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I have some background in complex analysis (a very minimal amount) but I did come up with a rather odd question.

Given a polynomial a + bx + cx^2 + dx^3... nx^n

There exists n or fewer solutions to the equation that each have a multiplicity of 1 to n.

Given that information suppose we take the exponential function e^x and break it down to its taylor series:

1 + x/1! + x^2/2! + x^3/3! + x^4/4!...

Doesn't that mean that there are infinite roots to the exponential function which may be equal to some type of infinity (or not)

The exponential-infinite roots of unity?

If they are positioned @ infinities then are there more "projective-like" relationships between them that allows you to differentiate between the roots of say e^x and 2^x?