The product of non-homogeneous linear forms. II The minimum of a class of non-homogeneous linear forms

1954 ◽  
Vol 50 (3) ◽  
pp. 380-390 ◽  
Author(s):  
P. A. Samet
Keyword(s):  
Linear Forms ◽  
Cubic Fields ◽  
Homogeneous Linear

In this paper we determine the first minimum of a class of linear forms associated with certain cubic fields that depend on a parameter.

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.

A Generalization of a Result of Birch and Swinnerton-Dyer

2018 ◽  
Vol 85 (3-4) ◽  
pp. 342
Author(s):  
Leetika Kathuria ◽  
Madhu Raka
Keyword(s):  
Linear Forms ◽  
Homogeneous Linear

In this paper, we give a proof of the generalization of a result of Birch and Swinnerton-Dyer [1956], which has been used by Hans-Gill, Sehmi and authors while obtaining estimates on the classical conjecture of Minkowski on the product of <em>n</em> non-homogeneous linear forms.

Asymmetric Inequalities for Non-Homogeneous Linear Forms

1947 ◽  
Vol s1-22 (1) ◽  
pp. 53-61 ◽  
Author(s):  
H. Davenport ◽  
H. Heilbronn
Keyword(s):  
Linear Forms ◽  
Homogeneous Linear

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 product of n real homogeneous linear forms

1950 ◽  
Vol 82 (0) ◽  
pp. 185-208 ◽  
Author(s):  
C. A. Rogers
Keyword(s):  
Linear Forms ◽  
Homogeneous Linear
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