On common zeros of a pair of quadratic forms over a finite field

2018 ◽  
Vol 51 ◽  
pp. 191-203 ◽  
Author(s):  
A.S. Sivatski
Keyword(s):  
Finite Field ◽  
Quadratic Forms ◽  
Common Zeros

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.

The Distribution of Zeros of Quadratic Forms over Finite Fields

2015 ◽  
Vol 7 (2) ◽  
pp. 18
Author(s):  
Ali H. Hakami
Keyword(s):  
Positive Integer ◽  
Quadratic Form ◽  
Finite Field ◽  
Finite Fields ◽  
Lower Bounds ◽  
Quadratic Forms ◽  
Distribution Of Zeros ◽  
Linearly Independent ◽  
Set Of Points

Let $m$ be a positive integer with $m < p/2$ and $p$ is a prime. Let $\mathbb{F}_q$ be the finite field in $q = p^f$ elements, $Q({\mathbf{x}})$ be a nonsinqular quadratic form over $\mathbb{F}_q$ with $q$ odd, $V$ be the set of points in $\mathbb{F}_q^n$ satisfying the equation $Q({\mathbf{x}}) = 0$ in which the variables are restricted to a box of points of the type\[\mathcal{B}(m) = \left\{ {{\mathbf{x}} \in \mathbb{F}_q^n \left| {x_i  = \sum\limits_{j = 1}^f {x_{ij} \xi _j } ,\;\left| {x_{ij} } \right| < m,\;1 \leqslant i \leqslant n,\;1 \leqslant j \leqslant f} \right.} \right\},\]where $\xi _1 , \ldots ,\xi _f$ is a basis for $\mathbb{F}_q$ over $\mathbb{F}_p$ and $n > 2$ even. Set $\Delta  = \det Q$ such that $\chi \left( {( - 1)^{n/2} \Delta } \right) = 1.$ We shall motivate work of (Cochrane, 1986) to obtain lower bounds on $m,$ size of the box $\mathcal{B},$ so that $\mathcal{B} \cap V$ is nonempty. For this we show that the box $\mathcal{B}(m)$ contains a zero of $Q({\mathbf{x}})$ provided that $m \geqslant p^{1/2}.$ We also show that the box $\mathcal{B}(m)$ contains $n$ linearly independent zeros of $Q({\mathbf{x}})$ provided that $m \geqslant 2^{n/2} p^{1/2} .$

scholarly journals Pairs of Quadratic Forms over Finite Fields

10.37236/4072 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Alexander Pott ◽  
Kai-Uwe Schmidt ◽  
Yue Zhou
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Boolean Functions ◽  
Quadratic Forms ◽  
Symmetric Matrices ◽  
Explicit Expressions

Let $\mathbb{F}_q$ be a finite field with $q$ elements and let $X$ be a set of matrices over $\mathbb{F}_q$. The main results of this paper are explicit expressions for the number of pairs $(A,B)$ of matrices in $X$ such that $A$ has rank $r$, $B$ has rank $s$, and $A+B$ has rank $k$ in the cases that (i) $X$ is the set of alternating matrices over $\mathbb{F}_q$ and (ii) $X$ is the set of symmetric matrices over $\mathbb{F}_q$ for odd $q$. Our motivation to study these sets comes from their relationships to quadratic forms. As one application, we obtain the number of quadratic Boolean functions that are simultaneously bent and negabent, which solves a problem due to Parker and Pott.

The Zeta Function of a Pair of Quadratic Forms

2001 ◽  
Vol 44 (2) ◽  
pp. 242-256
Author(s):  
Laura Mann Schueller
Keyword(s):  
Finite Field ◽  
Zeta Function ◽  
Function Field ◽  
Quadratic Forms ◽  
Arbitrary Characteristic ◽  
Number Of Zeros ◽  
Even Order ◽  
Hyperelliptic Function

AbstractThe zeta function of a nonsingular pair of quadratic forms defined over a finite field, k, of arbitrary characteristic is calculated. A. Weil made this computation when char k ≠ 2. When the pair has even order, a relationship between the number of zeros of the pair and the number of places of degree one in an appropriate hyperelliptic function field is established.

Association schemes of quadratic forms over a finite field of characteristic two

1998 ◽  
Vol 43 (23) ◽  
pp. 1965-1968 ◽  
Author(s):  
Yangxian Wang ◽  
Chunsen Wang ◽  
Changli Ma
Keyword(s):  
Finite Field ◽  
Quadratic Forms ◽  
Association Schemes ◽  
Characteristic Two

On the Number of Solutions of Some General Types of Equations in a Finite Field

1952 ◽  
Vol 4 ◽  
pp. 343-351 ◽  
Author(s):  
Olin B. Faircloth
Keyword(s):  
Number Theory ◽  
Finite Field ◽  
Riemann Hypothesis ◽  
Quadratic Forms ◽  
Function Fields ◽  
Commutative Rings ◽  
Number Of Solutions ◽  
Fermat's Last Theorem ◽  
Conditional Equation ◽  
And Algebra

The conditional equation f(x1, … , xs) = 0, where f is a polynomial in the x´s with coefficients in a finite field F(pn), is connected with many well-known developments in number theory and algebra, such as: Waring's problem, the arithmetical theory of quadratic forms, the Riemann hypothesis for function fields, Fermat's Last Theorem, cyclotomy, and the theory of congruences in commutative rings.

scholarly journals The existence of primitive normal elements of quadratic forms over finite fields

2020 ◽  
pp. 2250068
Author(s):  
Himangshu Hazarika ◽  
Dhiren Kumar Basnet ◽  
Stephen D. Cohen
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Finite Fields ◽  
Quadratic Forms ◽  
Primitive Element ◽  
Sufficient Condition ◽  
Extension Field ◽  
Normal Element

For [Formula: see text] ([Formula: see text]), denote by [Formula: see text] the finite field of order [Formula: see text] and for a positive integer [Formula: see text], let [Formula: see text] be its extension field of degree [Formula: see text]. We establish a sufficient condition for existence of a primitive normal element [Formula: see text] such that [Formula: see text] is a primitive element, where [Formula: see text], with [Formula: see text] satisfying [Formula: see text] in [Formula: see text].

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