proves that you can't > have a continuous branch over the entire parameter space, you can > certainly have such continuous sections over various parts of it. > > This fact seems intuitively quite obvious but I can't see any other > way to prove it that does not require some fairly advanced topology > (obstruction theory). If anyone knows of any other way I would like > to hear of it. > > Andrzej Kozlowski > > > On 19 Dec 2006, at 09:06, Andrzej Kozlowski wrote: > >> What you describe, including the fact that the numbering or roots >> changes is inevitable and none of it is not a bug. There cannot >> exist an ordering of complex roots that does not suffer from this >> problem. What happens is this. >> Real root objects are ordered in the natural way. A cubic can have >> either three real roots or one real root and two conjugate complex >> ones. Let's assume we have the latter situation. Then the real >> root will be counted as being earlier then the complex ones. Now >> suppose you start changing the coefficients continuously. The >> roots will start "moving in the complex plane", with the real root >> remaining on the real line the two complex roots always remaining >> conjugate (symmetric with respect to the real axis). Eventually >> they may collide and form a double real root. If this double real >> root is now smaller then the the "original real root" (actually >> than the root to which the original real root moved due the the >> changing of the parameter), there will be a jump in the ordering; >> the former root number 1 becoming number 3. >> This is completely unavoidable, not any kind of bug, and I am not >> complaining about it. It takes only elementary topology of >> configuration spaces to prove that this must always be so. >> >> What I am not really convinced of is that Limit really couldn't >> deal with this problem, at least partially, in the case of real >> roots of cubics and quartics. I have been told that it cannot be >> done because Limit relies on Series - and that I of course >> completely agree that Series can't possibly deal with this. Even >> worse problems of this type will inevitably happen in the case >> of non-real roots, which can switch order in intractable ways. But >> it seem to me that in the case of real roots one could use a >> simple numeric-symbolic method to get the right answer- which is >> indeed what I did by hand before posting this problem. The key >> point is, that problem can only occur at a double root, because >> two conjugate complex roots must first collide on the real line >> before a real root is formed. The double roots can be found by >> using the derivative and Resultant, as I did in this case: >> >> >> Reduce[Resultant[poly[b,x], D[poly[b,x]], x] == 0, b, Reals] >> >> where b is a parameter. Now, if a Limit is taken at a point that >> is not a double root continuity can be assumed and we are home. If >> the Limit point is a double root, then (in the case of a cubic) we >> that the only possibilities for the limit will be the value of the >> first or the third real root at this point (where there will be >> three real roots). We can use NLimit with significance arithmetic >> to settle this. Significance arithmetic is important since the two >> branches can be arbitrarily close (we may have a near tripe root). >> Actually, we would have to first check algebraically that we do >> not have a triple root at this point - if we do, there is no >> problem though. >> >> I can see that doing all this would considerably increase the >> complexity of Limit whenever any kind of Root objects were present >> in an expression - which may be strong argument against it. On the >> other hand, I think these kind of situations are quite interesting >> and it would be a good idea to be able to get them right without >> having to resort to manual computations, as I did in this case. >> Another argument for leaving things as they are, the same kind of >> phenomenon for complex roots (and perhaps polynomials of high >> degree) seems quite un-manageable so in that sense the problem >> would still remain. But I think real roots are interesting and >> important, and even if this could be done only for cubic and >> quartics, it would be worth while. >> If all this is not practical, I think a warning message should >> always be issued by Limit whenever such Root objects are >> encountered (I think Series already does that). >> >> >> Andrzej Kozlowski >> >> >> >> >> On 19 Dec 2006, at 01:22, dh wrote: >> >>> *This message was transferred with a trial version of CommuniGate >>> (tm) Pro* >>> Hi Andrzej, >>> It is definitly a bug. The reason for the bug may be that not >>> only the function is not continuous at b == -(3/2^(2/3)), but >>> also the numbering changes. There are three different real roots >>> for b<-(3/2^(2/3)), therefore, the first is the smallest. For b=- >>> (3/2^(2/3)) the two lowest roots merge and for b>-(3/2^(2/3)) the >>> two "former" lowest become complex. And now the first root is >>> the "former" highest. MMA seems to keep the number of the root in >>> the limit process. To make the bug even worse, the wrong first >>> root at b=-(3/2^(2/3)) is a double root and therefore, reduced to >>> a quadratic root object. >>> Daniel >>> >>> >>> Andrzej Kozlowski wrote: >>>> It is easy to check that the function >>>> f[b_] := Root[#1^3 + b*#1 - 1 & , 1] >>>> is discontinuous at b, where >>>> Reduce[Resultant[x^3 + b*x - 1, D[x^3 + b*x - 1, x], x] == 0, b, >>>> Reals] >>>> b == -(3/2^(2/3)) >>>> indeed this was not so long ago discussed in connection with a >>>> little argument about "usefulness' of Root objects. In view of >>>> this, isn't the following a bug? >>>> u = Limit[f[b], b -> -(3/2^(2/3)), Direction -> 1] >>>> Root[2*#1^3 + 1 & , 1] >>>> v = Limit[Root[#1^3 + b*#1 - 1 & , 1], b -> -(3/2^(2/3)), >>>> Direction - > -1] >>>> Root[2*#1^3 + 1 & , 1] >>>> u == v >>>> True >>>> It looks like Limit is making life too easy for itself by >>>> assuming continuity. >>>> Using NLimit shows that things are not as simple: >>>> w = NLimit[f[b], b -> -(3/2^(2/3)), Direction -> -1] >>>> 1.5874010343874532 >>>> z = NLimit[Root[#1^3 + b*#1 - 1 & , 1], b -> -(3/2^(2/3)), >>>> Direction - > 1] >>>> -0.7937180869283765 >>>> Andrzej Kozlowski >>> >> >