The inhomogeneous minimum of an indefinite quadratic form

1959 ◽  
Vol 55 (4) ◽  
pp. 368-370 ◽  
Author(s):  
G. L. Watson ◽  
B. J. Birch
Keyword(s):  
Quadratic Form ◽  
Inhomogeneous Minimum ◽  
Indefinite Quadratic Form ◽  
Indefinite Quadratic

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.

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 :

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

scholarly journals On Restriction Estimates for Discrete Quadratic Surfaces

2018 ◽  
Vol 2019 (23) ◽  
pp. 7139-7159 ◽  
Author(s):  
Kevin Henriot ◽  
Kevin Hughes
Keyword(s):  
Quadratic Form ◽  
Integer Matrix ◽  
Discrete Surfaces ◽  
Indefinite Quadratic Form ◽  
Indefinite Quadratic ◽  
Quadratic Surfaces

Abstract We obtain truncated restriction estimates of an unexpected form for discrete surfaces $$\begin{align*}S_N = \{\, ( n_1 , \dots , n_d , R( n_1 , \dots, n_d ) ) \,,\, n_i \in [-N,N] \cap {\mathbb{Z}} \,\},\end{align*}$$ where $R$ is an indefinite quadratic form with integer matrix.

scholarly journals The arithmetically reduced indefinite quadratic form in n-variables

1931 ◽  
Vol 131 (816) ◽  
pp. 99-108 ◽  
Keyword(s):  
Quadratic Form ◽  
Real Number ◽  
Reduced Form ◽  
Indefinite Quadratic Form ◽  
Indefinite Quadratic

ƒ ( x 1 , x 2 , ... , x n ) = Ʃ n r,s = 1 a rs x r x s , or for brevity, say ƒ ( x ), where a rs = a sr and a rs is any real number, rational or irrational, be a quadratic form in n -variables. Suppose that the deter­minant ∆ = │ a rs │≠ 0, so that ƒ ( x ) cannot be expressed as a quadratic form with fewer than n variables. From (1) can be derived an infinity of forms g ( y 1 , y 2 , ... , y n ) = Ʃ n r,s = 1 b rs y r y s , say g ( y ), with b rs = b sr , by means of the linear substitutions x r = Ʃ n s = 1 λ rs y s , ( r = 1, 2, ..., n ), where the λ’s are integers and the determinant | λ rs | = 1. We consider throughout only such substitutions. All the forms g ( y ) have the same deter­minant ∆. They are said to be equivalent to ƒ ( x ) and to define a class of forms, the class including all the forms equivalent to ƒ ( x ) and only these. The problem of selecting a particular form as representing the class, i. e ., the so-called reduced form, is fundamental.

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 Non-negative values of quadratic forms

1971 ◽  
Vol 12 (2) ◽  
pp. 224-238 ◽  
Author(s):  
R. T. Worley
Keyword(s):  
Quadratic Form ◽  
Upper Bound ◽  
Quadratic Forms ◽  
Simple Consequence ◽  
Ternary Quadratic Forms ◽  
Indefinite Quadratic Form ◽  
Indefinite Quadratic

In a paper [1] of the same title Barnes considered the problem of finding an upper bound for the infimum m+(f) of the non-negative values1 of an indefinite quadratic form f in n variables, of given determinant det(f) ≠ 0 and of signature s. In particular it was announced (and later proved — see [2]) that m+(f) ≦ (16/5)+ for ternary quadratic forms of determinant 1 and signature — 1. A simple consequence of this result is that m+(f) ≦ (256/135)+ for quaternary quadratic forms of determinant — 1 and signature — 2.

Bounded representations of the positive values of an indefinite quadratic form

1958 ◽  
Vol 54 (1) ◽  
pp. 14-17 ◽  
Author(s):  
B. Lawton
Keyword(s):  
Quadratic Form ◽  
Indefinite Quadratic Form ◽  
Indefinite Quadratic ◽  
Image Position ◽  
Real Quadratic

Letbe a real quadratic form in n variables (n ≥ 2) with integral coefficients and determinant D = |fij| ≠ 0. Cassels ((1),(2)) has recently proved that if the equation f = 0 is properly soluble in integers x1, …, xn, then there is a solution satisfyingwhere F = max | fij and cn depends only on n. An example given by Kneser (see (2)) shows that the exponent ½(n – 1) is best possible. A simpler proof of Cassels's result has since been given by Davenport(3), and the theorem has been improved in certain cases by Watson(4). Here I consider the inequality f(x1, …, xn) > 0, where f is an indefinite form, and obtain a result analogous to that of Cassels.

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