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Problem expanding algebraic function via Newton polygon

  1. Oct 22, 2012 #1
    Hi,

    I've run into a problem with expanding algebraic functions via Newton polygons. Consider the function:

    [tex]f(z,w)=a_0(z)+a_1(z)w+a_2(z^2)w^2+\cdots+a_{10}(z)w^{10}=0[/tex]

    and say the degree of each [itex]a_i(z)[/itex] is ten.

    Now suppose I wish to expand the function around some ramification point of the function, say [itex]r_i\neq 0[/itex]

    Now, in general, I won't be able to compute exactly that ramification point so that the standard means of expanding around this point, that is, by letting [itex]z\to z+r_i[/itex] and expanding around zero, would seem to fail because I won't be able to exactly compute the new expansion center. That is, the new center of expansion will be slightly off from the ramification point. This point would then be just a regular point and so the algorithm would just compute regular expansions (cycle-1 sheets) when the branching at this point may in fact be ramified.

    I just do not understand how it is possible, using this method, to compute ramified power series centered away from zero of not only this function, but any sufficiently high degree function in which the ramification points cannot be computed exactly because of this problem.

    Am I missing something with this?

    Thanks,
    Jack
     
    Last edited: Oct 22, 2012
  2. jcsd
  3. Oct 22, 2012 #2

    mathwonk

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    you seem to be asking how to proceed when you need to know the solution of some polynomial equation of high degree, when such an equation has no obvious solution formula. as far as i know there is no remedy for this. thats why all methods taught in calculus say for integration by partial fractions, e.g. that begin by saying: first find all the roots of this polynomial, are bogus.
     
  4. Oct 22, 2012 #3
    I find this disappointing as all the references I have about Newton polygon suggest making that substitution ([itex]z\to z+z_0[/itex]) to expand the function about the point [itex]z_0[/itex]. Then if there is no solution to this problem, such a substituion in a degree 4 or higher function likely will encounter round-off error and thus fail to pick up the ramifications about [itex]z_0[/itex] if [itex]z_0[/itex] is a ramification point -- basically that's saying the method is pretty much useless for expanding high degree functions about ramification points except the origin.

    I just cannot believe that's not mentioned anywhere. Not even Walker says anything about this.


    But thanks for replying. :)

    Edit:

    Dang it. I re-read some of my reference and this point is in fact emphasized in several of them. For example in:
    http://www.lifl.fr/~poteaux/fichiers/monodromy_snc07.pdf
    and in fact would lead to incorrect ramifications for [itex]c_k[/itex]. Basically, the calculations have to be done exactly in the standard Newton polygon but there appears to be modified methods which can overcome some of these problems.

    Guess I didn't understand before. Sorry guys.
     
    Last edited: Oct 22, 2012
  5. Oct 22, 2012 #4

    mathwonk

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    the substitution method is perfectly correct. this problem has nothing to do with that method, nor with newton polygons. the problem is with solving polynomials.
     
  6. Oct 23, 2012 #5
    Ok, I understand that now.
     
    Last edited: Oct 23, 2012
  7. Oct 23, 2012 #6

    mathwonk

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    this is why we have existence theorems in mathematics. e.g. every polynomial f of odd degree has a root, by the IVT, since f(x) goes to infinity and aLSO TO MINUS INFINITY as |x|-->infinity.

    but just try finding the roots of a complicated polynomial of degree 9. all you can do is give an iterative procedure for writing down any finite number of decimals places of a root.

    of course it also depends on what you mean by "writing down" a root. if your polynomial has integer coefficients and is irreducible then you could try to work in the field Q[X]/(f), where in fact the root is {X}!

    but an element of this field is represented by a vector in nine dimensional Q -space. i.e. an element of this field is represented by a polynomial of degree 8 with rational coefficients.
     
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