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Topic: math -- (f(x) - f(y))/(x - y) = (x^3 + y^3)*g(x,y)
Replies: 6   Last Post: Apr 12, 2008 3:43 PM

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Bill Daly

Posts: 69
Registered: 12/8/04
Re: math -- (f(x) - f(y))/(x - y) = (x^3 + y^3)*g(x,y)
Posted: Apr 12, 2008 12:40 PM
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On Apr 12, 12:04 am, quasi <qu...@null.set> wrote:
> Problem:
>
> Does there exist a nonconstant polynomial f in C[x] such that
>
>    f(x) - f(y) = (x - y)*(x^3 + y^3)*g(x,y)
>
> for some g in C[x,y]?
>
> quasi


If f(x) = f0 + f6*x^6 + f12*x^12 + ..., then f(x) - f(y) is
algebraically divisible by (x - y)*(x^3 + y^3), and the quotient
g(x,y) is given by

g(x,y) = (x^2 + x*y + y^2)*[f6 + f12*(x^6+y^6) + ...]

So, for example, if f(x) = 1 + x^6, then g(x,y) = x^2 + x*y + y^2.



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