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Given the similarity between polynomials and power series, I was wondering if there is any generalization of the fundamental theorem of algebra for power series. I understand that it doesn't make much sense to talk of multiplicity when the roots are supposed to be infinite, but maybe there is something like this :P

Gennaro Marco Devincenzis
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    The power series $\sum\limits_{n\ge0}\frac{x^n}{n!}$ defines a function that never assumes the value $0$. For analytic functions, the concept of multiplicity is surely well defined. – egreg Jun 04 '14 at 10:55
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    If you think of FTA as saying any nonconstant polynomial can be factored into linear parts, then there is a generalization of this to entire functions using infinite products. See the Weierstrass and Hadamard factorization theorems in complex analysis. The role of nonzero constants is replaced by functions $e^{g(z)}$ where $g(z)$ is a suitable entire function. – KCd Jun 04 '14 at 11:01
  • The introduction of the book Nevanlinna’s Theory of Value Distribution makes a cases that Nevanlinna’s theory is a generalization of the fundamental theorem of algebra to holomorphic and meromorphic functions. – lhf Jun 04 '14 at 11:07
  • @KCd, nice, but I think the OP is interested in conditions on entire functions that guarantee the existence of a zero. – lhf Jun 04 '14 at 11:08
  • If one viewpoint doesn't directly extend, sometimes a different one does. It might be good if the OP clarifies what is meant for the purpose of this question by "the fundamental theorem of algebra". – KCd Jun 04 '14 at 11:10
  • A generalization to power series of "any nonconstant polynomial over the complex numbers has a zero" is "any entire function not of the form $e^{g(z)}$ for some entire function $g(z)$ has a zero", but that may not be of much practical use since the way you typically know an entire function has that form is by knowing it has no zeros. – KCd Jun 04 '14 at 11:13
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    I was merely thinking of zeroes of a power series, honestly. But if there are other viewpoints of the FTA which can be generalised to power series then they would be interesting to know anyway. – Gennaro Marco Devincenzis Jun 04 '14 at 11:14

1 Answers1

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One generalization of the fundamental theorem of algebra to entire functions is given by the Little Picard theorem, which can be phrased as follows:

If $f$ is a non-constant entire function and $w\in \mathbb C$, then the equation $f(z)=w$ always has a solution, except perhaps for a single value of $w$.

This statement generalizes the fundamental theorem of algebra, which can be phrased as follows:

If $f$ is a non-constant polynomial function and $w\in \mathbb C$, then the equation $f(z)=w$ always has a solution.

lhf
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    However, I don't see right now how the Little Picard theorem implies the fundamental theorem of algebra. – lhf Jun 04 '14 at 11:19