Hi,(adsbygoogle = window.adsbygoogle || []).push({});

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

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Problem expanding algebraic function via Newton polygon

**Physics Forums | Science Articles, Homework Help, Discussion**