Inhomogeneous minimum of indefinite quadratic forms in five variables of type (3,2) or (2,3): A conjecture of watson

1979 ◽  
Vol 88 (4) ◽  
pp. 305-320 ◽  
Author(s):  
R. J. Hans-Gill ◽  
Madhu Raka
Keyword(s):  
Quadratic Forms ◽  
Inhomogeneous Minimum ◽  
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 The Equivalence of Quadratic Forms

1957 ◽  
Vol 9 ◽  
pp. 526-548 ◽  
Author(s):  
G. L. Watson
Keyword(s):  
Quadratic Forms ◽  
Indefinite Quadratic ◽  
Number Of Classes

The main object of this paper is to find the number of classes in a genus of indefinite quadratic forms, with integral coefficients, in k ≥ 4 variables, distinguishing for even k two cases, according as improper equivalence is or is not admitted.

scholarly journals Strongly indefinite quadratic forms and wild algebras

1990 ◽  
Vol 26 (1) ◽  
pp. 341-351
Author(s):  
Nikolaos Marmaridis
Keyword(s):  
Quadratic Forms ◽  
Indefinite Quadratic
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