Height function: Difference between revisions

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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. 2{{math|7}} for the coordinates {{math|(3/97, 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>

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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===

==Height functions in Diophantine geometry==

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}}

===History===

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>{{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 ''H'' of a [[rational number]] ''x'' = ''p''/''q'' (in lowest terms) is {{math|''H''(''x'') {{=}} log max(&#124;''p''&#124;, &#124;''q''&#124;)}}.<ref name="planetmath">{{PlanetMath |urlname=https://planetmath.org/canonicalheightonanellipticcurve |title=Canonical height on an elliptic curve }}</ref> Therefore, the naive height of {{math|4/10}} is {{math|log(5)}} for example.

* 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|y² {{=}} x² + Ax + B}} is defined to be {{math|''H(E)'' {{=}} log max(4&#124;''A''&#124;³, 27&#124;''B''&#124;²}}.<ref name="planetmath"/>

The naive height ''H'' of an [[elliptic curve]] ''E'' given by {{math|''y² {{=}} x²³ + Ax + B''}} is defined to be {{math|''H(E)'' {{=}} log max(4&#124;{{pipe}}''A''&#124;{{pipe}}³, 27&#124;{{pipe}}''B''&#124;{{pipe}}²)}}.<ref name="planetmath"/>

===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===

TheLet ''Weil heightX'' is defined onbe a [[projective variety]] ''X'' over a number field ''K''. equipped with a line bundleLet ''L'' on ''X''. Givenbe a [[Ample line bundle|very ample line bundle]] ''L₀'' on ''X'', one may define a height function using the naive height function ''h''. Since ''L₀''' is very ample, its complete linear system gives a map ''ϕ'' from ''X'' to projective space. Then for all points ''p'' on ''X'', define

One defines the ''Weil height'' on ''X'' with respect to ''L'' as follows.

<math>h_{L_0}(p) := h(\phi(p)).</math><ref name=Silverman/><ref name=Gubler/>

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)''.

OneIn maygeneral, writeone ancan arbitrary line bundlewrite ''L'' on ''X'' as the difference of two very ample line bundles ''L₁'' and ''L₂'' on ''X'', upand todefine [[Proj<math>h_{L} construction#The:= twistingh_{L_1} sheaf- of Serre|Serre's twisting sheaf]] ''O(1)''h_{L_2}, so one may define the Weil height ''h<sub>L</submath>'' on ''X'' with respect to ''L'' via

(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>

<math>h_{L} := h_{L_1} - h_{L_2},</math>

(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====

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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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====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>

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==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=20021998}}</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==

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*[[Lehmer conjecture#Elliptic analogues|Elliptic Lehmer conjecture]]

*[[Heath-Brown–Moroz constant]]

*[[Height of a formal group law]]

*[[Height zeta function]]

*[[Raynaud's isogeny theorem]]

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==Sources==

*{{cite journal| last1=Baker | first1=Alan | authorlinkauthor-link = Alan Baker (mathematician) | title=Linear forms in the logarithms of algebraic numbers. I | doi=10.1112/S0025579300003971 | mr=0220680 | year=1966 | journal=[[Mathematika. A Journal of Pure and Applied Mathematics]] | 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. A Journal of Pure and Applied Mathematics]] | 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. A Journal of Pure and Applied Mathematics]] | issn=0025-5793 | volume=14 | issue=2 | pages=220–228 }}

*{{cite book | first1=Alan | last1=Baker | first2=Gisbert | last2= Wüstholz | authorlink2author-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 | authorlink1author-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 | authorlinkauthor-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], 3,142,148 14148 }}

*{{cite book | first=Daniel | last=Bump| authorlink1author-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 |lastlast1=Cornell |firstfirst1=Gary |author2last2=Silverman, | first2=Joseph H. |authorlink2author-link2=Joseph H. Silverman |year=1986 |publisher=Springer |___location= New York |isbn=0387963111 |pages= }} → 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=Germande | ref=harv}}

*{{cite journal |last1=Faltings | first1=Gerd | author1linkauthor1-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 | refjstor=harv2944319 }}

*{{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 | authorlinkauthor-link=Kurt Mahler | title=On two extremum properties of polynomials | journal=[[Illinois J.Journal Math.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 | authorlinkauthor-link=André Néron | title=Quasi-fonctions et hauteurs sur les variétés abéliennes | journal=[[Ann.Annals of Math.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 | authorlinkauthor-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. | authorlinkauthor-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 | refjstor=harv1970824 }}

*{{cite book | first=Serge | last=Lang | authorlinkauthor-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|authorlinkauthor-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 |authorlast=Silverman, |first=Joseph H. |authorlinkauthor-link=Joseph H. Silverman |year=1994|publisher=Springer |___location= New York |isbn=978-1-4612-0851-8 }}

*{{cite book | last1=Vojta | first1=Paul | author1linkauthor1-link=Paul Vojta | title=Diophantine approximationsApproximations and valueValue distributionDistribution theoryTheory | 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 | ref=harv}}

*{{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]

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

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