Height function#Naive height

{{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 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.{{harvs|txt|last=Lang|authorlink=Serge Lang|year=1997|loc1=pp. 43–67}}

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. {{math|7}} for the coordinates {{math|(3/7, 1/2)}}), but in a logarithmic scale.

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.{{harvs|txt|last1=Bombieri|last2=Gubler|authorlink1=Enrico Bombieri|year=2006|loc1=pp. 15–21}} 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 {{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 hyperplanes and generalizes Siegel's theorem on integral points and solution of the S-unit equation.{{harvs|txt|last1=Bombieri|last2=Gubler|authorlink1=Enrico Bombieri|year=2006|loc1=pp. 176–230}}

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 equations, arithmetic geometry, and mathematical logic.{{harvs|txt|last1=Vojta|authorlink=Paul Vojta|year=1987}}{{harvs|txt|last1=Faltings|authorlink1=Gerd Faltings|year=1991}}

History

An early form of height function was proposed by Giambattista Benedetti (c. 1563), who argued that the consonance of a 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.{{harvs|txt||last=Weil|authorlink=André Weil|year=1929}} 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 bundles are in other parts of algebraic geometry. In the 1970s, Suren Arakelov developed Arakelov heights in Arakelov theory.{{harvs|txt|last=Lang|authorlink=Serge Lang|year=1988}} In 1983, Faltings developed his theory of Faltings heights in his proof of Faltings's theorem.{{harvs|txt|last=Faltings|authorlink=Gerd Faltings|year=1983}}

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 bits needed to store a point. 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.{{harvs|txt|last1=Baker | authorlink1=Alan Baker (mathematician)|last2= Wüstholz | authorlink2=Gisbert Wüstholz|year=2007|loc1=p. 3}}

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

multiplicative height $H(p/q) = \max\{|p|,|q|\}$
logarithmic height: $h(p/q) = \log H (p/q)$ [https://mathoverflow.net/q/203852 mathoverflow question: average-height-of-rational-points-on-a-curve]

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 {{=}} x³ + Ax + B}} is defined to be {{math|H(E) {{=}} log max(4{{pipe}}A{{pipe}}³, 27{{pipe}}B{{pipe}}²)}}.

=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,{{harvs|txt|last=Néron|authorlink=André Néron|year=1965}} and John Tate, who defined it globally in an unpublished work.{{harvs|txt|last=Lang|authorlink=Serge Lang|year=1997|page=72}}

=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 very ample. A choice of basis of the space $\Gamma(X,L)$ of global sections defines a morphism ϕ from X to projective space, and for all points p on X, one defines

$h_L(p) := h(\phi(p))$ , where h is the naive height on projective space. For fixed X and L, choosing a different basis of global sections changes $h_L$ , but only by a bounded function of p. Thus $h_L$ 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 L₁ and L₂ on X and define $h_{L} := h_{L_1} - h_{L_2},$

which again is well-defined up to O(1).{{harvs|txt|last=Silverman|authorlink=Joseph H. Silverman|year=1994|loc1=III.10}}{{harvs|txt|last1=Bombieri|last2=Gubler|authorlink1=Enrico Bombieri|year=2006|loc1=Sections 2.2–2.4}}

==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 metrics on the Archimedean fields and the usual metric on the non-Archimedean fields.{{harvs|txt|last1=Bombieri|last2=Gubler|authorlink1=Gerd Faltings|year=2006|loc1=pp. 66–67}}{{harvs|txt|last=Lang|authorlink=Serge Lang|year=1988|loc1=pp. 156–157}} It is the usual Weil height equipped with a different metric.{{harvs|txt|last1=Fili|last2=Petsche|last3=Pritsker|year=2017|loc1=p. 441}}

=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 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

=Height of a polynomial=

For a polynomial P of degree n given by

: $P = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n ,$

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

: $H(P) = \underset{i}{\max} \,|a_i|.$

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

: $L(P) = \sum_{i=0}^n |a_i|.$

==Relation to Mahler measure==

The Mahler measure M(P) of P is also a measure of the complexity of P.{{harvs|txt|last=Mahler|authorlink=Kurt Mahler|year=1963}} The three functions H(P), L(P) and M(P) are related by the inequalities

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

: $L(p) \le 2^n M(p) \le 2^n L(p) ;$

: $H(p) \le L(p) \le (n+1) H(p)$

where $\scriptstyle \binom{n}{\lfloor n/2 \rfloor}$ 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.{{harvs|txt|last=Bump|authorlink=Daniel Bump|year=1998}}

Other height functions

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

References

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=349–366 |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}}