scholarly journals Counting Points of Slope Varieties over Finite Fields

10.37236/490 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Thomas Enkosky
Keyword(s):  
Finite Fields ◽  
Complete Graph ◽  
Algebraic Set ◽  
Reducible Graph ◽  
Counting Points

The slope variety of a graph is an algebraic set whose points correspond to drawings of that graph. A complement-reducible graph (or cograph) is a graph without an induced four-vertex path. We construct a bijection between the zeroes of the slope variety of the complete graph on $n$ vertices over $\mathbb{F}_2$, and the complement-reducible graphs on $n$ vertices.

scholarly journals Atomic Latin Squares based on Cyclotomic Orthomorphisms

10.37236/1919 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Ian M. Wanless
Keyword(s):  
Finite Fields ◽  
Complete Graph ◽  
Prime Order ◽  
Automorphism Groups ◽  
Latin Square ◽  
Latin Squares ◽  
Established Method ◽  
Cyclic Groups ◽  
Complete Bipartite Graphs ◽  
First Time

Atomic latin squares have indivisible structure which mimics that of the cyclic groups of prime order. They are related to perfect $1$-factorisations of complete bipartite graphs. Only one example of an atomic latin square of a composite order (namely 27) was previously known. We show that this one example can be generated by an established method of constructing latin squares using cyclotomic orthomorphisms in finite fields. The same method is used in this paper to construct atomic latin squares of composite orders 25, 49, 121, 125, 289, 361, 625, 841, 1369, 1849, 2809, 4489, 24649 and 39601. It is also used to construct many new atomic latin squares of prime order and perfect $1$-factorisations of the complete graph $K_{q+1}$ for many prime powers $q$. As a result, existence of such a factorisation is shown for the first time for $q$ in $\big\{$529, 2809, 4489, 6889, 11449, 11881, 15625, 22201, 24389, 24649, 26569, 29929, 32041, 38809, 44521, 50653, 51529, 52441, 63001, 72361, 76729, 78125, 79507, 103823, 148877, 161051, 205379, 226981, 300763, 357911, 371293, 493039, 571787$\big\}$. We show that latin squares built by the 'orthomorphism method' have large automorphism groups and we discuss conditions under which different orthomorphisms produce isomorphic latin squares. We also introduce an invariant called the train of a latin square, which proves to be useful for distinguishing non-isomorphic examples.

Counting points on double octic Calabi–Yau threefolds over finite fields

2020 ◽  
Vol 125 (3) ◽  
pp. 215-227
Author(s):  
Aleksander Czarnecki
Keyword(s):  
Finite Fields ◽  
Counting Points

scholarly journals Counting Points on Curves and Abelian Varieties Over Finite Fields

2001 ◽  
Vol 32 (3) ◽  
pp. 171-189 ◽  
Author(s):  
Leonard M. Adleman ◽  
Ming-Deh Huang
Keyword(s):  
Finite Fields ◽  
Abelian Varieties ◽  
Counting Points

scholarly journals Equivariant Poincaré Polynomials and Counting Points over Finite Fields

2002 ◽  
Vol 247 (2) ◽  
pp. 435-451 ◽  
Author(s):  
M. Kisin ◽  
G.I. Lehrer
Keyword(s):  
Finite Fields ◽  
Counting Points

Counting Points on Curves over Finite Fields

1999 ◽  
pp. 143-172
Author(s):  
Serguei A. Stepanov
Keyword(s):  
Finite Fields ◽  
Curves Over Finite Fields ◽  
Counting Points

Certain 3-decompositions of complete graphs, with an application to finite fields

1985 ◽  
Vol 99 (3-4) ◽  
pp. 277-281 ◽  
Author(s):  
Peter Rowlinson
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Complete Graph ◽  
Disjoint Union ◽  
Necessary Condition ◽  
Complete Graphs ◽  
Characteristic P ◽  
Strongly Regular

SynopsisA necessary condition is obtained for a complete graph to have a decomposition as the line-disjoint union of three isomorphic strongly regular subgraphs. The condition is used to determine the number of non-trivial solutions of the equation x3+y3 = z3 in a finite field of characteristic p ≡ 2 mod 3.

scholarly journals Counting points over finite fields and hypergeometric functions

2013 ◽  
Vol 49 (1) ◽  
pp. 137-157 ◽  
Author(s):  
Adriana Salerno
Keyword(s):  
Finite Fields ◽  
Hypergeometric Functions ◽  
Counting Points
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