scholarly journals Polynomial maps on vector spaces over a finite field

2015 ◽  
Vol 31 ◽  
pp. 1-7 ◽  
Author(s):  
Michiel Kosters
Keyword(s):  
Finite Field ◽  
Vector Spaces ◽  
Polynomial Maps

The genus of graphs associated with vector spaces

2019 ◽  
Vol 19 (05) ◽  
pp. 2050086 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
K. Prabha Ananthi
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Undirected Graph ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Vertex Connectivity ◽  
Dimensional Vector Space ◽  
Nonzero Component ◽  
Vertex Set ◽  
Component Graph

Let [Formula: see text] be a k-dimensional vector space over a finite field [Formula: see text] with a basis [Formula: see text]. The nonzero component graph of [Formula: see text], denoted by [Formula: see text], is a simple undirected graph with vertex set as nonzero vectors of [Formula: see text] such that there is an edge between two distinct vertices [Formula: see text] if and only if there exists at least one [Formula: see text] along which both [Formula: see text] and [Formula: see text] have nonzero scalars. In this paper, we find the vertex connectivity and girth of [Formula: see text]. We also characterize all vector spaces [Formula: see text] for which [Formula: see text] has genus either 0 or 1 or 2.

scholarly journals Restriction Operators Acting on Radial Functions on Vector Spaces over Finite Fields

2014 ◽  
Vol 57 (4) ◽  
pp. 834-844
Author(s):  
Doowon Koh
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Algebraic Varieties ◽  
Test Functions ◽  
Radial Functions ◽  
Zero Radius ◽  
Intersection Points ◽  
Field Setting

AbstractWe study Lp → Lr restriction estimates for algebraic varieties V in the case when restriction operators act on radial functions in the finite field setting. We show that if the varieties V lie in odd dimensional vector spaces over finite fields, then the conjectured restriction estimates are possible for all radial test functions. In addition, assuming that the varieties V are defined in even dimensional spaces and have few intersection points with the sphere of zero radius, we also obtain the conjectured exponents for all radial test functions.

scholarly journals FUSION OVER A VECTOR SPACE

2006 ◽  
Vol 06 (02) ◽  
pp. 141-162 ◽  
Author(s):  
ANDREAS BAUDISCH ◽  
AMADOR MARTIN-PIZARRO ◽  
MARTIN ZIEGLER
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Vector Spaces

Let T1 and T2 be two countable strongly minimal theories with the DMP whose common theory is the theory of vector spaces over a fixed finite field. We show that T1 ∪ T2 has a strongly minimal completion.

scholarly journals An analogue of Vosper's theorem for extension fields

2017 ◽  
Vol 163 (3) ◽  
pp. 423-452 ◽  
Author(s):  
CHRISTINE BACHOC ◽  
ORIOL SERRA ◽  
GILLES ZÉMOR
Keyword(s):  
Finite Field ◽  
Linear Span ◽  
Field Extension ◽  
Small Dimension ◽  
Vector Spaces ◽  
Prime Extension ◽  
Linear Subspaces ◽  
Formula Group ◽  
Central Result ◽  
Image Position

AbstractWe are interested in characterising pairs S, T of F-linear subspaces in a field extension L/F such that the linear span ST of the set of products of elements of S and of elements of T has small dimension. Our central result is a linear analogue of Vosper's Theorem, which gives the structure of vector spaces S, T in a prime extension L of a finite field F for which \begin{linenomath}$$ \dim_FST =\dim_F S+\dim_F T-1, $$\end{linenomath} when dimFS, dimFT ⩾ 2 and dimFST ⩽ [L : F] − 2.

scholarly journals Fast Jacobian Group Operations for C3,4 Curves over a Large Finite Field

2007 ◽  
Vol 10 ◽  
pp. 307-328 ◽  
Author(s):  
Fatima K. Abu Salem ◽  
Kamal khuri-makdisi
Keyword(s):  
Finite Field ◽  
Linear Algebra ◽  
Algebraic Curve ◽  
Vector Spaces ◽  
Explicit Formulae ◽  
Group Law

Let C be an arbitrary smooth algebraic curve of genus g over a large finite field F. The authors of this paper revisit fast addition algorithms in the Jacobian of C due to Khuri-Makdisi [math.NT/0409209, to appear in Mathematics of Computation]. The algorithms, which reduce to linear algebra in vector spaces of dimension O(g) once |K| ≫ g and which asymptotically require O(g2.376) field operations using fast linear algebra, are shown to perform efficiently even for certain low genus curves. Specifically, the authors provide explicit formulae for performing the group law on Jacobians of C3,4 curves of genus 3. They show show that, typically, the addition of two distinct elements in the Jacobian of a C3,4 curve requires 117 multiplications and 2 inversions in K, and an element can be doubled using 129 multiplications and 2 inversions in K. This represents an improvement of approximately 20% over previous methods.

scholarly journals On $k$-simplexes in $(2k-1)$-dimensional vector spaces over finite fields

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Le Anh Vinh
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Finite Fields ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Nous Montrons ◽  
International Audience

International audience We show that if the cardinality of a subset of the $(2k-1)$-dimensional vector space over a finite field with $q$ elements is $\gg q^{2k-1-\frac{1}{ 2k}}$, then it contains a positive proportional of all $k$-simplexes up to congruence. Nous montrons que si la cardinalité d'un sous-ensemble de l'espace vectoriel à $(2k-1)$ dimensions sur un corps fini à $q$ éléments est $\gg q^{2k-1-\frac{1}{ 2k}}$, alors il contient une proportion non-nulle de tous les $k$-simplexes de congruence.

scholarly journals ON POINT SETS IN VECTOR SPACES OVER FINITE FIELDS THAT DETERMINE ONLY ACUTE ANGLE TRIANGLES

2009 ◽  
Vol 81 (1) ◽  
pp. 114-120 ◽  
Author(s):  
IGOR E. SHPARLINSKI
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Upper Bound ◽  
Dimensional Space ◽  
Real Space ◽  
Acute Angle ◽  
Vector Spaces ◽  
Point Sets ◽  
Similar Question ◽  
The Real

AbstractFor three points$\vec {u}$,$\vec {v}$and$\vec {w}$in then-dimensional space 𝔽nqover the finite field 𝔽qofqelements we give a natural interpretation of an acute angle triangle defined by these points. We obtain an upper bound on the size of a set 𝒵 such that all triples of distinct points$\vec {u}, \vec {v}, \vec {w} \in \cZ $define acute angle triangles. A similar question in the real space ℛndates back to P. Erdős and has been studied by several authors.

scholarly journals Corrigendum: ‘On certain algebraic curves related to polynomial maps, Compositio Math. 103 (1996), 319–350’

2010 ◽  
Vol 147 (1) ◽  
pp. 332-334 ◽  
Author(s):  
Patrick Morton
Keyword(s):  
Finite Field ◽  
Laurent Series ◽  
Algebraic Curves ◽  
Periodic Points ◽  
Series Expansions ◽  
Rational Field ◽  
Polynomial Maps ◽  
Polynomial Map ◽  
Hensel’S Lemma ◽  
Compositio Math

AbstractAn argument is given to fill a gap in a proof in the author’s article On certain algebraic curves related to polynomial maps, Compositio Math. 103 (1996), 319–350, that the polynomial Φn(x,c), whose roots are the periodic points of period n of a certain polynomial map x→f(x,c), is absolutely irreducible over the finite field of p elements, provided that f(x,1) has distinct roots and that the multipliers of the orbits of period n are also distinct over $\mathbb { F}_p$. Assuming that Φn(x,c) is reducible in characteristic p, we show that Hensel’s lemma and Laurent series expansions of the roots can be used to obtain a factorization of Φn(x,c) in characteristic 0, contradicting the absolute irreducibility of this polynomial over the rational field.

The analog of the Erdös distance problem in finite fields

2017 ◽  
Vol 13 (09) ◽  
pp. 2319-2333
Author(s):  
S. D. Adhikari ◽  
Anirban Mukhopadhyay ◽  
M. Ram Murty
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Kloosterman Sums ◽  
Vector Spaces ◽  
Characteristic 2 ◽  
Distance Problem ◽  
Erdős Distance Problem ◽  
Erdos Distance Problem

In this paper, we give a proof of the result of Iosevich and Rudnev [Erdös distance problem in vector spaces over finite fields, Trans. Amer. Math. Soc. 359 (2007) 6127–6142] on the analog of the Erdös–Falconer distance problem in the case of a finite field of characteristic [Formula: see text], where [Formula: see text] is an odd prime, without using estimates for Kloosterman sums. We also address the case of characteristic 2.

scholarly journals NONCOMMUTATIVE POLYNOMIAL MAPS

2012 ◽  
Vol 11 (04) ◽  
pp. 1250076 ◽  
Author(s):  
ANDRÉ LEROY
Keyword(s):  
Finite Field ◽  
Polynomial Ring ◽  
Ore Extension ◽  
Linear Transformations ◽  
Frobenius Automorphism ◽  
Polynomial Maps ◽  
Noncommutative Polynomial

Polynomial maps attached to polynomials of an Ore extension are naturally defined. In this setting we show the importance of pseudo-linear transformations and give some applications. In particular, factorizations of polynomials in an Ore extension over a finite field 𝔽q[t;θ], where θ is the Frobenius automorphism, are translated into factorizations in the usual polynomial ring 𝔽q[x].

Sign in / Sign up

Export Citation Format

Share Document