✦ LIBER ✦

On Representations of Integers by Indefinite Ternary Quadratic Forms

✍ Scribed by Mikhail Borovoi

Publisher: Elsevier Science
Year: 2001
Tongue: English
Weight: 143 KB
Volume: 90
Category: Article
ISSN: 0022-314X
DOI: 10.1006/jnth.2001.2662

No coin nor oath required. For personal study only.

✦ Synopsis

Let f be an indefinite ternary integral quadratic form and let q be a nonzero integer such that &q det( f ) is not a square. Let N(T, f, q) denote the number of integral solutions of the equation f (x)=q where x lies in the ball of radius T centered at the origin. We are interested in the asymptotic behavior of N(T, f, q) as T Ä . We deduce from the results of our joint paper with Z. Rudnick that N(T, f, q)tcE HL (T, f, q) as T Ä , where E HL (T, f, q) is the Hardy Littlewood expectation (the product of local densities) and 0 c 2. We give examples of f and q such that c takes the values 0, 1, 2.

📜 SIMILAR VOLUMES

On the Representation of Integers by the

On the Representation of Integers by the Lorentzian Quadratic Form

✍ John G. Ratcliffe; Steven T. Tschantz 📂 Article 📅 1997 🏛 Elsevier Science 🌐 English ⚖ 413 KB

In this paper, we establish an asymptotic formula, for large radius r, for the number of representations of a nonzero integer k by the Lorentzian quadratic form x 2 1 +x 2 2 + } } } +x 2 n &x 2 n+1 that are contained in the ball of radius r centered at the origin in Euclidean (n+1)-space. 1997 Acade

Simultaneous Representation of Integers

Simultaneous Representation of Integers by a Pair of Ternary Quadratic Forms—With an Application to Index Form Equations in Quartic Number Fields

✍ István Gaál; Attila Petho&#x030B;; Michael Pohst 📂 Article 📅 1996 🏛 Elsevier Science 🌐 English ⚖ 411 KB

Let Q 1 , Q 2 # Z[X, Y, Z] be two ternary quadratic forms and u 1 , u 2 # Z. In this paper we consider the problem of solving the system of equations (1) Q 2 (x, y, z)=u 2 in x, y, z # Z with gcd(x, y, z)=1. According to Mordell [12] the coprime solutions of can be presented by finitely many expr

Representations of Codimension ⩾3 by Def

Representations of Codimension ⩾3 by Definite Quadratic Forms

✍ Wai Kiu Chan; Dennis R. Estes; M. Jöchner 📂 Article 📅 1998 🏛 Elsevier Science 🌐 English ⚖ 185 KB

Let M be a positive definite quadratic Z-lattice of rank n+3. If N is a quadratic Z-lattice of rank n which is primitively represented by the genus of M and if all the successive minima of N increase sufficiently quickly, then there exists a global primitive representation of N by M with approximati

Representations of Binary Forms by Certa

Representations of Binary Forms by Certain Quinary Positive Integral Quadratic Forms

✍ Myung-Hwan Kim; Ja Kyung Koo; Byeong-Kweon Oh 📂 Article 📅 2001 🏛 Elsevier Science 🌐 English ⚖ 150 KB

In this article, we provide the complete answer to a question raised by Kitaoka in his book. (1999, ``Arithmetic of Quadratic Forms,'' Cambridge Univ. Press, Cambridge, UK). More precisely, we prove that A 4 = ( 4) represents all but one and D 4 20[2 1 2 ] represents all but three binary positive ev

Local Densities of Representations of Qu

Local Densities of Representations of Quadratic Forms over p-Adic Integers (The Non-Dyadic Case)

✍ Fumihiro Sato; Yumiko Hironaka 📂 Article 📅 2000 🏛 Elsevier Science 🌐 English ⚖ 235 KB

We give an explicit formula for local densities of integral representations of nondegenerate integral symmetric matrices of arbitrary size in the case p{2, in terms of invariants of quadratic forms.

Representations by Quadratic Forms in a

Representations by Quadratic Forms in a Finite Field of Characteristic Two

✍ John D. Fulton 📂 Article 📅 1977 🏛 John Wiley and Sons 🌐 English ⚖ 392 KB

Let F be a finite field of cliaracteristic two and'let F'xm and FIXn denote vector spaces of m-tuples and n-tuples, respectively, over P. Let Q be a quadratic form of rank m defined on FIXm and let Q, be a quadratic form of rank n defined on F I X n . Then relative to given ordered bases for .FIXm a