THE QUADRATIC FORM IN NINE PRIME VARIABLES

2016 ◽  
Vol 223 (1) ◽  
pp. 21-65 ◽  
Author(s):  
LILU ZHAO
Keyword(s):  
Quadratic Form ◽  
Positive Real ◽  
Integral Quadratic Form ◽  
Real Solutions ◽  
Indefinite Integral ◽  
Local Solutions

Let $f(x_{1},\ldots ,x_{n})$ be a regular indefinite integral quadratic form with $n\geqslant 9$, and let $t$ be an integer. Denote by $\mathbb{U}_{p}$ the set of $p$-adic units in $\mathbb{Z}_{p}$. It is established that $f(x_{1},\ldots ,x_{n})=t$ has solutions in primes if (i) there are positive real solutions, and (ii) there are local solutions in $\mathbb{U}_{p}$ for all prime $p$.

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].

scholarly journals Integral quadratic forms avoiding arithmetic progressions

2020 ◽  
Vol 16 (10) ◽  
pp. 2141-2148
Author(s):  
A. G. Earnest ◽  
Ji Young Kim
Keyword(s):  
Positive Integer ◽  
Quadratic Form ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Arithmetic Progressions ◽  
Integral Quadratic Form ◽  
Positive Integers

For every positive integer [Formula: see text], it is shown that there exists a positive definite diagonal quaternary integral quadratic form that represents all positive integers except for precisely those which lie in [Formula: see text] arithmetic progressions. For [Formula: see text], all forms with this property are determined.

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 ON NUMBER FIELDS WITH EQUIVALENT INTEGRAL TRACE FORMS

2012 ◽  
Vol 08 (07) ◽  
pp. 1569-1580 ◽  
Author(s):  
GUILLERMO MANTILLA-SOLER
Keyword(s):  
Quadratic Form ◽  
Real Number ◽  
Number Field ◽  
Number Fields ◽  
Trace Form ◽  
Integral Quadratic Form ◽  
Trace Forms ◽  
Fundamental Discriminant ◽  
Totally Real

Let K be a number field. The integral trace form is the integral quadratic form given by tr k/ℚ(x2)|OK. In this article we study the existence of non-conjugated number fields with equivalent integral trace forms. As a corollary of one of the main results of this paper, we show that any two non-totally real number fields with the same signature and same prime discriminant have equivalent integral trace forms. Additionally, based on previous results obtained by the author and the evidence presented here, we conjecture that any two totally real quartic fields of fundamental discriminant have equivalent trace zero forms if and only if they are conjugated.

The Laplacian quadratic form and edge connectivity of a graph

2015 ◽  
Vol 30 ◽  
Author(s):  
William Watkins
Keyword(s):  
Quadratic Form ◽  
Connected Graph ◽  
Positive Semidefinite ◽  
Positive Definite ◽  
Edge Connectivity ◽  
Integral Quadratic Form ◽  
Simple Connected Graph

Let G be a simple connected graph with associated positive semidefinite integral quadratic form Q(x) = \sum (x(i) − x(j))^2, where the sum is taken over all edges ij of G. It is showed that the minimum positive value of Q(x) for x ∈ Z_n equals the edge connectivity of G. By restricting Q(x) to x ∈ Z_{n−1} × {0}, the quadratic form becomes positive definite. It is also showed that the number of minimal disconnecting sets of edges of G equals twice the number of vectors x ∈ Z_{n−1} ×{0} for which the form Q attains its minimum positive value.

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)).

The convexity of the function y = E(x) defined by xy = yx

2020 ◽  
Vol 104 (559) ◽  
pp. 36-43
Author(s):  
Alan F. Beardon ◽  
Russell A. Gordon
Keyword(s):  
Rational Solutions ◽  
Positive Real ◽  
Real Solutions ◽  
The Past ◽  
Leonhard Euler ◽  
Integer Solutions ◽  
Complex Solutions ◽  
Distinct Integer ◽  
The Web

The set of solutions to the equation xy = yx has been studied extensively over the past three centuries, including work by well known mathematicians such as Daniel Bernoulli (1700–1782), Leonhard Euler (1707–1783), and Christian Goldbach (1690–1764). Various mathematicians have focused on the integer, rational, real, and complex solutions. For example, it has been shown (see [1]) that the equality 24 = 42 gives the only distinct integer solutions. Our exposition below presents some of the key ideas behind the positive real solutions to this equation and illustrates how rational solutions can be found. To learn more about the various solutions, the reader can consult the articles listed at the end of this paper, as well as the extensive references given in these articles. It is also possible to find some of this material on the Web.

scholarly journals Sums of squares of integral linear forms

2000 ◽  
Vol 69 (3) ◽  
pp. 298-302 ◽  
Author(s):  
Hideyo Sasaki
Keyword(s):  
Quadratic Form ◽  
Positive Definite ◽  
Sum Of Squares ◽  
Sums Of Squares ◽  
Linear Forms ◽  
Integral Quadratic Form

AbstractIn this paper we prove that every positive definite n-ary integral quadratic form with 12 < n < 13 (respectively 14 ≦ n ≤ 20) that can be represented by a sum of squares of integral linear forms is represented by a sum of 2 · 3n + n + 6 (respectively 3 · 4n + n + 3) squares. We also prove that every positive definite 7-ary integral quadratic form that can be represented by a sum of squares is represented by a sum of 25 squares.

scholarly journals Brauer–Manin obstruction for integral points of homogeneous spaces and representation by integral quadratic forms

2009 ◽  
Vol 145 (2) ◽  
pp. 309-363 ◽  
Author(s):  
Jean-Louis Colliot-Thélène ◽  
Fei Xu
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Homogeneous Spaces ◽  
Algebraic Groups ◽  
Integral Points ◽  
Integral Quadratic Form ◽  
Linear Algebraic Groups ◽  
Global Principle

AbstractAn integer may be represented by a quadratic form over each ring ofp-adic integers and over the reals without being represented by this quadratic form over the integers. More generally, such failure of a local-global principle may occur for the representation of one integral quadratic form by another integral quadratic form. We show that many such examples may be accounted for by a Brauer–Manin obstruction for the existence of integral points on schemes defined over the integers. For several types of homogeneous spaces of linear algebraic groups, this obstruction is shown to be the only obstruction to the existence of integral points.

scholarly journals An inequality for an integral quadratic form

1995 ◽  
Vol 8 (2) ◽  
pp. 81-84 ◽  
Author(s):  
Sui Sun Cheng ◽  
Tzon Tzer Lu
Keyword(s):  
Quadratic Form ◽  
Integral Quadratic Form
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