Sofia Kovalevskaya

Leigh Ellison

Abelian Integrals

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The second of these papers saw Sofia Investigating a certain class of Abelian integrals of the third rank which reduce to elliptic integrals. It was again Weierstrass who assigned this problem to her. Although her results were considered to be of little real importance at the time, Weierstrass still felt that the paper, demonstrated proof of a high level of mathematical competence.

Sofia was to begin this paper with a corollary of a theorem by Abel which states that if an Abelian integral reduces to elliptic integrals, then there is an integral of the form
R(s)(ϕ(s)ds\int_{-\infty}^\infty \large\frac {R(s)}{\sqrt {(\phi(s)}}\normalsize ds
to which the original integral can be reduced and such that s, R(s), φ(s) and ‚àöφ(s) are all rational functions of the original variables x and y, φ being a polynomial of degree three or four in s. By doing some work with this corollary she reduced the question of degeneracy to a question of whether or not there are Abelian integrals of first kind associated with f(x, y) = 0 which reduce to elliptic integrals. The next part of the paper was inspired by Weierstrass with Sofia using a number of arguments which had been made by him originally. One particular line of argument which she followed resulted in Weierstrass's transcendental condition for degeneracy. This states that among the theta functions associated with the Abelian integrals in question, there must be one which has the simple form that it can be broken up into the product of a theta function of v1 and a theta function of v2 , ... , vp.

She then went on to use another theorem by Weierstrass which allowed her to work the transcendental criterion into something more easy to use. The remainder of the paper then saw Sofia developing the argument that degeneracy of an integral implies relations for the theta functions, which then place restrictions on and . To do so she selected theta functions for which the following holds:
g1θ(v1 , v2 , v3)5θ(v1 , v2 , v3)125+ g2θ(v1 , v2 , v3)345θ(v1 , v2 , v3)06
+ g3θ(v1 , v2 , v3)46θ(v1 , v2 , v3)035+ g4θ(v1, v2, v3)3θ(v1, v2, v3)123= 0
This equation was then differentiated to obtain a symmetrical solution in x and y:
h1‚√(ξ1ξ1') + h2‚√(ξ2ξ2') + h2‚√(ξ3ξ3')
where h1, h2, h3 are constants, and the ξj' are homogeneous linear functions of ξ1, ξ2 and ξ3 . Thus, when the variables are suitably replaced the result is a homogeneous equation of degree four:
F(X1, X2, X3) = 0
She had thus proved that with the exception of one case,
If y is an algebraic function of x of rank 3, then the equation which holds between x and y can be transformed (in infinitely many ways) into a homogeneous equation of degree four, F = 0, among three quantities X1 , X2 , X3 which are rational functions of x and y.
She then converted F(X1, X2, X3) = 0 into G(p, q, r) = 0 where p, q, r are homogeneous linear functions. By analysing this further she showed that,
Among the double tangents of this equation four must coincide, i.e., their eight points of tangency must lie in a single conic section; they intersect in one point if among the Abelian integrals depending on if there is at least one which can be reduced to an elliptic integral by a transformation of the type under condideration.
This purely algebraic condition is far easier to work with than the transcendental ones given by Weierstrass and is the main achievement of this paper which was otherwise of little interest.

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