Existence of Non-Trivial Solutions to Homogeneous Equations

The size of a polynomial with integer coefficients is computed by replacing each coefficient by its absolute value and evaluating the resulting polynomial at 2. This chapter investigates homogenous equations. The chapter starts with equations that can be studied by elementary methods like the Hasse principle, before proceeding to more advanced methods such as the Mordell–Weil sieve. The chapter ends with a discussion of which integers can be represented as a sum or a difference of two rational d-th powers.