Abstract :
Let image be an algebraic number field with non-zero α, βset membership, variantimage. Siegel showed in 1929 that there are only finitely many units var epsilon1, var epsilon2 in K which satisfy the unit equation αvar epsilon1+βvar epsilon2=1. In this article we present a new algorithm for solving unit equations which utilizes methods from the geometry of numbers. For the fist time unit equations in number fields up to unit rank 10 and with more than 100,000 solutions are solved. By applying our algorithm to index form equations we compute all power integral bases in the cyclotomic number fields up to degree 12 and in image(ζ17), image(ζ19), image(ζ23).
