Abstract
This paper is based on a talk given at the bicentenary celebration of the birth of Niels Henrik Abel held in Oslo in June, 2002. The objectives of the talk were first to recall Abel’s theorem in more or less its original form, secondly to discuss two of the perhaps less well known converses to the theorem, and thirdly to present two (from among the many) interesting issues in modern algebraic geometry that may at least in part be traced to the work of Abel. Finally, in the reprise I will suggest that the arithmetic aspects of Abel’s theorem may be a central topic for the 21st century.
Keywords
Algebraic Geometry Algebraic Curve Maximum Rank Partial Fraction Expansion Lagrange Interpolation Formula
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Notes
- 1.The term “highly transcendental” needs care in interpretation-cf. the reprise below. Again Abel, in a paper published in 1826, showed the existence of polynomials R, F such that \( \int {\frac{{F dx}} {{\sqrt R }} = \ln \left( {\frac{{P + \sqrt {RQ} }} {{R - \sqrt {RQ} }}} \right)} \) has solutions for relatively prime polynomials P, Q. Here, R is a polynomial of degree 2n with distinct roots and F is a polynomial of degree n-1, so that the integrand is a differential of the 3rd kind. This is an “exceptional” case where the integral is transcendental but expressible in terms of elementary functions.Google Scholar
- 2.Integrating a DE means finding a solution by an iterative process. Since there are no derivations of ℚ the methods of calculus break down-one must break the problem into “increments” by some other means, perhaps either by an iteration process that at each stage decreases the “arithmetic complexity” of the 0-cycle, or by analyzing the DE’s (4.15) and (4.20) in the completions of ℚ under all valuations.Google Scholar
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© Springer-Verlag Berlin Heidelberg 2004