Geometry of numbers

Geometry of numbers is the part of number theory which uses geometry for the study of algebraic numbers. Typically, a ring of algebraic integers is viewed as a lattice in $\mathbb {R} ^{n},$ and the study of these lattices provides fundamental information on algebraic numbers.^[1] The geometry of numbers was initiated by Hermann Minkowski (1910).

The geometry of numbers has a close relationship with other fields of mathematics, especially functional analysis and Diophantine approximation, the problem of finding rational numbers that approximate an irrational quantity.^[2]

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Enrico Bombieri

& Walter Gubler (2006). Heights in Diophantine Geometry. Cambridge U. P.

Enrico Bombieri

. An Introduction to the Geometry of Numbers. Springer Classics in Mathematics, Springer-Verlag 1997 (reprint of 1959 and 1971 Springer-Verlag editions).

J. W. S. Cassels

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P. M. Gruber

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Lovász, L., A. Schrijver: Geometric Algorithms and Combinatorial Optimization, Springer, 1988

M. Grötschel

Hancock, Harris (1939). Development of the Minkowski Geometry of Numbers. Macmillan. (Republished in 1964 by Dover.)

Johannes Schoißengeier, Rudolf Taschner. Geometric and Analytic Number Theory. Universitext. Springer-Verlag, 1991.

Edmund Hlawka

; Peck, N. Tenney; Roberts, James W. (1984), An F-space sampler, London Mathematical Society Lecture Note Series, 89, Cambridge: Cambridge University Press, pp. xii+240, ISBN 0-521-27585-7, MR 0808777

Kalton, Nigel J.

. Geometry of Numbers. Wolters-Noordhoff, North Holland, Wiley. 1969.

C. G. Lekkerkererker

; Lenstra, H. W. Jr.; Lovász, L. (1982). "Factoring polynomials with rational coefficients" (PDF). Mathematische Annalen. 261 (4): 515–534. doi:10.1007/BF01457454. hdl:1887/3810. MR 0682664. S2CID 5701340.

Lenstra, A. K.

: An Algorithmic Theory of Numbers, Graphs, and Convexity, CBMS-NSF Regional Conference Series in Applied Mathematics 50, SIAM, Philadelphia, Pennsylvania, 1986

Lovász, L.

Malyshev, A.V. (2001) [1994], , Encyclopedia of Mathematics, EMS Press

"Geometry of numbers"

(1910), Geometrie der Zahlen, Leipzig and Berlin: R. G. Teubner, JFM 41.0239.03, MR 0249269, retrieved 2016-02-28

Minkowski, Hermann

. Diophantine approximation. Lecture Notes in Mathematics 785. Springer. (1980 [1996 with minor corrections])

Wolfgang M. Schmidt

(1996). Diophantine approximations and Diophantine equations. Lecture Notes in Mathematics. Vol. 1467 (2nd ed.). Springer-Verlag. ISBN 3-540-54058-X. Zbl 0754.11020.

Schmidt, Wolfgang M.

(1989). Lectures on the Geometry of Numbers. Springer-Verlag.

Siegel, Carl Ludwig

Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Cambridge University Press, Cambridge, 1993.

Anthony C. Thompson, Minkowski geometry, Cambridge University Press, Cambridge, 1996.

. Theory of reduction for arithmetical equivalence . Trans. Amer. Math. Soc. 48 (1940) 126–164. doi:10.1090/S0002-9947-1940-0002345-2

Hermann Weyl

Hermann Weyl. Theory of reduction for arithmetical equivalence. II . Trans. Amer. Math. Soc. 51 (1942) 203–231. :10.2307/1989946