Existence of Integer Solutions

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 whether Diophantine equations have any integer solutions at all (Hilbert 10th Problem). The chapter starts with elementary methods like the integral Hasse principle, but then proceed to more advanced methods, such as the use of Jacobi symbol or the analysis of possible prime divisors of cubic polynomials. The chapter ends with a discussion of which equations are solvable in positive integers.