“Height function”的意思、由来-开放百科全书

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.^[1]

Significance

Height functions allow mathematicians to count objects 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. 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.^[2]

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.^[3]^[4]

Height functions in Diophantine geometry

History

Heights in Diophantine geometry were initially developed by André Weil and Douglas Northcott beginning in the 1920s.^[5] 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.^[6] In 1983, Faltings developed his theory of Faltings heights in his proof of Faltings's theorem.^[7]

Naive height

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.^[9]

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 and John Tate.

Weil height

The Weil height is defined on a projective variety X over a number field K equipped with a line bundle L on X. Given a 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 may write an arbitrary line bundle L on X as the difference of two very ample line bundles L₁ and L₂ on X, up to Serre's twisting sheaf O(1), so one may define the Weil height h_L on X with respect to L via

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.^[12]^[13] It is the usual Weil height equipped with a different metric.^[14]

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

the height H(P) is defined to be the maximum of the magnitudes of its coefficients:^[15]

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

Relation to Mahler measure

The Mahler measure M(P) of P is also a measure of the complexity of P.^[16] The three functions H(P), L(P) and M(P) are related by the inequalities

See also

References

1. ^{{harvs|txt|last=Lang|authorlink=Serge Lang|year=1997|loc1=pp. 43–67}}
2. ^{{harvs|txt|last1=Bombieri|last2=Gubler|authorlink1=Enrico Bombieri|year=2006|loc1=pp. 176–230}}
3. ^{{harvs|txt|last1=Vojta|authorlink=Paul Vojta|year=1987}}
4. ^{{harvs|txt|last1=Faltings|authorlink1=Gerd Faltings|year=1991}}
5. ^{{harvs|txt||last=Weil|authorlink=André Weil|year=1929}}
6. ^{{harvs|txt|last=Lang|authorlink=Serge Lang|year=1988}}
7. ^{{harvs|txt|last=Faltings|authorlink=Gerd Faltings|year=1983}}
8. ^{{harvs|txt|last1=Bombieri|last2=Gubler|authorlink1=Enrico Bombieri|year=2006|loc1=pp. 15–21}}
9. ^{{harvs|txt|last1=Baker | authorlink1=Alan Baker (mathematician)|last2= Wüstholz | authorlink2=Gisbert Wüstholz|year=2007|loc1=p. 3}}
10. ^{{harvs|txt|last=Silverman|authorlink=Joseph H. Silverman|year=1994|loc1=III.10}}
11. ^{{harvs|txt|last1=Bombieri|last2=Gubler|authorlink1=Enrico Bombieri|year=2006|loc1=Sections 2.2–2.4}}
12. ^{{harvs|txt|last1=Bombieri|last2=Gubler|authorlink1=Gerd Faltings|year=2006|loc1=pp. 66–67}}
13. ^{{harvs|txt|last=Lang|authorlink=Serge Lang|year=1988|loc1=pp. 156–157}}
14. ^{{harvs|txt|last1=Fili|last2=Petsche|last3=Pritsker|year=2017|loc1=p. 441}}
15. ^{{harvs|txt|last=Borwein|authorlink=Peter Borwein|year=2002}}
16. ^{{harvs|txt|last=Mahler|authorlink=Kurt Mahler|year=1963}}