Height function

A height function is a function that quantifies the complexity of mathematical objects. In Diophantine geometry, height functions quantify the size of solutions to Diophantine equations and are typically functions from a set of points on algebraic varieties (or a set of algebraic varieties) to the real numbers. For instance, the classical or naive height over the rational numbers is typically defined to be the maximum of the numerators and denominators of the coordinates (e.g. 7 for the coordinates (3/7, 1/2)), but in a logarithmic scale. Height functions allow mathematicians to count objects, such as rational points, that are otherwise infinite in quantity. For instance, the set of rational numbers of naive height (the maximum of the numerator and denominator when expressed in lowest terms) below any given constant is finite despite the set of rational numbers being infinite. In this sense, height functions can be used to prove asymptotic results such as Baker's theorem in transcendental number theory which was proved by . In other cases, height functions can distinguish some objects based on their complexity. For instance, the subspace theorem proved by demonstrates that points of small height (i.e. small complexity) in projective space lie in a finite number of hyperplanes and generalizes Siegel's theorem on integral points and solution of the S-unit equation. Height functions were crucial to the proofs of the Mordell–Weil theorem and Faltings's theorem by and respectively. Several outstanding unsolved problems about the heights of rational points on algebraic varieties, such as the Manin conjecture and Vojta's conjecture, have far-reaching implications for problems in Diophantine approximation, Diophantine equations, arithmetic geometry, and mathematical logic. An early form of height function was proposed by Giambattista Benedetti (c. 1563), who argued that the consonance of a musical interval could be measured by the product of its numerator and denominator (in reduced form); see .