Height function

Height function, Mathematical functions that quantify complexity.

Mathematical functions that quantify complexity This article is about mathematical functions that quantify complexity. For other uses of height, see Height (disambiguation).

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. 3 for the coordinates (3/9, 1/2)), but in a logarithmic scale.

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Significance

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 Alan Baker:(1966, 1967a, 1967b).

In other cases, height functions can distinguish some objects based on their complexity. For instance, the subspace theorem proved by Wolfgang M. Schmidt:(1972) demonstrates that points of small height (i.e. small complexity) in projective space lie in a finite nu... ...read more

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