Estimates For Integer Points on Algebraic Variety by using Diophantine Approxiwatic

对于超越数x，我们考虑用代数数beta在beta的高度和beta的次数方面的丢番图逼近。通过使用这种近似，称为超越（或超越）措施，我们观察代数曲线上的整数点，例如，椭圆曲线。当x来自数域上定义的代数群的有理点的指数映射的逆象的坐标时，我们得到了这种测度的一个加细，代数群定义为简单交换簇的非平凡扩张.我们还得到了一类指数丢番图方程整数解的有限性结果。

For transcendental number x, we consider Diophantine approximations by algebraic number beta in terms of height of beta and degree of beta. By using such approximation, called transcendental (or transcendence) measure, we observe integer points on algebraic curves, for example, elliptic curves. We obtain here a refinement for such measure when x comes from coordinates of inverse image by exponential map of rational points of an algebraic group defined over a number field, which is defined as a non trivial extention of a simple abelian variety by the additine group. We get also some finiteness result concerning with integer solutions to certain exponential Diophantine equations.