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 CONGRUENCES CONCERNING LEGENDRE POLYNOMIALS III

2013 ◽  
Vol 09 (04) ◽  
pp. 965-999 ◽  
Author(s):  
ZHI-HONG SUN
Keyword(s):  
Positive Integer ◽  
Quadratic Forms ◽  
Legendre Polynomials ◽  
Binary Quadratic Forms ◽  
Modulo P ◽  
Mod P

Suppose that p is an odd prime and d is a positive integer. Let x and y be integers given by p = x2+dy2 or 4p = x2+dy2. In this paper we determine x( mod p) for many values of d. For example, [Formula: see text] where x is chosen so that x ≡ 1 ( mod 3). We also pose some conjectures on supercongruences modulo p2 concerning binary quadratic forms.

scholarly journals Primitive zeros of quadratic forms mod $p^2$

2015 ◽  
Vol 46 (3) ◽  
pp. 349-364
Author(s):  
Ali H. Hakami
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Integer Coefficients ◽  
Mod P

Let $Q({\bf{x}}) = Q(x_1 ,x_2 ,\ldots,x_n )$ be a quadratic form with integer coefficients, $p$ be an odd prime and $\left\| \bf{x} \right\| = \max _i \left| {x_i } \right|.$ A solution of the congruence $Q({\mathbf{x}}) \equiv {\mathbf{0}}\;(\bmod\; p^2 )$ is said to be a primitive solution if $p\nmid x_i $ for some $i$. In this paper, we seek to obtain primitive solutions of this congruence in small rectangular boxes of the type $ \mathcal{B} = \{ {\mathbf{x}} \in \mathbb{Z}^n : |x_i| \le M_i ,\;1 \leqslant i \leqslant n\} $ where for $1 \le i \le l$ we have $M_i \le p$, while for $i>l$ we have $M_i>p$. In particular, we show that if $n \ge 4$, $n$ even, $l \le \frac n2-2$, and $Q$ is nonsingular $\pmod p$, then there exists a primitive solution with $x_i = 0$, $1 \le i \le l$, and $|x_i| \le 2^{\frac {4n+3}{n-l}} p^{\frac n{n-l}} +1$, for $l<i \le n$.

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.

SYSTEMS OF CONGRUENCES WITH PRODUCTS OF VARIABLES FROM SHORT INTERVALS

2015 ◽  
Vol 93 (3) ◽  
pp. 364-371
Author(s):  
IGOR E. SHPARLINSKI
Keyword(s):  
Upper Bound ◽  
Number Of Solutions ◽  
Short Intervals ◽  
Image Position ◽  
Mod P

We obtain an upper bound for the number of solutions to the system of $m$ congruences of the type $$\begin{eqnarray}\displaystyle \mathop{\prod }_{i=1}^{{\it\nu}}(x_{i}+s_{i})\equiv {\it\lambda}_{j}~(\text{mod }p)\quad j=1,\ldots ,m, & & \displaystyle \nonumber\end{eqnarray}$$ modulo a prime $p$, with variables $1\leq x_{i}\leq h$, $i=1,\ldots ,{\it\nu}$ and arbitrary integers $s_{j},{\it\lambda}_{j}$, $j=1,\ldots ,m$, for a parameter $h$ significantly smaller than $p$. We also mention some applications of this bound.

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.

scholarly journals A BOUND FOR THE INDEX OF A QUADRATIC FORM AFTER SCALAR EXTENSION TO THE FUNCTION FIELD OF A QUADRIC

2018 ◽  
Vol 19 (2) ◽  
pp. 421-450 ◽  
Author(s):  
Stephen Scully
Keyword(s):  
Quadratic Form ◽  
Upper Bound ◽  
Function Field ◽  
The Other ◽  
Direct Generalization ◽  
Other Hand ◽  
General Field ◽  
The One ◽  
Anisotropic Quadratic Form

Let $q$ be an anisotropic quadratic form defined over a general field $F$. In this article, we formulate a new upper bound for the isotropy index of $q$ after scalar extension to the function field of an arbitrary quadric. On the one hand, this bound offers a refinement of an important bound established in earlier work of Karpenko–Merkurjev and Totaro; on the other hand, it is a direct generalization of Karpenko’s theorem on the possible values of the first higher isotropy index. We prove its validity in two key cases: (i) the case where $\text{char}(F)\neq 2$, and (ii) the case where $\text{char}(F)=2$ and $q$ is quasilinear (i.e., diagonalizable). The two cases are treated separately using completely different approaches, the first being algebraic–geometric, and the second being purely algebraic.

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.

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