scholarly journals On Directions Determined by Subsets of Vector Spaces over Finite Fields

Integers ◽  
2011 ◽  
Vol 11 (6) ◽  
Author(s):  
Alex Iosevich ◽  
Hannah Morgan ◽  
Jonathan Pakianathan
Keyword(s):  
Finite Fields ◽  
Vector Spaces

AbstractWe prove that if a subset of a

scholarly journals A note on extensions of multilinear maps defined on multilinear varieties

2021 ◽  
pp. 1-26
Author(s):  
W. T. Gowers ◽  
L. Milićević
Keyword(s):  
Finite Fields ◽  
Vector Spaces ◽  
Restriction Map ◽  
Zero Set ◽  
Dimensional Vector ◽  
Prime Field ◽  
Finite Dimensional ◽  
Multilinear Maps ◽  
Multilinear Map

Abstract Let $G_1, \ldots , G_k$ be finite-dimensional vector spaces over a prime field $\mathbb {F}_p$ . A multilinear variety of codimension at most $d$ is a subset of $G_1 \times \cdots \times G_k$ defined as the zero set of $d$ forms, each of which is multilinear on some subset of the coordinates. A map $\phi$ defined on a multilinear variety $B$ is multilinear if for each coordinate $c$ and all choices of $x_i \in G_i$ , $i\not =c$ , the restriction map $y \mapsto \phi (x_1, \ldots , x_{c-1}, y, x_{c+1}, \ldots , x_k)$ is linear where defined. In this note, we show that a multilinear map defined on a multilinear variety of codimension at most $d$ coincides on a multilinear variety of codimension $O_{k}(d^{O_{k}(1)})$ with a multilinear map defined on the whole of $G_1\times \cdots \times G_k$ . Additionally, in the case of general finite fields, we deduce similar (but slightly weaker) results.

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.

On groupoids representable by vector spaces over finite fields

1973 ◽  
Vol 24 (1) ◽  
pp. 14-20 ◽  
Author(s):  
A. Mitschke ◽  
H. Werner
Keyword(s):  
Finite Fields ◽  
Vector Spaces

Distances in vector spaces over finite fields

2010 ◽  
pp. 119-125
Author(s):  
Julia Garibaldi ◽  
Alex Iosevich ◽  
Steven Senger
Keyword(s):  
Finite Fields ◽  
Vector Spaces

scholarly journals Subspace Codes Based on Partial Injective Maps of Vector Spaces Over Finite Fields

IEEE Access ◽  
2020 ◽  
Vol 8 ◽  
pp. 192608-192615
Author(s):  
Hong-Li Wang ◽  
Gang Wang ◽  
You Gao
Keyword(s):  
Finite Fields ◽  
Vector Spaces ◽  
Subspace Codes ◽  
Injective Maps

scholarly journals LONG PATHS IN THE DISTANCE GRAPH OVER LARGE SUBSETS OF VECTOR SPACES OVER FINITE FIELDS

2016 ◽  
Vol 53 (1) ◽  
pp. 115-126 ◽  
Author(s):  
MICHAEL BENNETT ◽  
JEREMY CHAPMAN ◽  
DAVID COVERT ◽  
DERRICK HART ◽  
ALEX IOSEVICH ◽  
...  
Keyword(s):  
Finite Fields ◽  
Distance Graph ◽  
Vector Spaces ◽  
Long Paths
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