The inhomogeneous minimum of quadratic forms of signature ± 1

1981 ◽  
Vol 89 (2) ◽  
pp. 225-235 ◽  
Author(s):  
Madhu Raka
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Real Numbers ◽  
Inhomogeneous Minimum ◽  
Indefinite Quadratic Form ◽  
Indefinite Quadratic ◽  
Image Position

Let Qr be a real indefinite quadratic form in r variables of determinant D ≠ 0 and of type (r1, r2), 0 < r1 < r, r = r1 + r2, S = r1 − r2 being the signature of Qr. It is known (e.g. Blaney (3)) that, given any real numbers c1, c2,…, cr, there exists a constant C depending only on r and s such that the inequalityhas a solution in integers x1, x2, …, xr.

Inhomogeneous minimum of indefinite quadratic forms in six variables: A conjecture of Watson

1983 ◽  
Vol 94 (1) ◽  
pp. 1-8 ◽  
Author(s):  
Madhu Raka
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Real Numbers ◽  
Famous Conjecture ◽  
Inhomogeneous Minimum ◽  
Indefinite Quadratic Form ◽  
History Of ◽  
Indefinite Quadratic ◽  
Image Position

The famous conjecture of Watson(11) on the minima of indefinite quadratic forms in n variables has been proved for n ≤ 5, n ≥ 21 and for signatures 0 and ± 1. For the details and history of the conjecture the reader is referred to the author's paper(8). In the succeeding paper (9), we prove Watson's conjecture for signature ± 2 and ± 3 and for all n. Thus only one case for n = 6 (i.e. forms of type (1, 5) or (5, 1)) remains to he proved which we do here; thereby completing the case n = 6. This result is also used in (9) for proving the conjecture for all quadratic forms of signature ± 4. More precisely, here we prove:Theorem 1. Let Q6(x1, …, x6) be a real indefinite quadratic form in six variables of determinant D ( < 0) and of type (5, 1) or (1, 5). Then given any real numbers ci, 1 ≤ i ≤ 6, there exist integers x1,…, x6such that

Inhomogeneous minimum of indefinite quaternary quadratic forms

1967 ◽  
Vol 63 (2) ◽  
pp. 277-290 ◽  
Author(s):  
Vishwa Chander Dumir
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Real Numbers ◽  
Inhomogeneous Minimum ◽  
Indefinite Quadratic Form ◽  
Indefinite Quadratic ◽  
Image Position

Let Q (x1, …, xn) be a real indefinite quadratic form in n-variables x1,…, xn with signature (r, s),r + s = n and determinant D ≠ 0. Then it is known (see Blaney (2)) that there exists constant Cr, s depending only on r, s such that given any real numbers c1, …,cn we can find integers x1, …, xn satisfying

Integral Bases for Quadratic Forms

1963 ◽  
Vol 15 ◽  
pp. 412-421 ◽  
Author(s):  
J. H. H. Chalk
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Integer Variables ◽  
Inhomogeneous Minimum ◽  
Integral Bases ◽  
Indefinite Quadratic Form ◽  
Homogeneous Minimum ◽  
Indefinite Quadratic ◽  
Image Position

Letbe an indefinite quadratic form in the integer variables x1, . . . , xn with real coefficients of determinant D = ||ars||(n) ≠ 0. The homogeneous minimum MH(Qn) and the inhomogeneous minimum MI(Qn) of Qn(x) are defined as follows :

Positive values of inhomogeneous quaternary quadratic forms, I

1968 ◽  
Vol 8 (1) ◽  
pp. 87-101 ◽  
Author(s):  
Vishwa Chander Dumir
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Real Numbers ◽  
Indefinite Quadratic Form ◽  
Indefinite Quadratic ◽  
Image Position

Let Q(x1, …, xn) be an indefinite quadratic form in n-variables with real coefficients, determinant D ≠ 0 and signature (r, s), r+s = n. Then it is known (e.g. see Blaney [2]) that there exist constants Γr, s depending only on r and s such for any real numbers c1, …, cn we can find integers x1, …, xn satisfying

On a conjecture of Watson

1983 ◽  
Vol 94 (1) ◽  
pp. 9-22 ◽  
Author(s):  
Madhu Raka
Keyword(s):  
Quadratic Form ◽  
Classical Result ◽  
Preceding Paper ◽  
Real Numbers ◽  
Linear Forms ◽  
Indefinite Quadratic Form ◽  
Indefinite Quadratic ◽  
Image Position

Let Qn be a real indefinite quadratic form in n variables x1, x2,…, xn, of determinant D ≠ 0 and of type (r, s), 0 < r < n, n = r + s. Let σ denote the signature of Qn so that σ = r − s. It is known (see e.g. Blaney(4)) that, given any real numbers c1 c2, …, cn, there exists a constant C depending upon n and σ only such that the inequalityhas a solution in integers x1, x2, …, xn. Let Cr, s denote the infimum of all such constants. Clearly Cr, s = Cs, r, so we need consider non-negative signatures only. For n = 2, C1, 1 = ¼ follows from a classical result of Minkowski on the product of two linear forms. When n = 3, Davenport (5) proved that C2, 1 = 27/100. For all n and σ = 0, Birch (3) proved that Cr, r = ¼. In 1962, Watson(18) determined the values of Cr, s for all n ≥ 21 and for all signatures σ. He proved thatWatson also conjectured that (1·2) holds for all n ≥ 4. Dumir(6) proved Watson's conjecture for n = 4. For n = 5, it was proved by Hans-Gill and Madhu Raka(7, 8). The author (12) has proved the conjecture for σ = 1 and all n. In the preceding paper (13) we proved that C5, 1 = 1. In this paper we prove Watson's conjecture for σ = 2, 3 and 4.

scholarly journals Positive values of inhomogeneous quaternary ouadratic forms, II

1968 ◽  
Vol 8 (2) ◽  
pp. 287-303 ◽  
Author(s):  
Vishwa Chander Dumir
Keyword(s):  
Quadratic Form ◽  
Real Numbers ◽  
Indefinite Quadratic Form ◽  
Indefinite Quadratic ◽  
Definition Of ◽  
Image Position ◽  
Quaternary Quadratic Form

In a previous paper [4] we showed that Γ3,1 = 16/. For the definition of Γr, s for an indefinite quadratic form in n = r + s variables of the type (r, s) see the above paper. Here we shall show that Γ2,2 = 16. More precisely we prove: Theorem. Let Q (x, y, z, t) be an indefinite quaternary quadratic form with determinant D > 0 and signature (2, 2). Then given any real numbers x0, y0, z0, t0 we can find integers x, y, z, t such thatEquality is necessary if and only if either where ρ ≠ 0. For Q1 equality occurs if and only if

scholarly journals The inhomogeneous minima of indefinite quadratic forms

1961 ◽  
Vol 2 (1) ◽  
pp. 9-10 ◽  
Author(s):  
E. S. Barnes
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Indefinite Quadratic Form ◽  
Indefinite Quadratic ◽  
Image Position

If f(x) is a real indefinite quadratic form in n variables with determinant d ≠ 0, we set for any real α.

The lattice properties of asymmetric hyperbolic regions

1948 ◽  
Vol 44 (4) ◽  
pp. 457-462
Author(s):  
J. W. S. Cassels
Keyword(s):  
Quadratic Form ◽  
Classical Theorem ◽  
Real Numbers ◽  
Indefinite Quadratic Form ◽  
Indefinite Quadratic ◽  
Image Position

1. Letwhere a > 0, be an indefinite quadratic form, so that d = b2 − 4ac > 0. A classical theorem of Minkowski states that, if (x0, y0) is any pair of real numbers, there are numbers (x, y) congruent (mod 1) to (x0, y0), such thatand, more recently, Davenport has shown that this theorem can be sharpened for certain special f, for instance that it is always possible to satisfy

scholarly journals An Inhomogeneous Minimum For Non-Convex Star-Regions With Hexagonal Symmetry

1955 ◽  
Vol 7 ◽  
pp. 337-346 ◽  
Author(s):  
R. P. Bambah ◽  
K. Rogers
Keyword(s):  
Quadratic Form ◽  
Binary Quadratic Form ◽  
Elementary Geometry ◽  
Hexagonal Symmetry ◽  
Real Numbers ◽  
Inhomogeneous Minimum ◽  
Image Position ◽  
Indefinite Binary Quadratic Form

1. Introduction. Several authors have proved theorems of the following type:Let x0, y0 be any real numbers. Then for certain functions f(x, y), there exist numbers x, y such that1.1 x ≡ x0, y ≡ y0 (mod 1),and1.2 .The first result of this type, but with replaced by min , was given by Barnes (3) for the case when the function is an indefinite binary quadratic form. A generalisation of this was proved by elementary geometry by K. Rogers (6).

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