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Re: Limit and Root Objects
*To*: mathgroup at smc.vnet.net
*Subject*: [mg72322] Re: Limit and Root Objects
*From*: Adam Strzebonski <adams at wolfram.com>
*Date*: Tue, 26 Dec 2006 05:46:34 -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>
*Reply-to*: adams at wolfram.com
I think I have a somewhat simpler, or at least more elementary proof.
Suppose there exists a continuous map Root[_, 1] : C^d - S --> C.
Let us define a map F: [0, 1] --> C^d - S by putting
F(t) := (-E^(2 I Pi t), 0, ..., 0)
(F(t) corresponds to the polynomial x^d-E^(2 I Pi t)).
The the composition G of F followed by Root[_, 1] is a continuous map
from [0, 1] to the unit circle such that G(t)^d == E^(2 I Pi t).
For t in [0, 1], let H(t) be k iff G(t) == E^((2 I Pi t+2 I Pi k)/d).
H is a continuous mapping from [0, 1] to the discrete set
{0, 1, ..., k-1}, hence it is constant, say H(t)==k.
Then G(0)==E^((2 I Pi k)/d) and G(1)==E^((2 I Pi+2 I Pi k)/d),
but this is is impossible, because F(0)==F(1), and so G(0)==G(1).
Best Regards,
Adam Strzebonski
Wolfram Research
Andrzej Kozlowski wrote:
> *This message was transferred with a trial version of CommuniGate(tm) Pro*
> 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
>>>
>>
>
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