Some applications of decomposable form equations to resultant equations

On Some Arithmetical Properties of Solutions of Decomposable Form Equations

We study some fine arithmetic properties of the components of solutions of a decomposable form equation. Lower growth rates for the greatest prime factor of a component are obtained for density 1 of the solutions. Also, high pure powers are shown to occur rarely. Computations illustrate the applicability of our results; for example, to the study of units in abelian group rings.

Counting Solutions of Decomposable Form Equations

Solving resultant form equations over number fields

We give an efficient algorithm for solving resultant form equations over number fields. This is the first time that such equations are completely solved by reducing them to unit equations in two variables.

On the spatial distribution of solutions of decomposable form equations

We study the distribution in space of the integral solutions to an integral decomposable form equation, by considering the images of these solutions under central projection onto a unit ball. If we think of the solutions as stars in the night sky, we ask what constellations are visible from the earth (the unit ball). Answers are given for a large class of examples which are then illustrated usi...

Applications of Decomposable Measures on Nonlinear Difference Equations

A class of nonlinear difference equations important in the theory of the architecture of computers with parallel processing (neural nets) is examined. The integral with respect to decomposable measure is used as main tool. By a generalization of the theory of generalized functions the limit behavior of the considered difference equation is characterized. AMS Mathematics Subject Classification (...