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

Isolated minima of the product of n linear forms

1953 ◽  
Vol 49 (1) ◽  
pp. 59-62 ◽  
Author(s):  
E. S. Barnes
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Linear Forms ◽  
Image Position

Letbe n linear forms with real coefficients and determinant Δ = ∥ aij∥ ≠ 0; and denote by M(X) the lower bound of | X1X2 … Xn| over all integer sets (u) ≠ (0). It is well known that γn, the upper bound of M(X)/|Δ| over all sets of forms Xi, is finite, and the value of γn has been determined when n = 2 and n = 3.

Analogue of a theorem of Khintchine in fields of formal power series

1966 ◽  
Vol 62 (4) ◽  
pp. 637-642 ◽  
Author(s):  
T. W. Cusick
Keyword(s):  
Power Series ◽  
Positive Constant ◽  
Classical Result ◽  
Real Numbers ◽  
Linear Forms ◽  
Absolute Value ◽  
The Difference ◽  
Image Position ◽  
Formal Power ◽  
Positive Integers

For a real number λ, ‖λ‖ is the absolute value of the difference between λ and the nearest integer. Let X represent the m-tuple (x1, x2, … xm) and letbe any n linear forms in m variables, where the Θij are real numbers. The following is a classical result of Khintchine (1):For all pairs of positive integers m, n there is a positive constant Г(m, n) with the property that for any forms Lj(X) there exist real numbers α1, α2, …, αn such thatfor all integers x1, x2, …, xm not all zero.

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 Diophantine approximations of the solutions of q-functional equations

2002 ◽  
Vol 132 (3) ◽  
pp. 639-659 ◽  
Author(s):  
Tapani Matala-aho
Keyword(s):  
Lower Bound ◽  
Vector Space ◽  
Linear Independence ◽  
Number Fields ◽  
Growth Conditions ◽  
Linear Forms ◽  
Dimension Estimate ◽  
Linearly Independent ◽  
Diophantine Approximations ◽  
Image Position

Given a sequence of linear forms in m ≥ 2 complex or p-adic numbers α1, …,αm ∈ Kv with appropriate growth conditions, Nesterenko proved a lower bound for the dimension d of the vector space Kα1 + ··· + Kαm over K, when K = Q and v is the infinite place. We shall generalize Nesterenko's dimension estimate over number fields K with appropriate places v, if the lower bound condition for |Rn| is replaced by the determinant condition. For the q-series approximations also a linear independence measure is given for the d linearly independent numbers. As an application we prove that the initial values F(t), F(qt), …, F(qm−1t) of the linear homogeneous q-functional equation where N = N(q, t), Pi = Pi(q, t) ∈ K[q, t] (i = 1, …, m), generate a vector space of dimension d ≥ 2 over K under some conditions for the coefficient polynomials, the solution F(t) and t, q ∈ K*.

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

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

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

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