Simple and Double Integrals on an Algebraic Surface

Abstract

a. Simple integrals. A simple integral, or integral of total differential, attached to an algebraic surface F \(f\left( {x,y,z} \right) = 0\) is an integral of the form $$I = \int\limits_{\left( {{x_0},{y_0},{z_0}} \right)}^{\left( {x,y,z} \right)} Q \left( {x,y,z} \right)dx + R\left( {x,y,z} \right)dy$$ (1) where Q and R are rational functions of x, y, z satisfying the integrability condition \(\frac{{\partial Q}}{{\partial y}} = \frac{{\partial R}}{{\partial x}}\), the derivatives being evaluated by considering z as an implicit function of x and y. These integrals have been introduced by Picard (a, 2, 3, 4). The integral I may possess on F either polar or logarithmic singularities, and their locus is referred to respectively as the polar or the logarithmic curve of I. The integral is of the first, second or third kind according as it possesses no singularities, polar singularities only, or logarithmic singularities. An integral I of the first or second kind is also characterized by the condition that \(\int {d{\text{ }}I = 0} \) for any 1-cycle Г1 on F which is homologous to zero. If I is any simple integral and if Г1 is any 1-cycle on F, the integral \(\int\limits_{{\Gamma _1}} {d{\text{ }}I} \) is called the period of I relative to Г1 if Г1 ≁ 0 on F, a logarithmic period if Г1 ∽ 0. A simple integral I without periods (i. e. whose periods all vanish) is a constant, a rational function or a logarithmo-rational function of F, according as I is of the first, second or third kind.KeywordsSpectral SequenceAlgebraic CurveAlgebraic SurfaceDouble IntegralSimple IntegralThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.