Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Polynomial Roots and Open Mappings   The "openness" of a complex polynomial mapping is discussed and applied to
the Fundamental Theorem of Algebra. In this category fall proofs of S.
Wolfenstein, R.L. Thompson, J. Milnor, and S. Reich-S. Smale. These proofs take
into account the critical points of the polynomial. New elementary proofs of
openness due to D. Reem and F.S. Cater make possible a very short proof of FTA
without reference to zeros of the derivative. We regard Gauss's Helmstedt
Thesis (1799) as an exercise in applied differential topology, and fill out a
synopsis published by S. Gersten and J. Stallings. The polynomial function may
be perturbed or re-aligned (work of Martin, Savitt and Singer) to eliminate
critical values so we work with a configuration of real, plane algebraic
curves. The treatment we give to the Implicit Function Theorem uses contractive
operators on the Banach algebra of convergent power series according to W.
Walter. In addition to Functional Analysis, the key geometric insight is a
topological transversality result that leads to a rigorous demonstration that
the plane curves intersect as Gauss and A. Ostrowski said they would. Working
in the category of analytic mappings makes uniqueness of "implicit" solutions
more transparent, and the polynomial (and harmonic) nature of the curves
imposes finiteness on the number of extrema, coming from a well-known
application of the Be'zout Theorem. We think this provides a clarification of
Gauss's statement that an algebraic curve entering a domain, must leave again,
that is more concrete than declaring such a curve to be a "one-dimensional
manifold".

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The categories of this paper are: bayes-an
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The categories of this paper are: astro-ph
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The categories of this paper are: cs
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The categories of this paper are: solv-int
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The categories of this paper are: cond-mat
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The categories of this paper are: physics
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-ex
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The categories of this paper are: patt-sol
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The categories of this paper are: supr-con
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The categories of this paper are: mtrl-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: quant-ph
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The categories of this paper are: eess
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The categories of this paper are: ao-sci
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The categories of this paper are: q-bio