On an Application of Elimination Theory to the Theory of Abelian Integrals

Abstract

An algebraic equation F(x,y) = 0 is given, of degree μ with respect to x and degree ν with respect to y. We are required to replace x and y by new quantities ε and η by means of the equations. It is required to replace x and y with new quantities ɛ and ɳ by means of equations. ɛ = j(x,y), ɳ = f(x,y), j and f are rational functions. With a few exceptions, which should be the subject of special research in each particular case, transformation (2) will be birational. This transformation can be carried out through rational actions in the following ways. Let's consider a way to reduce a hyperelliptic integral.

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