scholarly journals An upper bound on the number of perfect quadratic forms

2020 ◽  
Vol 365 ◽  
pp. 107031
Author(s):  
W.P.J. van Woerden
Keyword(s):  
Upper Bound ◽  
Quadratic Forms

TOTALLY GEODESIC SURFACES AND QUADRATIC FORMS

2013 ◽  
Vol 22 (13) ◽  
pp. 1350072
Author(s):  
PRADTHANA JAIPONG
Keyword(s):  
Upper Bound ◽  
Quadratic Forms ◽  
Incompressible Surface ◽  
Totally Geodesic ◽  
Geodesic Surfaces ◽  
Figure Eight Knot ◽  
Dehn Fillings

Let M be a compact, connected, irreducible, orientable 3-manifold with torus boundary. A closed, orientable, immersed, incompressible surface F in M with no incompressible annulus joining F and ∂M compresses in at most finitely many Dehn fillings M(α). It is known that there is no universal upper bound on the number of such fillings, independent of the surface, and the figure-eight knot complement is the first example of a manifold where this phenomenon occurs. In this paper, we show that the same behavior of the figure-eight knot complement is shared by other two cusped manifolds.

scholarly journals An Upper Bound for the Size of $s$-Distance Sets in Real Algebraic Sets

10.37236/9712 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Gábor Hegedüs ◽  
Lajos Rónyai
Keyword(s):  
Simple Proof ◽  
Upper Bound ◽  
Quadratic Forms ◽  
Short Proof ◽  
Algebraic Method ◽  
Upper Bounds ◽  
Additive Combinatorics ◽  
Algebraic Sets ◽  
Distance Sets ◽  
Real Quadratic

In a recent paper, Petrov and Pohoata developed a new algebraic method which combines the Croot-Lev-Pach Lemma from additive combinatorics and Sylvester’s Law of Inertia for real quadratic forms. As an application, they gave a simple proof of the Bannai-Bannai-Stanton bound on the size of $s$-distance sets (subsets $\mathcal{A}\subseteq \mathbb{R}^n$ which determine at most $s$ different distances). In this paper we extend their work and prove upper bounds for the size of $s$-distance sets in various real algebraic sets. This way we obtain a novel and short proof for the bound of Delsarte-Goethals-Seidel on spherical s-distance sets and a generalization of a bound by Bannai-Kawasaki-Nitamizu-Sato on $s$-distance sets on unions of spheres. In our arguments we use the method of Petrov and Pohoata together with some Gröbner basis techniques.

On the space of generalized theta-series for certain quadratic forms in any number of variables

2019 ◽  
Vol 69 (1) ◽  
pp. 87-98
Author(s):  
Ketevan Shavgulidze
Keyword(s):  
Upper Bound ◽  
Quadratic Forms ◽  
Theta Series ◽  
Vector Spaces ◽  
Generalized Theta Series

Abstract An upper bound of the dimension of vector spaces of generalized theta-series corresponding to some nondiagonal quadratic forms in any number of variables is established. In a number of cases, an upper bound of the dimension of the space of theta-series with respect to the quadratic forms of five variables is improved and the basis of this space is constructed.

On Positive Integer Solutions of the Equation xy + yz + xz = n

1996 ◽  
Vol 39 (2) ◽  
pp. 199-202 ◽  
Author(s):  
Al-Zaid Hassan ◽  
B. Brindza ◽  
Á. Pintér
Keyword(s):  
Positive Integer ◽  
Upper Bound ◽  
Class Number ◽  
Quadratic Forms ◽  
Negative Integer ◽  
Number Of Solutions ◽  
Integer Solutions

AbstractAs it had been recognized by Liouville, Hermite, Mordell and others, the number of non-negative integer solutions of the equation in the title is strongly related to the class number of quadratic forms with discriminant —n. The purpose of this note is to point out a deeper relation which makes it possible to derive a reasonable upper bound for the number of solutions.

scholarly journals Non-negative values of quadratic forms

1971 ◽  
Vol 12 (2) ◽  
pp. 224-238 ◽  
Author(s):  
R. T. Worley
Keyword(s):  
Quadratic Form ◽  
Upper Bound ◽  
Quadratic Forms ◽  
Simple Consequence ◽  
Ternary Quadratic Forms ◽  
Indefinite Quadratic Form ◽  
Indefinite Quadratic

In a paper [1] of the same title Barnes considered the problem of finding an upper bound for the infimum m+(f) of the non-negative values1 of an indefinite quadratic form f in n variables, of given determinant det(f) ≠ 0 and of signature s. In particular it was announced (and later proved — see [2]) that m+(f) ≦ (16/5)+ for ternary quadratic forms of determinant 1 and signature — 1. A simple consequence of this result is that m+(f) ≦ (256/135)+ for quaternary quadratic forms of determinant — 1 and signature — 2.

Polynomial bounds for equivalence of quadratic forms with cube-free determinant

2007 ◽  
Vol 143 (3) ◽  
pp. 521-532 ◽  
Author(s):  
RAINER DIETMANN
Keyword(s):  
Upper Bound ◽  
Quadratic Forms ◽  
The Other ◽  
Unimodular Matrix ◽  
Polynomial Bounds

AbstractGiven two integrally equivalent integral quadratic forms in at least three variables and with cube-free determinant, we establish an upper bound on the smallest unimodular matrix transforming one of the forms into the other. This bound is polynomial in the height of the two forms involved, confirming a conjecture of Masser for the class of forms considered.

scholarly journals CONCENTRATION OF POINTS ON MODULAR QUADRATIC FORMS

2011 ◽  
Vol 07 (07) ◽  
pp. 1835-1839 ◽  
Author(s):  
ANA ZUMALACÁRREGUI
Keyword(s):  
Quadratic Form ◽  
Upper Bound ◽  
Quadratic Forms ◽  
Irreducible Polynomial ◽  
Number Of Solutions ◽  
Modulo P ◽  
Absolutely Irreducible Polynomial ◽  
Mod P

Let Q(x, y) be a quadratic form with discriminant D ≠ 0. We obtain non-trivial upper bound estimates for the number of solutions of the congruence Q(x, y) ≡ λ ( mod p), where p is a prime and x, y lie in certain intervals of length M, under the assumption that Q(x, y) - λ is an absolutely irreducible polynomial modulo p. In particular, we prove that the number of solutions to this congruence is Mo(1) when M ≪ p1/4. These estimates generalize a previous result by Cilleruelo and Garaev on the particular congruence xy ≡ λ( mod p).

scholarly journals On pairs of quadratic forms in five variables

2021 ◽  
pp. 1-16
Author(s):  
Kummari Mallesham
Keyword(s):  
Upper Bound ◽  
Quadratic Forms ◽  
Circle Method ◽  
Integral Solutions

In this paper, we obtain an upper bound for the number of integral solutions, of given height, of system of two quadratic forms in five variables. Our bound is an improvement over the bound given in [H. Iwaniec and R. Munshi, The circle method and pairs of quadratic forms, J. Théor. Nr. Bordx. 22 (2010) 403–419].

Small Zeros of Quadratic Forms Avoiding a Finite Number of Prescribed Hyperplanes

2009 ◽  
Vol 52 (1) ◽  
pp. 63-65 ◽  
Author(s):  
Rainer Dietmann
Keyword(s):  
Quadratic Form ◽  
Finite Number ◽  
Number Field ◽  
Upper Bound ◽  
Quadratic Forms ◽  
Additional Restriction

AbstractWe prove a new upper bound for the smallest zero x of a quadratic form over a number field with the additional restriction that x does not lie in a finite number of m prescribed hyperplanes. Our bound is polynomial in the height of the quadratic form, with an exponent depending only on the number of variables but not on m.

On the Dimension of Some Spaces of Generalized Ternary Theta-Series

2002 ◽  
Vol 9 (1) ◽  
pp. 167-178
Author(s):  
K. Shavgulidze
Keyword(s):  
Upper Bound ◽  
Quadratic Forms ◽  
Theta Series ◽  
Vector Spaces ◽  
Ternary Quadratic Forms ◽  
Generalized Theta Series

Abstract The upper bound of dimension of vector spaces of generalized theta-series corresponding to some ternary quadratic forms is established. In a number of cases, the dimension of vector spaces of generalized theta-series is established and bases of these spaces are constructed.

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