scholarly journals Distinct Triangle Areas in a Planar Point Set over Finite Fields

10.37236/700 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Le Anh Vinh
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Prime Power ◽  
Finite Plane ◽  
Expected Number ◽  
Planar Point ◽  
Point Set ◽  
Distinct Triangle

Let $\mathcal{P}$ be a set of $n$ points in the finite plane $\mathbb{F}_q^2$ over the finite field $\mathbb{F}_q$ of $q$ elements, where $q$ is an odd prime power. For any $s \in \mathbb{F}_q$, denote by $A (\mathcal{P}; s)$ the number of ordered triangles whose vertices in $\mathcal{P}$ having area $s$. We show that if the cardinality of $\mathcal{P}$ is large enough then $A (\mathcal{P}; s)$ is close to the expected number $|\mathcal{P}|^3/q$.

scholarly journals Incidences between points and generalized spheres over finite fields and related problems

2017 ◽  
Vol 29 (2) ◽  
pp. 449-456 ◽  
Author(s):  
Nguyen D. Phuong ◽  
Pham Thang ◽  
Le A. Vinh
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Prime Power ◽  
Combinatorial Geometry ◽  
Random Point ◽  
Large Subset ◽  
Point Set ◽  
Distance Problem ◽  
Generalized Distances ◽  
Isosceles Triangles

AbstractLet ${\mathbb{F}_{q}}$ be a finite field of q elements, where q is a large odd prime power and${Q=a_{1}x_{1}^{c_{1}}+\cdots+a_{d}x_{d}^{c_{d}}\in\mathbb{F}_{q}[x_{1},\ldots,% x_{d}]},$where ${2\leq c_{i}\leq N}$, ${\gcd(c_{i},q)=1}$, and ${a_{i}\in\mathbb{F}_{q}}$ for all ${1\leq i\leq d}$. A Q-sphere is a set of the form ${\bigl{\{}\boldsymbol{x}\in\mathbb{F}_{q}^{d}\mid Q(\boldsymbol{x}-\boldsymbol% {b})=r\bigr{\}}},$where ${\boldsymbol{b}\in\mathbb{F}_{q}^{d},r\in\mathbb{F}_{q}}$. We prove bounds on the number of incidences between a point set ${{{\mathcal{P}}}}$ and a Q-sphere set ${{{\mathcal{S}}}}$, denoted by ${I({{\mathcal{P}}},{{\mathcal{S}}})}$, as the following:$\Biggl{|}I({{\mathcal{P}}},{{\mathcal{S}}})-\frac{|{{\mathcal{P}}}||{{\mathcal% {S}}}|}{q}\Biggr{|}\leq q^{d/2}\sqrt{|{{\mathcal{P}}}||{{\mathcal{S}}}|}.$We also give a version of this estimate over finite cyclic rings ${\mathbb{Z}/q\mathbb{Z}}$, where q is an odd integer. As a consequence of the above bounds, we give an estimate for the pinned distance problem and a bound on the number of incidences between a random point set and a random Q-sphere set in ${\mathbb{F}_{q}^{d}}$. We also study the finite field analogues of some combinatorial geometry problems, namely, the number of generalized isosceles triangles, and the existence of a large subset without repeated generalized distances.

Zeros of random polynomials over finite fields

1992 ◽  
Vol 111 (2) ◽  
pp. 193-197 ◽  
Author(s):  
R. W. K. Odoni
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Prime Power ◽  
Random Polynomials ◽  
Polynomials Over Finite Fields ◽  
Image Position

Let be the finite field with q elements (q a prime power), let r 1 and let X1, , Xr be independent indeterminates over . We choose an arbitrary and a d 1 and consider

scholarly journals Counting Non-Convex 5-Holes in a Planar Point Set

Symmetry ◽  
2022 ◽  
Vol 14 (1) ◽  
pp. 78
Author(s):  
Young-Hun Sung ◽  
Sang Won Bae
Keyword(s):  
General Position ◽  
Brute Force ◽  
Expected Number ◽  
Planar Point ◽  
Point Set ◽  
Empty Triangles ◽  
Bounded Body

Let S be a set of n points in the general position, that is, no three points in S are collinear. A simple k-gon with all corners in S such that its interior avoids any point of S is called a k-hole. In this paper, we present the first algorithm that counts the number of non-convex 5-holes in S. To our best knowledge, prior to this work there was no known algorithm in the literature except a trivial brute force algorithm. Our algorithm runs in time O(T+Q), where T denotes the number of 3-holes, or empty triangles, in S and Q that denotes the number of non-convex 4-holes in S. Note that T+Q ranges from Ω(n2) to O(n3), while its expected number is Θ(n2logn) when the points in S are chosen uniformly and independently at random from a convex and bounded body in the plane.

scholarly journals ON SUM OF PRODUCTS AND THE ERDŐS DISTANCE PROBLEM OVER FINITE FIELDS

2011 ◽  
Vol 84 (1) ◽  
pp. 1-9
Author(s):  
LE ANH VINH
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Prime Power ◽  
Sum Of Products ◽  
Distance Problem ◽  
Erdős Distance Problem ◽  
Erdos Distance Problem

AbstractFor a prime powerq, let 𝔽qbe the finite field ofqelements. We show that 𝔽*q⊆d𝒜2for almost every subset 𝒜⊂𝔽qof cardinality ∣𝒜∣≫q1/d. Furthermore, ifq=pis a prime, and 𝒜⊆𝔽pof cardinality ∣𝒜∣≫p1/2(logp)1/d, thend𝒜2contains both large and small residues. We also obtain some results of this type for the Erdős distance problem over finite fields.

Almost empty polygons

2003 ◽  
Vol 40 (3) ◽  
pp. 269-286 ◽  
Author(s):  
H. Nyklová
Keyword(s):  
Exact Results ◽  
Point Sets ◽  
Planar Point ◽  
Convex Position ◽  
Vertex Set ◽  
Point Set ◽  
Empty Polygons

In this paper we study a problem related to the classical Erdos--Szekeres Theorem on finding points in convex position in planar point sets. We study for which n and k there exists a number h(n,k) such that in every planar point set X of size h(n,k) or larger, no three points on a line, we can find n points forming a vertex set of a convex n-gon with at most k points of X in its interior. Recall that h(n,0) does not exist for n = 7 by a result of Horton. In this paper we prove the following results. First, using Horton's construction with no empty 7-gon we obtain that h(n,k) does not exist for k = 2(n+6)/4-n-3. Then we give some exact results for convex hexagons: every point set containing a convex hexagon contains a convex hexagon with at most seven points inside it, and any such set of at least 19 points contains a convex hexagon with at most five points inside it.

Maximal Sets of Pairwise Orthogonal Vectors in Finite Fields

2012 ◽  
Vol 55 (2) ◽  
pp. 418-423 ◽  
Author(s):  
Le Anh Vinh
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Finite Fields ◽  
Bilinear Form ◽  
Symmetric Bilinear Form

AbstractGiven a positive integern, a finite fieldofqelements (qodd), and a non-degenerate symmetric bilinear formBon, we determine the largest possible cardinality of pairwiseB-orthogonal subsets, that is, for any two vectorsx,y∈ Ε, one hasB(x,y) = 0.

scholarly journals Short Kloosterman Sums for Polynomials over Finite Fields

2003 ◽  
Vol 55 (2) ◽  
pp. 225-246 ◽  
Author(s):  
William D. Banks ◽  
Asma Harcharras ◽  
Igor E. Shparlinski
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Small Degree ◽  
Kloosterman Sums ◽  
Degree Relative ◽  
Irreducible Polynomials ◽  
Arbitrary Polynomial ◽  
Polynomials Over Finite Fields ◽  
Titchmarsh Theorem

AbstractWe extend to the setting of polynomials over a finite field certain estimates for short Kloosterman sums originally due to Karatsuba. Our estimates are then used to establish some uniformity of distribution results in the ring [x]/M(x) for collections of polynomials either of the form f−1g−1 or of the form f−1g−1 + afg, where f and g are polynomials coprime to M and of very small degree relative to M, and a is an arbitrary polynomial. We also give estimates for short Kloosterman sums where the summation runs over products of two irreducible polynomials of small degree. It is likely that this result can be used to give an improvement of the Brun-Titchmarsh theorem for polynomials over finite fields.

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