Computing Rational Points on Curves
Bjorn Poonen · 2023
The solution of diophantine equations (such as https://www.w3.org/1998/Math/MathML" display="inline"> x 13 + y 13 = z 13 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780138747022/abb99e87-ffb7-4196-a4c4-acb0033e3d6a/content/eq2155.tif "/> ) over the integers often reduces to the problem of determining the rational number solutions to a single polynomial equation in two variables. Such an equation describes a curve, and the problem of finding rational number solutions can be interpreted geometrically as finding the rational points on the curve, i.e., the points on the curve with rational coordinates.