Alternative Orderings of Diophantine Equations

This chapter discusses various natural ways to order Diophantine equations. For example, one may order equations by length, defined as the sum of total degrees of its monomials plus the base 2 logarithm of the product of the absolute values of its coefficients. The chapter also introduces the so-called “canonical” ordering of Diophantine equations. For each of the discussed orderings, the chapter studies all polynomial Diophantine equations in that order and investigates which of them are solvable by the methods discussed in this book. The first open equations for each ordering are presented. The chapter ends with a big table collecting all open equations from the book in one place.