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It is well known that the complex plane is algebraically closed: Every polynomial has a zero. The relationship seems, to me, to run deeper: For every complex-differentiable function, there exists a power series which converges to it; power series could be regarded as a sort of a generalized polynomial. Every entire function with a pole at infinity is a polynomial. Etc.

Complex analysis can in some ways be regarded as the successor to Euclidean plane geometry. Many proofs exist of the aforementioned facts, which seem to link the polynomials much more tightly to the complex plane than to the real line, but is there an intuitive, geometric reason as to why these transformations seem so fundamentally linked to what is essentially plane geometry?

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    $\begingroup$ The function does not converge to a power series; rather, the power series converges to the function. $\endgroup$ – Michael Hardy May 16 '13 at 5:48
  • $\begingroup$ Fair point. Sorry for the poor choice of words. $\endgroup$ – Dan M. Katz May 16 '13 at 6:04
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Well, nice question.

First thing to mention is probably the Gauss correspondence b/w ruler and compass constructions and roots of quadratic polynomials.

Second connection: Every polynomial is a characteristic polynomial of a companion matrix; the number of real roots of a polynomial can be read from the determinants of the corresponding Hurwitz matrix; Jordan blocks of a matrix correspond to multiplicities of eigenvalues, which are roots of its characteristic polynomial, etc... So, theory of polynomials and special matrices are intrinsically connected. Now, matrices can be viewed as 2D/plane objects, because they have rows and columns, and special matrices can be described geometrically... Also, taking limits one gets complex functions defined on the complex plane from both theories to get a unified self-contained theory.

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  • $\begingroup$ Would you mind elaborating on the matrices as 2D/plane objects theme? I'm not totally convinced that the matrix being described by "two dimensions" of data makes it an inherently "two dimensional" object in the same sense that the plane is...After all, an nxn matrix describes a transformation on n space, not 2 space. $\endgroup$ – Dan M. Katz May 16 '13 at 6:54
  • $\begingroup$ Well, it's a speculation: the king of matrices, Pascal triangle can be viewed as a function on a grid that satisfies an identity given by a planar graph (the roots of its characteristic polynomial have a nice structure). Also, generalizations of matrices are kernels of linear transformations, which in the case of functions of one variable are functions/distributions on a plane. $\endgroup$ – DVD May 16 '13 at 7:20
  • $\begingroup$ Also, Pascal triangle generalizes to modified Gamma function. $\endgroup$ – DVD May 17 '13 at 0:44
  • $\begingroup$ Alright, you've lost me haha $\endgroup$ – Dan M. Katz May 17 '13 at 4:53
  • $\begingroup$ Sorry, did not mean to:( $\endgroup$ – DVD May 17 '13 at 4:55

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