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{{short description|Mathematical functions that quantify complexity}}
{{About|mathematical functions that quantify complexity|other uses of height|Height (disambiguation)}}
A '''height function''' is a [[function (mathematics)|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 variety|algebraic varieties]] (or a set of algebraic varieties) to the [[real numbers]].<ref>{{harvs|txt|last=Lang|authorlink=Serge Lang|year=1997|loc1=pp. 43–67}}</ref>

For instance, the ''classical'' or ''naive height'' over the [[rational number]]s is typically defined to be the maximum of the numerators and denominators of the coordinates (e.g. {{math|7}} for the coordinates {{math|(3/7, 1/2)}}), but in a [[logarithmic scale]].

{{TOC limit|3}}

==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>

Height functions were crucial to the proofs of the [[Mordell–Weil theorem]] and [[Faltings's theorem]] by {{harvs|txt||last=Weil|authorlink=André Weil|year=1929}} and {{harvs|txt|last=Faltings|authorlink=Gerd Faltings|year=1983}} 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 equation]]s, [[arithmetic geometry]], and [[mathematical logic]].<ref>{{harvs|txt|last1=Vojta|authorlink=Paul Vojta|year=1987}}</ref><ref>{{harvs|txt|last1=Faltings|authorlink1=Gerd Faltings|year=1991}}</ref>

==History==
An early form of height function was proposed by [[Giambattista Benedetti]] (c. 1563), who argued that the [[consonance]] of a [[interval (music)|musical interval]] could be measured by the product of its numerator and denominator (in reduced form); see {{slink|Giambattista Benedetti|Music}}.{{cn|date=November 2022}}

Heights in Diophantine geometry were initially developed by [[André Weil]] and [[Douglas Northcott]] beginning in the 1920s.<ref>{{harvs|txt||last=Weil|authorlink=André Weil|year=1929}}</ref> Innovations in 1960s were the [[Néron–Tate height]] and the realization that heights were linked to projective representations in much the same way that [[ample line bundle]]s are in other parts of [[algebraic geometry]]. In the 1970s, [[Suren Arakelov]] developed Arakelov heights in [[Arakelov theory]].<ref>{{harvs|txt|last=Lang|authorlink=Serge Lang|year=1988}}</ref> In 1983, Faltings developed his theory of Faltings heights in his proof of Faltings's theorem.<ref>{{harvs|txt|last=Faltings|authorlink=Gerd Faltings|year=1983}}</ref>

==Height functions in Diophantine geometry==

===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 name="ReferenceA"/> 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 
* multiplicative height <math> H(p/q) = \max\{|p|,|q|\}</math>
* logarithmic height: <math> h(p/q) = \log H (p/q)</math><ref>[https://mathoverflow.net/q/203852 mathoverflow question: average-height-of-rational-points-on-a-curve]</ref>

Therefore, the naive multiplicative and logarithmic heights of {{math|4/10}} are {{math|5}} and {{math|log(5)}}, for example.

The naive height ''H'' of an [[elliptic curve]] ''E'' given by {{math|''y<sup>2</sup> {{=}} x<sup>3</sup> + Ax + B''}} is defined to be {{math|''H(E)'' {{=}} log max(4{{pipe}}''A''{{pipe}}<sup>3</sup>, 27{{pipe}}''B''{{pipe}}<sup>2</sup>)}}.

===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===
Let ''X'' be a [[projective variety]] over a number field ''K''.  Let ''L'' be a line bundle on ''X''.
One defines the ''Weil height'' on ''X'' with respect to ''L'' as follows.

First, suppose that ''L'' is [[Ample line bundle|very ample]].  A choice of basis of the space <math>\Gamma(X,L)</math> of global sections defines a morphism ''ϕ'' from ''X'' to projective space, and for all points ''p'' on ''X'', one defines
<math>h_L(p) := h(\phi(p))</math>, where ''h'' is the naive height on projective space.<ref name=Silverman/><ref name=Gubler/>  For fixed ''X'' and ''L'', choosing a different basis of global sections changes <math>h_L</math>, but only by a bounded function of ''p''.  Thus <math>h_L</math> is well-defined up to addition of a function that is ''O(1)''.

In general, one can write ''L'' as the difference of two very ample line bundles ''L<sub>1</sub>'' and ''L<sub>2</sub>'' on ''X'' and define <math>h_{L} := h_{L_1} - h_{L_2},</math>
which again is well-defined up to ''O(1)''.<ref name=Silverman>{{harvs|txt|last=Silverman|authorlink=Joseph H. Silverman|year=1994|loc1=III.10}}</ref><ref name=Gubler>{{harvs|txt|last1=Bombieri|last2=Gubler|authorlink1=Enrico Bombieri|year=2006|loc1=Sections 2.2–2.4}}</ref>

====Arakelov height====
The ''Arakelov height'' on a projective space over the field of algebraic numbers is a global height function with local contributions coming from [[Fubini–Study metric]]s on the [[Archimedean field]]s and the usual metric on the [[non-Archimedean field]]s.<ref>{{harvs|txt|last1=Bombieri|last2=Gubler|authorlink1=Gerd Faltings|year=2006|loc1=pp. 66–67}}</ref><ref>{{harvs|txt|last=Lang|authorlink=Serge Lang|year=1988|loc1=pp. 156–157}}</ref> It is the usual Weil height equipped with a different metric.<ref>{{harvs|txt|last1=Fili|last2=Petsche|last3=Pritsker|year=2017|loc1=p. 441}}</ref>

===Faltings height===
The ''Faltings height'' of an [[abelian variety]] defined over a [[number field]] is a measure of its arithmetic complexity. It is defined in terms of the height of a [[Hermitian form|metrized]] [[line bundle]]. It was introduced by {{harvs|txt|last=Faltings|authorlink=Gerd Faltings|year=1983}} in his proof of the [[Mordell conjecture]].

==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

:<math>P = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n , </math>

the '''height''' ''H''(''P'') is defined to be the maximum of the magnitudes of its coefficients:<ref>{{harvs|txt|last=Borwein|authorlink=Peter Borwein|year=2002}}</ref>

:<math>H(P) = \underset{i}{\max} \,|a_i|. </math>

One could similarly define the '''length''' ''L''(''P'') as the sum of the magnitudes of the coefficients:

:<math>L(P) = \sum_{i=0}^n |a_i|.</math>

====Relation to Mahler measure====
The [[Mahler measure]] ''M''(''P'')  of ''P'' is also a measure of the complexity of ''P''.<ref>{{harvs|txt|last=Mahler|authorlink=Kurt Mahler|year=1963}}</ref> The three functions ''H''(''P''), ''L''(''P'') and ''M''(''P'') are related by the [[inequality (mathematics)|inequalities]]

:<math>\binom{n}{\lfloor n/2 \rfloor}^{-1} H(P) \le M(P) \le H(P) \sqrt{n+1} ; </math>

:<math>L(p) \le 2^n M(p) \le 2^n L(p) ; </math>

:<math>H(p) \le L(p) \le (n+1) H(p) </math>

where <math>\scriptstyle \binom{n}{\lfloor n/2 \rfloor}</math> is the [[binomial coefficient]].

==Height functions in automorphic forms==
One of the conditions in the definition of an [[automorphic form]] on the [[general linear group]] of an [[adelic algebraic group]] is ''moderate growth'', which is an asymptotic condition on the growth of a height function on the general linear group viewed as an [[affine variety]].<ref>{{harvs|txt|last=Bump|authorlink=Daniel Bump|year=1998}}</ref>

==Other height functions==
The height of an irreducible [[rational number]] ''x'' = ''p''/''q'', ''q'' > 0 is <math>|p|+q</math> (this function is used for constructing a [[bijection]] between <math>\mathbb{N}</math> and <math>\mathbb{Q}</math>).<ref>{{harvs|txt|last1=Kolmogorov | authorlink1=Andrey Kolmogorov |last2= Fomin | authorlink2=Sergei Fomin|year=1957| loc1=p. 5}}</ref>

==See also==
*[[abc conjecture]]
*[[Birch and Swinnerton-Dyer conjecture]]
*[[Lehmer conjecture#Elliptic analogues|Elliptic Lehmer conjecture]]
*[[Heath-Brown–Moroz constant]]
*[[Height of a formal group law]]
*[[Height zeta function]]
*[[Raynaud's isogeny theorem]]

==References==
{{reflist|30em}}

==Sources==
*{{cite journal| last1=Baker | first1=Alan | author-link = Alan Baker (mathematician) | title=Linear forms in the logarithms of algebraic numbers. I | doi=10.1112/S0025579300003971 | mr=0220680 | year=1966 | journal=[[Mathematika]] | issn=0025-5793 | volume=13 | issue=2 | pages=204–216 }}
*{{cite journal| last1=Baker | first1=Alan | title=Linear forms in the logarithms of algebraic numbers. II | doi=10.1112/S0025579300008068 | mr=0220680 | year=1967a | journal=[[Mathematika]] | issn=0025-5793 | volume=14 | pages=102–107 }}
*{{cite journal| last1=Baker | first1=Alan | title=Linear forms in the logarithms of algebraic numbers. III | doi=10.1112/S0025579300003843  | mr=0220680 | year=1967b | journal=[[Mathematika]] | issn=0025-5793 | volume=14 | issue=2 | pages=220–228 }}
*{{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 }}
*{{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}}
*{{cite journal |last=Faltings |first=Gerd |year=1983 |title=Endlichkeitssätze für abelsche Varietäten über Zahlkörpern |journal=Inventiones Mathematicae |volume=73 |issue=3 |pages=349&ndash;366 |doi=10.1007/BF01388432 |bibcode=1983InMat..73..349F | mr=0718935 |s2cid=121049418 | trans-title=Finiteness theorems for abelian varieties over number fields | language=de }}
*{{cite journal |last1=Faltings | first1=Gerd | author1-link=Gerd Faltings | title=Diophantine approximation on abelian varieties | journal=Annals of Mathematics | mr=1109353| year=1991 | volume=123 | pages=549–576 | doi=10.2307/2944319 | issue=3 | jstor=2944319 }}
*{{cite journal|title=Energy integrals and small points for the Arakelov height|journal=Archiv der Mathematik|last1=Fili|first1=Paul|last2=Petsche|first2=Clayton|last3=Pritsker|first3=Igor|volume=109|issue=5|year=2017|pages=441–454 |doi=10.1007/s00013-017-1080-x|arxiv=1507.01900|s2cid=119161942}}
*{{cite journal | first=K. | last=Mahler | author-link=Kurt Mahler | title=On two extremum properties of polynomials | journal=[[Illinois Journal of Mathematics]] | volume=7 | pages=681–701 | year= 1963 | issue=4 | zbl=0117.04003 | doi=10.1215/ijm/1255645104| doi-access=free }}
*{{cite journal | first=André | last=Néron | author-link=André Néron | title=Quasi-fonctions et hauteurs sur les variétés abéliennes | journal=[[Annals of Mathematics]] | volume=82 | year=1965 | issue=2 | pages=249–331 | doi=10.2307/1970644 | jstor=1970644 | mr=0179173 | language=fr }}
*{{cite book | last=Schinzel | first=Andrzej | author-link=Andrzej Schinzel | title=Polynomials with special regard to reducibility | zbl=0956.12001 | series=Encyclopedia of Mathematics and Its Applications | volume=77 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2000 | isbn=0-521-66225-7 | page=[https://archive.org/details/polynomialswiths0000schi/page/212 212] | url=https://archive.org/details/polynomialswiths0000schi/page/212 }}
*{{cite journal | last1=Schmidt | first1=Wolfgang M. | author-link=Wolfgang M. Schmidt | title=Norm form equations | mr=0314761 | year=1972 | journal=[[Annals of Mathematics]] |series=Second Series | volume=96 | pages=526–551 | issue=3 | doi=10.2307/1970824 | jstor=1970824 }}
*{{cite book | first=Serge | last=Lang | author-link=Serge Lang | title=Introduction to Arakelov theory | publisher=[[Springer-Verlag]] | place=New York | year=1988 | isbn=0-387-96793-1 | mr=0969124 | zbl=0667.14001 }}
*{{cite book | first=Serge | last=Lang | title=Survey of Diophantine Geometry | publisher=[[Springer-Verlag]] | year=1997 | isbn=3-540-61223-8 | zbl=0869.11051 }}
*{{cite journal|last=Weil|first=André|author-link=André Weil|title=L'arithmétique sur les courbes algébriques|journal=[[Acta Mathematica]]|volume=52|year=1929|pages=281–315|issue=1|doi=10.1007/BF02592688|mr=1555278 |doi-access=free}}
*{{cite book |title=Advanced Topics in the Arithmetic of Elliptic Curves |last=Silverman |first=Joseph H. |author-link=Joseph H. Silverman |year=1994|publisher=Springer |location= New York |isbn=978-1-4612-0851-8 }}
*{{cite book | last1=Vojta | first1=Paul | author1-link=Paul Vojta | title=Diophantine approximations and value distribution theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics | isbn=978-3-540-17551-3 | doi=10.1007/BFb0072989 | zbl=0609.14011 | mr=883451 | year=1987 | volume=1239 }}
*{{cite book | first1=Andrey | last1=Kolmogorov | author-link1=Andrey Kolmogorov | first2=Sergei | last2= Fomin | author-link2=Sergei Fomin | title=Elements of the Theory of Functions and Functional Analysis |location= New York | publisher=Graylock Press | year=1957}}

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

[[Category:Polynomials]]
[[Category:Abelian varieties]]
[[Category:Elliptic curves]]
[[Category:Diophantine geometry]]
[[Category:Algebraic number theory]]
[[Category:Abstract algebra]]