The positive values of inhomogeneous ternary quadratic forms

1961 ◽  
Vol 2 (2) ◽  
pp. 127-132 ◽  
Author(s):  
E. S. Barnes
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Ternary Quadratic Form ◽  
Equality Sign ◽  
Ternary Quadratic Forms ◽  
Positive Multiple ◽  
Indefinite Ternary Quadratic Form ◽  
Image Position

Let f(x, y, z) be an indefinite ternary quadratic form of signature (2, 1) and determinant d ≠ 0. Davenport [3] has shown that there exist integral x, y, z with, the equality sign being necessary if and only if f is a positive multiple of f1(x, y, z) = x2 + yz.

scholarly journals On indefinite ternary quadratic forms

1974 ◽  
Vol 18 (4) ◽  
pp. 388-401 ◽  
Author(s):  
R. T. Worley
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Ternary Quadratic Form ◽  
Ternary Quadratic Forms ◽  
Indefinite Ternary Quadratic Form ◽  
Image Position

In a paper [1] with the same title Barnes has shown that if Q(x, y, z) is an indefinite ternary quadratic form of determinant d ≠ 0 then there exist integers x1, y1, z1, x2,···z3 satisfying for which Furthermore, unless Q is equivalent to a multiple of or two other forms Q2, Q3 then the constant ⅔ in (1.2) can be replaced by 1/2.2. For Q1 equality is needed on at least one side of (1.2) while for Q2, Q3 the constant ⅔ can be reduced to 12/25 but no further.

On the existence of integral ternary quadratic forms without automorphs of finite order

2017 ◽  
Vol 26 (14) ◽  
pp. 1750102 ◽  
Author(s):  
José María Montesinos-Amilibia
Keyword(s):  
Quadratic Form ◽  
Finite Order ◽  
Quadratic Forms ◽  
Ternary Quadratic Form ◽  
Ternary Quadratic Forms

An example of an integral ternary quadratic form [Formula: see text] such that its associated orbifold [Formula: see text] is a manifold is given. Hence, the title is proved.

The number of representations of integers by generalized Bell ternary quadratic forms

2020 ◽  
pp. 1-15
Author(s):  
Kyoungmin Kim ◽  
Yeong-Wook Kwon
Keyword(s):  
Quadratic Form ◽  
Class Number ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Cusp Forms ◽  
Ternary Quadratic Form ◽  
Closed Formula ◽  
Integral Quadratic Form ◽  
Ternary Quadratic Forms ◽  
Representations Of Integers

For a positive definite ternary integral quadratic form [Formula: see text], let [Formula: see text] be the number of representations of an integer [Formula: see text] by [Formula: see text]. A ternary quadratic form [Formula: see text] is said to be a generalized Bell ternary quadratic form if [Formula: see text] is isometric to [Formula: see text] for some nonnegative integers [Formula: see text]. In this paper, we give a closed formula for [Formula: see text] for a generalized Bell ternary quadratic form [Formula: see text] with [Formula: see text] and class number greater than [Formula: see text] by using the Minkowski–Siegel formula and bases for spaces of cusp forms of weight [Formula: see text] and level [Formula: see text] with [Formula: see text] consisting of eta-quotients.

scholarly journals Spinor representations of positive definite ternary quadratic forms

2018 ◽  
Vol 14 (02) ◽  
pp. 581-594 ◽  
Author(s):  
Jangwon Ju ◽  
Kyoungmin Kim ◽  
Byeong-Kweon Oh
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Ternary Quadratic Form ◽  
Definite Integral ◽  
Constant Multiple ◽  
Ternary Quadratic Forms ◽  
Spinor Genus ◽  
Local Densities ◽  
Spinor Representations

For a positive definite integral ternary quadratic form [Formula: see text], let [Formula: see text] be the number of representations of an integer [Formula: see text] by [Formula: see text]. The famous Minkowski–Siegel formula implies that if the class number of [Formula: see text] is one, then [Formula: see text] can be written as a constant multiple of a product of local densities which are easily computable. In this paper, we consider the case when the spinor genus of [Formula: see text] contains only one class. In this case the above also holds if [Formula: see text] is not contained in a set of finite number of square classes which are easily computable. By using this fact, we prove some extension of the recent results on both the representations of generalized Bell ternary forms and the representations of ternary quadratic forms with some congruence conditions.

scholarly journals Inequalities for inert primes and their applications

2020 ◽  
Vol 16 (08) ◽  
pp. 1819-1832
Author(s):  
Zilong He
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Ternary Quadratic Form ◽  
Kronecker Symbol ◽  
Ternary Quadratic Forms ◽  
New Criterion

For any given non-square integer [Formula: see text], we prove Euclid’s type inequalities for the sequence [Formula: see text] of all primes satisfying the Kronecker symbol [Formula: see text], [Formula: see text] and give a new criterion on a ternary quadratic form to be irregular as an application, which simplifies Dickson and Jones’s argument in the classification of regular ternary quadratic forms to some extent.

An Extension of Meyer's Theorem on Indefinite Ternary Quadratic Forms

1952 ◽  
Vol 4 ◽  
pp. 120-128 ◽  
Author(s):  
Burton W. Jones
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Ternary Quadratic Form ◽  
Primitive Form ◽  
Ternary Quadratic Forms

Let f be a ternary quadratic form whose matrix F has integral elements with g.c.d. 1, that is, an improperly or properly primitive form according as all diagonal elements are even or not. Let d be the determinant of f (denoted by ) Ω, the g.c.d. of the 2-rowed minors of F. Then d = Ω2 Δ determines an integer Δ. Two forms f in the same genus have the same invariants Ω, Δ, d. The form whose matrix is adj F/Ω, is called the reciprocal form of f.

XII. On the orders and genera of ternary quadratic forms

1867 ◽  
Vol 157 ◽  
pp. 255-298 ◽  
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Ternary Quadratic Form ◽  
Greatest Common Divisor ◽  
Ternary Quadratic Forms ◽  
The Matrix

Eisenstein, in a Memoir entitled "Neue Theoreme der höheren Arithmetik", has defined the ordinal and generic characters of ternary quadratic forms of an uneven determinant; and, in the case of definite forms, has assigned the weight of any given order or genus. But he has not considered forms of an even determinant, neither has he given any demonstrations of his results. To supply these omissions, and so far to complete the work of Eisenstein, is the object of the present memoir. Art. 2. We represent by f the ternary quadratic form a x 2 + a ' y 2 + a '' z 2 +2 byz +2 b ' xz +2 b '' xy ; . . . . . . . (1) we suppose that f is primitive ( i. e . that the six integral numbers a , a ', a '', b , b ', b " admit of no common divisor other than unity), and that its discriminant is different from zero; this discriminant, or the determinant of the matrix a , b ", b ' b ", a ', b b ', b , a " . . . . . . . (2) we represent by D ; by Ω we denote the greatest common divisor of the minor determinants of the matrix (2); by Ω F the contravariant of f , or the form ( a ' a ''— b 2 ) x 2 + ( a " a — b ' 2 ) y 2 + ( a a '— b '' 2 ) z 2 + 2( b ' b "— ab ) yz + 2( b " b — a ' b ') zx + 2( bb ' — a '' b ") xy ; }. . . . . . . . (3) we shall term F the primitive contravariant of f , and we shall write F = A x 2 + A' y 2 + 2B yz + 2B' xy + 2B'' xy . . . . . . (4)

FINITENESS RESULTS FOR REGULAR DEFINITE TERNARY QUADRATIC FORMS OVER ${\mathbb Q}(\sqrt{5})$

2007 ◽  
Vol 03 (04) ◽  
pp. 541-556 ◽  
Author(s):  
WAI KIU CHAN ◽  
A. G. EARNEST ◽  
MARIA INES ICAZA ◽  
JI YOUNG KIM
Keyword(s):  
Quadratic Form ◽  
Number Field ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Integral Quadratic Form ◽  
Ring Of Integers ◽  
Ternary Quadratic Forms ◽  
Totally Positive ◽  
Finiteness Result

Let 𝔬 be the ring of integers in a number field. An integral quadratic form over 𝔬 is called regular if it represents all integers in 𝔬 that are represented by its genus. In [13,14] Watson proved that there are only finitely many inequivalent positive definite primitive integral regular ternary quadratic forms over ℤ. In this paper, we generalize Watson's result to totally positive regular ternary quadratic forms over [Formula: see text]. We also show that the same finiteness result holds for totally positive definite spinor regular ternary quadratic forms over [Formula: see text], and thus extends the corresponding finiteness results for spinor regular quadratic forms over ℤ obtained in [1,3].

Ternary Quadratic Forms and Eta Quotients

2015 ◽  
Vol 58 (4) ◽  
pp. 858-868 ◽  
Author(s):  
Kenneth S. Williams
Keyword(s):  
Quadratic Forms ◽  
Fourier Coefficients ◽  
Arithmetic Progressions ◽  
Dedekind Eta Function ◽  
Ternary Quadratic Forms ◽  
Sum Formula ◽  
Image Position ◽  
Positive Integers

AbstractLet denote the Dedekind eta function. We use a recent productto- sum formula in conjunction with conditions for the non-representability of integers by certain ternary quadratic forms to give explicitly ten eta quotientssuch that the Fourier coefficients c(n) vanish for all positive integers n in each of infinitely many non-overlapping arithmetic progressions. For example, we show that if we have c(n) = 0 for all n in each of the arithmetic progressions

REPRESENTATIONS OF ARITHMETIC PROGRESSIONS BY POSITIVE DEFINITE QUADRATIC FORMS

2011 ◽  
Vol 07 (06) ◽  
pp. 1603-1614 ◽  
Author(s):  
BYEONG-KWEON OH
Keyword(s):  
Positive Integer ◽  
Quadratic Form ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Arithmetic Progressions ◽  
Negative Integer ◽  
Definite Integral ◽  
Integral Quadratic Form ◽  
Ternary Quadratic Forms

For a positive integer d and a non-negative integer a, let Sd,a be the set of all integers of the form dn + a for any non-negative integer n. A (positive definite integral) quadratic form f is said to be Sd,a-universal if it represents all integers in the set Sd, a, and is said to be Sd,a-regular if it represents all integers in the non-empty set Sd,a ∩ Q((f)), where Q(gen(f)) is the set of all integers that are represented by the genus of f. In this paper, we prove that there is a polynomial U(x,y) ∈ ℚ[x,y] (R(x,y) ∈ ℚ[x,y]) such that the discriminant df for any Sd,a-universal (Sd,a-regular) ternary quadratic forms is bounded by U(d,a) (respectively, R(d,a)).

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