Height function

This article is about mathematical functions that quantify complexity. For other uses of height, see 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.

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 Alan Baker (1966, 1967a, 1967b).

In other cases, height functions can distinguish some objects based on their complexity. For instance, the subspace theorem proved by Wolfgang M. Schmidt (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.

Height functions were crucial to the proofs of the Mordell–Weil theorem and Faltings's theorem. 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.

Height functions in Diophantine geometry

History

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

Naive height

Classical or naive height is defined in terms of ordinary absolute value on homogeneous coordinates: it is now usual to take a logarithmic scale, that is, height is 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.

Néron–Tate height

Main article: 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.

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. It is prominently used in Arakelov theory and named after Suren Arakelov.

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 Faltings (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:

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

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. The three functions H(P), L(P) and M(P) are related by the inequalities

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

$L(p)\leq 2^{n}M(p)\leq 2^{n}L(p);$

$H(p)\leq L(p)\leq (n+1)H(p)$

where $\scriptstyle {\binom {n}{\lfloor n/2\rfloor }}$ is the binomial coefficient.

See also

• Baker's theorem
• Heath-Brown–Moroz constant
• Manin conjecture
• Raynaud's isogeny theorem
• Subspace theorem
• Vojta's conjecture

References

• Baker, Alan (1966), "Linear forms in the logarithms of algebraic numbers. I", Mathematika. A Journal of Pure and Applied Mathematics, 13: 204–216, doi:10.1112/S0025579300003971, ISSN 0025-5793, MR 0220680
• Baker, Alan (1967a), "Linear forms in the logarithms of algebraic numbers. II", Mathematika. A Journal of Pure and Applied Mathematics, 14: 102–107, doi:10.1112/S0025579300008068, ISSN 0025-5793, MR 0220680
• Baker, Alan (1967b), "Linear forms in the logarithms of algebraic numbers. III", Mathematika. A Journal of Pure and Applied Mathematics, 14: 220–228, doi:10.1112/S0025579300003843, ISSN 0025-5793, MR 0220680
• Baker, Alan; Wüstholz, Gisbert (2007). Logarithmic Forms and Diophantine Geometry. New Mathematical Monographs. Vol. 9. Cambridge University Press. p. 3. ISBN 978-0-521-88268-2. Zbl 1145.11004.
• Bombieri, Enrico; Gubler, Walter (2006). Heights in Diophantine Geometry. New Mathematical Monographs. Vol. 4. Cambridge University Press. doi:10.2277/0521846153. ISBN 978-0-521-71229-3. Zbl 1130.11034.
• Borwein, Peter (2002). Computational Excursions in Analysis and Number Theory. CMS Books in Mathematics. Springer-Verlag. pp. 2, 3, 142, 148. ISBN 0-387-95444-9. Zbl 1020.12001.
• Cornell, Gary; Silverman, Joseph H. (1986). Arithmetic geometry. New York: Springer. ISBN 0387963111. → Contains an English translation of Faltings (1983)
• Faltings, Gerd (1983). "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern" [Finiteness theorems for abelian varieties over number fields]. Inventiones Mathematicae (in German). 73 (3): 349–366. doi:10.1007/BF01388432. MR 0718935. {{cite journal}}: Invalid |ref=harv (help)
• Faltings, Gerd (1991). "Diophantine approximation on abelian varieties". Annals of Mathematics. 123 (3): 549–576. doi:10.2307/2944319. MR 1109353. {{cite journal}}: Invalid |ref=harv (help)
• Mahler, K. (1963). "On two extremum properties of polynomials". Illinois J. Math. 7: 681–701. Zbl 0117.04003.
• Schinzel, Andrzej (2000). Polynomials with special regard to reducibility. Encyclopedia of Mathematics and Its Applications. Vol. 77. Cambridge: Cambridge University Press. p. 212. ISBN 0-521-66225-7. Zbl 0956.12001.
• Schmidt, Wolfgang M. (1972). "Norm form equations". Annals of Mathematics. Second Series. 96 (3): 526–551. doi:10.2307/1970824. MR 0314761. {{cite journal}}: Invalid |ref=harv (help)
• Lang, Serge (1988). Introduction to Arakelov theory. New York: Springer-Verlag. ISBN 0-387-96793-1. MR 0969124. Zbl 0667.14001.
• Lang, Serge (1997). Survey of Diophantine Geometry. Springer-Verlag. ISBN 3-540-61223-8. Zbl 0869.11051.
• Weil, André (1929). "L'arithmétique sur les courbes algébriques". Acta Mathematica. Vol. 52, no. 1. pp. 281–315. doi:10.1007/BF02592688. MR 1555278.
• Vojta, Paul (1987). Diophantine approximations and value distribution theory. Lecture Notes in Mathematics. Vol. 1239. Berlin, New York: Springer-Verlag. doi:10.1007/BFb0072989. ISBN 978-3-540-17551-3. MR 0883451. Zbl 0609.14011. {{cite book}}: Invalid |ref=harv (help)

External links

• Polynomial height at Mathworld