Height function: Difference between revisions

Revision as of 13:53, 2 February 2021

Revision as of 14:58, 19 September 2021

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==Significance==

Height functions allow mathematicians to count objects, such as [[rational point]]s, that are otherwise infinite in quantity. For instance, the set of rational numbers of naive height (the maximum of the numerator and denominator when [[Irreducible fraction|expressed in lowest terms]]) below any given constant is finite despite the set of rational numbers being infinite.<ref name="ReferenceA">{{harvs|txt|last1=Bombieri|last2=Gubler|authorlink1=Enrico Bombieri|year=2006|loc1=pp. 15–21}}</ref> In this sense, height functions can be used to prove [[Asymptotic analysis|asymptotic results]] such as [[Baker's theorem]] in [[transcendental number theory]] which was proved by {{harvs|txt|authorlink=Alan Baker (mathematician)|first=Alan|last= Baker|year1=1966|year2=1967a|year3=1967b}}.

In other cases, height functions can distinguish some objects based on their complexity. For instance, the [[subspace theorem]] proved by {{harvs|txt|authorlink=Wolfgang M. Schmidt|first=Wolfgang M. |last=Schmidt|year= 1972}} demonstrates that points of small height (i.e. small complexity) in [[projective space]] lie in a finite number of [[hyperplane]]s and generalizes [[Siegel's theorem on integral points]] and solution of the [[S-unit equation]].<ref>{{harvs|txt|last1=Bombieri|last2=Gubler|authorlink1=Enrico Bombieri|year=2006|loc1=pp. 176–230}}</ref>

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===Naive height===

''Classical'' or ''naive height'' is defined in terms of ordinary absolute value on [[homogeneous coordinates]]. It is typically a logarithmic scale and therefore can be viewed as being proportional to the "algebraic complexity" or number of [[bit]]s needed to store a point.<ref>{{harvs|txt|last1=Bombieri|last2=Gubler|authorlink1=Enrico Bombieri|yearname=2006|loc1=pp. 15–21}}<"ReferenceA"/ref> It is typically defined to be the [[logarithm]] of the maximum absolute value of the vector of coprime integers obtained by multiplying through by a [[lowest common denominator]]. This may be used to define height on a point in projective space over '''Q''', or of a polynomial, regarded as a vector of coefficients, or of an algebraic number, from the height of its minimal polynomial.<ref>{{harvs|txt|last1=Baker | authorlink1=Alan Baker (mathematician)|last2= Wüstholz | authorlink2=Gisbert Wüstholz|year=2007|loc1=p. 3}}</ref>

The naive height of a [[rational number]] ''x'' = ''p''/''q'' (in lowest terms) is

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===Néron–Tate height===

{{Main|Néron–Tate height}}

The ''Néron–Tate height'', or ''canonical height'', is a [[quadratic form]] on the [[Mordell–Weil group]] of [[rational points]] of an abelian variety defined over a [[global field]]. It is named after [[André Néron]], who first defined it as a sum of local heights,<ref>{{harvs|txt|last=Néron|authorlink=André Néron|year=1965}}</ref> and [[John Tate (mathematician)|John Tate]], who defined it globally in an unpublished work.<ref>{{harvs|txt|last=Lang|authorlink=Serge Lang|year=1997|page=72}}</ref>

===Weil height===

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==Height functions in algebra==

{{see also|Height (abelian group)|Height (ring theory)}}

===Height of a polynomial===

For a [[polynomial]] ''P'' of degree ''n'' given by

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*{{cite book | first1=Alan | last1=Baker | first2=Gisbert | last2= Wüstholz | author-link2=Gisbert Wüstholz | title=Logarithmic Forms and Diophantine Geometry | series=New Mathematical Monographs | volume=9 | publisher=[[Cambridge University Press]] | year=2007 | isbn=978-0-521-88268-2 | zbl=1145.11004 | page=3 }}

*{{cite book | first1=Enrico | last1=Bombieri | author-link1=Enrico Bombieri | first2=Walter | last2=Gubler | title=Heights in Diophantine Geometry | series=New Mathematical Monographs | volume=4 | publisher=[[Cambridge University Press]] | year=2006 | isbn=978-0-521-71229-3 | zbl=1130.11034 | doi=10.2277/0521846153 }}

*{{cite book | first=Peter | last=Borwein | author-link=Peter Borwein | title=Computational Excursions in Analysis and Number Theory | url=https://archive.org/details/computationalexc00borw | url-access=limited | series=CMS Books in Mathematics | publisher=[[Springer-Verlag]] | year=2002 | isbn=0-387-95444-9 | zbl=1020.12001 | pages=[https://archive.org/details/computationalexc00borw/page/n5 2], 3, 14148 }}

*{{cite book | first=Daniel | last=Bump| author-link1=Daniel Bump | title=Automorphic Forms and Representations | series=Cambridge Studies in Advanced Mathematics | volume=55 | publisher=Cambridge University Press | year=1998 | isbn=9780521658188 | page=300 }}

*{{cite book |title=Arithmetic geometry |last1=Cornell |first1=Gary |last2=Silverman | first2=Joseph H. |author-link2=Joseph H. Silverman |year=1986 |publisher=Springer |___location= New York |isbn=0387963111 }} → Contains an English translation of {{harvtxt|Faltings|1983}}

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==External links==

* [http://mathworld.wolfram.com/PolynomialHeight.html Polynomial height at Mathworld]