Complex homogeneous linear forms

1956 ◽  
Vol 52 (1) ◽  
pp. 35-38 ◽  
Author(s):  
K. Rogers
Keyword(s):  
Complex Space ◽  
Rational Numbers ◽  
Quadratic Equation ◽  
Complex Numbers ◽  
Geometry Of Numbers ◽  
Systematic Method ◽  
Linear Forms ◽  
Rings Of Integers ◽  
Image Position ◽  
Homogeneous Linear

Let Z, Q, C denote respectively the ring of rational integers, the field of rational numbers and the field of complex numbers. Minkowski (4) solved the problem of minimizingfor x, y ∈ Z(i) or Z(ρ), where a, b, c, d ∈ C have fixed determinant Δ ≠ 0. Here ρ = exp 2/3πi, and Z(i) and Z(p) are the rings of integers in Q(i) and Q(ρ) respectively. In fact he found the best possible resultsfor Z(i), andfor Z(ρ), wherewhile Buchner (1) used Minkowski's method to show thatfor Z(i√2). Hlawka(3) has also proved (1·2), and Cassels, Ledermann and Mahler (2) have proved both (1·2) and (1·3). In a paper being prepared jointly by H. P. F. Swinnerton-Dyer and the author, general problems of the geometry of numbers in complex space are discussed and a systematic method given for solving the above problem for all complex quadratic fields Q(ϑ). Here, ϑ is a non-real number satisfying. an irreduc7ible quadratic equation with rational coefficients. The above problem is solved in detail for Q(i√5), for whichand the ‘critical forms’ can be reduced to

On the product of n linear forms

1953 ◽  
Vol 49 (2) ◽  
pp. 190-193 ◽  
Author(s):  
H. Davenport
Keyword(s):  
Complex Conjugate ◽  
Linear Forms ◽  
Image Position ◽  
Complex Coefficients ◽  
Homogeneous Linear

Let L1, …, Ln be n homogeneous linear forms in n variables u1, …, un, with non-zero determinant Δ. Suppose that L1, …, Lr have real coefficients, that Lr+1, …, Lr+s have complex coefficients, and that the form Lr+s+j is the complex conjugate of the form Lr+j for j = 1, …, s, where r + 2s = n. Letfor integral u1, …, un, not all zero. For any n numbers α1, …, αn of the same ‘type’ as the forms L1, …, Ln (that is, α1, …, αr real, αr+1, …, αr+s complex, αr+s+j = ᾱr+j), let

On the product of three homogeneous linear forms

1951 ◽  
Vol 47 (2) ◽  
pp. 251-259 ◽  
Author(s):  
J. H. H. Chalk ◽  
C. A. Rogers
Keyword(s):  
Dimensional Space ◽  
General Point ◽  
Linear Forms ◽  
3 Dimensional ◽  
Image Position ◽  
Homogeneous Linear

Let X denote the general point with coordinates (x1, x2, x3) in 3-dimensional space; and let P(X) be the function defined by

On the product of three non-homogeneous linear forms

1947 ◽  
Vol 43 (2) ◽  
pp. 137-152 ◽  
Author(s):  
H. Davenport
Keyword(s):  
Real Numbers ◽  
Linear Forms ◽  
Image Position ◽  
The Right ◽  
Homogeneous Linear

Let ξ, η, ζ be linear forms in u, v, w with real coefficients and determinant Δ ≠ 0. A conjecture of Minkowski, which was subsequently proved by Remak, tells us that for any real numbers a, b, c there exist integral values of u, v, w for whichand the constant ⅛ on the right is best possible.

On the Product of Three Homogeneous Linear Forms. IV

1943 ◽  
Vol 39 (1) ◽  
pp. 1-21 ◽  
Author(s):  
H. Davenport
Keyword(s):  
Lower Bound ◽  
Real Number ◽  
Linear Transformation ◽  
Linear Forms ◽  
Special Linear ◽  
Absolute Value ◽  
Cubic Equation ◽  
Image Position ◽  
Homogeneous Linear

Let L1, L2, L3 be three homogeneous linear forms in u, v, w with real coefficients and determinant 1. Let M denote the lower bound offor integral values of u, v, w, not all zero. I proved a few years ago (1) thatmore precisely, thatexcept when L1, L2, L3 are of a special type, in which case If we denote by θ, ø, ψ the roots of the cubic equation t3+t2-2t-1 = 0, the special linear forms are equivalent, by an integral unimodular linear transformation, to(in any order), where λ1,λ2,λ3 are real number whose product is In this case, L1L2L3|λ1λ2λ3 is a non-zero integer, and the minimum of its absolute value is 1, giving

On the product of three homogeneous linear forms and indefinite ternary quadratic forms

1955 ◽  
Vol 248 (940) ◽  
pp. 73-96 ◽  
Keyword(s):  
Lower Bound ◽  
Quadratic Forms ◽  
Type A ◽  
Strong Type ◽  
Geometry Of Numbers ◽  
Linear Forms ◽  
Integer Coefficients ◽  
Ternary Quadratic Forms ◽  
Cubic Forms ◽  
Homogeneous Linear

Isolation theorems for the minima of factorizable homogeneous ternary cubic forms and of indefinite ternary quadratic forms of a new strong type are proved. The problems whether there exist such forms with positive minima other than multiples of forms with integer coefficients are shown to be equivalent to problems in the geometry of numbers of a superficially different type. A contribution is made to the study of the problem whether there exist real <j>, ijr such that x(f>x—y | y[rx — z | has a positive lower bound for all integers x > 0, y , z . The methods used have wide validity.

Note on non-homogeneous linear forms

1953 ◽  
Vol 49 (2) ◽  
pp. 360-362 ◽  
Author(s):  
E. S. Barnes
Keyword(s):  
Algebraic Number ◽  
Linear Forms ◽  
Image Position ◽  
Homogeneous Linear

Let θ be an algebraic number of degree n, with conjugates θ(1), …, θ(n), where θ(1), …, θ(r) are real and θ(r+j), θ(r+s+j) are complex conjugates for j = 1, …, s. [Here r ≥ 0, s ≥ 0, r + 2s = n.] Let ω1, …, ωn be a basis for the integers of k(θ), and set

scholarly journals The Packing of Spheres in the Space lp

1958 ◽  
Vol 4 (1) ◽  
pp. 22-25 ◽  
Author(s):  
Jane A. C. Burlak ◽  
R. A. Rankin ◽  
A. P. Robertson
Keyword(s):  
Infinite Number ◽  
Unit Sphere ◽  
Complex Space ◽  
Infinite Sequence ◽  
Complex Numbers ◽  
The Real ◽  
The Space Lp ◽  
Fixed Radius ◽  
Image Position

A point x in the real or complex space lpis an infinite sequence,(x1, x2, x3,…) of real or complex numbers such that is convergent. Here p ≥ 1 and we writeThe unit sphere S consists of all points x ε lp for which ¶ x ¶ ≤ 1. The sphere of radius a≥ ≤ 0 and centre y is denoted by Sa(y) and consists of all points x ε lp such that ¶ x - y ¶ ≤ a. The sphere Sa(y) is contained in S if and only if ¶ y ¶≤1 - a, and the two spheres Sa(y) and Sa(z) do not overlap if and only if¶ y- z ¶≥ 2aThe statement that a finite or infinite number of spheres Sa (y) of fixed radius a can be packed in S means that each sphere Sa (y) is contained in S and that no two such spheres overlap.

On lower estimates for linear forms involving certain transcendental numbers

1976 ◽  
Vol 14 (2) ◽  
pp. 161-179 ◽  
Author(s):  
Keijo Väänänen
Keyword(s):  
Rational Numbers ◽  
Linear Forms ◽  
Transcendental Numbers ◽  
Image Position

Letwhere λ is rational and not an integer. The author investigates lower estimates for example forwhere the αi are distinct rational numbers not 0, and where x1, …, xk, are integers and

Yet another proof of Minkowski's theorem on the product of two inhomogeneous linear forms

1953 ◽  
Vol 49 (2) ◽  
pp. 365-366 ◽  
Author(s):  
J. W. S. Cassels
Keyword(s):  
Linear Forms ◽  
Inhomogeneous Linear Forms ◽  
Image Position ◽  
Homogeneous Linear

Theorem 1. Let ξ = αx + βy, η = γx + δy be two homogeneous linear forms in x, y with real coefficients and determinant αδ − βγ = Δ ≠ 0. Then for any real constants p, q there are integers x, y such that

A ‘Waring's problem’ for homogeneous forms

1969 ◽  
Vol 65 (3) ◽  
pp. 663-672 ◽  
Author(s):  
W. J. Ellison
Keyword(s):  
Positive Definite ◽  
Necessary Condition ◽  
Complex Numbers ◽  
Linear Forms ◽  
Waring’S Problem ◽  
Waring's Problem ◽  
Homogeneous Form ◽  
Explicit Bound ◽  
Image Position

Professor H. Davenport has raised the following question. Let f(X) be a homogeneous form in n variables X, of degree k, with coefficients in a field K. Is it possible to write f(X) in the formwhere the Li(X) are linear forms over K and N depends only on K, n, and k? In particular, what can be said when K = R, the reals; K = Q, the rationals; K = C, the complex numbers ?We shall show that when k is odd such a representation is always possible and we shall obtain an explicit bound for N. However, if k is even, no such result is possible when K = R or Q, even when we impose the obvious necessary condition that f(X) be positive semi-definite. In fact we shall exhibit a construction for obtaining forms f(X) which are positive definite but cannot be expressed in the formr even, where φi(X) is a homogeneous form of degree s and sr = k. We shall determine precisely which definite forms can be expressed in the required form. Some natural generalizations of the problem are then considered.

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