Quadratic forms in normal open induction

1993 ◽  
Vol 58 (2) ◽  
pp. 456-476
Author(s):  
Margarita Otero
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Quotient Group ◽  
Pell Equation ◽  
Gaussian Integers ◽  
Nonzero Integer ◽  
Quadratic Extensions ◽  
Integrally Closed ◽  
Open Induction

AbstractModels of normal open induction (NOI) are those discretely ordered rings, integrally closed in their fraction field whose nonnegative part satisfy Peano's induction axioms for open formulas in the language of ordered semirings.Here we study the problem of representability of an element a of a model M of NOI (in some extension of M) by a quadratic form of the type X2 + b Y2 where b is a nonzero integer. Using either a trigonometric or a hyperbolic parametrization we prove that except in some trivial cases, M[x, y] with x2 + by2 = a can be embedded in a model of NOI.We also study quadratic extensions of a model M of NOI; we first prove some properties of the ring of Gaussian integers of M. Then we study the group of solutions of a Pell equation in NOI; we construct a model in which the quotient group by the squares has size continuum.

Quadratic forms over Quadratic Extensions of Fields with Two Quaternion Algebras

1979 ◽  
Vol 31 (5) ◽  
pp. 1047-1058 ◽  
Author(s):  
Craig M. Cordes ◽  
John R. Ramsey
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Square Root ◽  
Real Numbers ◽  
Quaternion Algebras ◽  
The Real ◽  
Quadratic Extensions

In this paper, we analyze what happens with respect to quadratic forms when a square root is adjoined to a field F which has exactly two quaternion algebras. There are many such fields—the real numbers and finite extensions of the p-adic numbers being two familiar examples. For general quadratic extensions, there are many unanswered questions concerning the quadratic form structure, but for these special fields we can clear up most of them.It is assumed char F ≠ 2 and K = F (√a) where a ∊ Ḟ – Ḟ2. Ḟ denotes the non-zero elements of F. Generally the letters a, b, c, … and α, β, … refer to elements from Ḟ and x, y, z, … come from .

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 REPRESENTATIONS OF ALTERNATIVE CLIFFORD ALGEBRAS OF QUADRATIC FORMS

2014 ◽  
Vol 57 (3) ◽  
pp. 579-590 ◽  
Author(s):  
STACY MARIE MUSGRAVE
Keyword(s):  
Quadratic Form ◽  
Clifford Algebra ◽  
Algebraic Structure ◽  
Quadratic Forms ◽  
Clifford Algebras ◽  
Future Research ◽  
Octonion Algebras

AbstractThis work defines a new algebraic structure, to be called an alternative Clifford algebra associated to a given quadratic form. I explored its representations, particularly concentrating on connections to the well-understood octonion algebras. I finished by suggesting directions for future research.

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.

scholarly journals New Estimates of the Singular Series Corresponding to Positive Quaternary Quadratic Forms

2006 ◽  
Vol 13 (4) ◽  
pp. 687-691
Author(s):  
Guram Gogishvili
Keyword(s):  
Quadratic Form ◽  
General Result ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Precise Estimate ◽  
Definite Integral ◽  
Singular Series ◽  
Prime Factors ◽  
Quaternary Quadratic Form ◽  
Best Estimates

Abstract Let 𝑚 ∈ ℕ, 𝑓 be a positive definite, integral, primitive, quaternary quadratic form of the determinant 𝑑 and let ρ(𝑓,𝑚) be the corresponding singular series. When studying the best estimates for ρ(𝑓,𝑚) with respect to 𝑑 and 𝑚 we proved in [Gogishvili, Trudy Tbiliss. Univ. 346: 72–77, 2004] that where 𝑏(𝑘) is the product of distinct prime factors of 16𝑘 if 𝑘 ≠ 1 and 𝑏(𝑘) = 3 if 𝑘 = 1. The present paper proves a more precise estimate where 𝑑 = 𝑑0𝑑1, if 𝑝 > 2; 𝑕(2) ⩾ –4. The last estimate for ρ(𝑓,𝑚) as a general result for quaternary quadratic forms of the above-mentioned type is unimprovable in a certain sense.

scholarly journals Quadratic form Approach for the Number of Zeros of Homogeneous Linear Recurring Sequences over Finite Fields

2017 ◽  
Vol 9 (3) ◽  
pp. 8
Author(s):  
Yasanthi Kottegoda
Keyword(s):  
Quadratic Form ◽  
Finite Field ◽  
Characteristic Polynomial ◽  
Finite Fields ◽  
Quadratic Forms ◽  
Linear Recurring Sequences ◽  
Exact Number ◽  
Number Of Zeros ◽  
Homogeneous Linear ◽  
Form Approach

We consider homogeneous linear recurring sequences over a finite field $\mathbb{F}_{q}$, based on an irreducible characteristic polynomial of degree $n$ and order $m$. Let $t=(q^{n}-1)/ m$. We use quadratic forms over finite fields to give the exact number of occurrences of zeros of the sequence within its least period when $t$ has q-adic weight 2. Consequently we prove that the cardinality of the set of zeros for sequences from this category is equal to two.

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.

An Elementary Proof of a Theorem About the Representation of Primes by Quadratic Forms

1954 ◽  
Vol 6 ◽  
pp. 353-363 ◽  
Author(s):  
W. E. Briggs
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Elementary Proof ◽  
Prime Numbers ◽  
Binary Quadratic Form ◽  
First Case

The theorem that every properly primitive binary quadratic form is capable of representing infinitely many prime numbers was first proved completely by H. Weber (5). The purpose of this paper is to give an elementary proof of the case where the form is ax2 + 2bxy + cy2, with a > 0, (a, 2b, c) = 1, and D = b2 — ac not a square. The cases where the form is ax2 + bxy + cy2 with b odd, and the case where the form is ax2+ 2bxy + cy2 with D a square, can be settled very simply once the first case is taken care of, and this is done in a page and a half in the Weber paper.

scholarly journals The minimum and the primitive representation of positive definite quadratic forms II

1996 ◽  
Vol 141 ◽  
pp. 1-27 ◽  
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Positive Definite Quadratic Form ◽  
Primitive Representation ◽  
Quadratic Lattices

We are concerned with representation of positive definite quadratic forms by a positive definite quadratic form. Let us consider the following assertion Am, n : Let M, N be positive definite quadratic lattices over Z with rank(M) = m and rank(N) = n respectively. We assume that the localization Mp is represented by Np for every prime p, that is there is an isometry from Mp to Np. Then there exists a constant c(N) dependent only on N so that M is represented by N if min(M) > c(N), where min(M) denotes the least positive number represented by M.

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