About: Height function

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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.

Attributes	Values
rdf:type	Thing
rdfs:label	Height function (en)
rdfs:comment	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. (en)
rdfs:seeAlso	Height (abelian group)
dcterms:subject	Algebra Diophantine geometry Abelian varieties Algebraic number theory Polynomials Elliptic curves
Wikipage page ID	3141855 (xsd:integer)
Wikipage revision ID	1124001783 (xsd:integer)
Link from a Wikipage to another Wikipage	Cambridge University Press Projective space Hermitian form Mordell conjecture Mordell–Weil group Mordell–Weil theorem Non-Archimedean field Binomial coefficient Birch and Swinnerton-Dyer conjecture Annals of Mathematics Arakelov theory John Tate (mathematician) Consonance Mathematical logic S-unit equation General linear group Elliptic curve Function (mathematics) Giambattista Benedetti André Néron André Weil Arithmetic geometry Logarithm Logarithmic scale Fubini–Study metric Archimedean field Suren Arakelov Mathematika Néron–Tate height Acta Mathematica Adelic algebraic group Heath-Brown–Moroz constant Height zeta function Irreducible fraction Line bundle Abc conjecture Affine variety Algebraic geometry Algebraic variety Ample line bundle Algebra Faltings's theorem Diophantine geometry Number field Diophantine approximation Diophantine geometry Global field Quadratic form Projective variety Rational point Raynaud's isogeny theorem Interval (music) Baker's theorem Hyperplane Asymptotic analysis Abelian variety Abelian varieties Algebraic number theory Polynomials Bijection Bit Homogeneous coordinates Height of a formal group law Diophantine equation Diophantine equations Douglas Northcott Automorphic form Polynomial Inequality (mathematics) Elliptic curves Rational number Real numbers Mahler measure Illinois Journal of Mathematics Lowest common denominator Vojta's conjecture Subspace theorem Manin conjecture Siegel's theorem on integral points Transcendental number theory Proj construction Tree height Springer-Verlag Rational points Lehmer conjecture
Link from a Wikipage to an external page	http://mathworld.wolfram.com/PolynomialHeight.html https://archive.org/details/computationalexc00borw https://archive.org/details/computationalexc00borw/page/n5

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