Old and New Conjectured Diophantine Inequalities

The original meaning of diophantine problems is to find all solutions of equations in integers or rational numbers, and to give a bound for these solutions. One may expand the domain of coefficients and solutions to include algebraic integers, algebraic numbers, polynomials, rational functions, or algebraic functions. In the case of polynomial solutions, one tries to bound their degrees. Inequalities concerning the size of solutions of diophantine problems are called diophantine inequalities. During the past few years, new insights have been gained in old problems combined with new ones, and great coherence has been achieved in understanding a number of diophantine inequalities. Some of these results, notably the first section, can be formulated in very simple terms, almost at the level of high school algebra.