Ovals in the Desarguesian plane of order 16

1975 ◽  
Vol 102 (1) ◽  
pp. 159-176 ◽  
Author(s):  
Marshall Hall
Keyword(s):  
Desarguesian Plane

scholarly journals The characterization of elliptic curves over finite fields

1988 ◽  
Vol 45 (2) ◽  
pp. 275-286 ◽  
Author(s):  
J. W. P. Hirschfeld ◽  
J. F. Voloch
Keyword(s):  
Elliptic Curves ◽  
Finite Fields ◽  
Sufficient Conditions ◽  
Maximum Size ◽  
Desarguesian Plane ◽  
Cubic Curve ◽  
Odd Order ◽  
Even Order ◽  
Curves Over Finite Fields

AbstractIn a finite Desarguesian plane of odd order, it was shown by Segre thirty years ago that a set of maximum size with at most two points on a line is a conic. Here, in a plane of odd or even order, sufficient conditions are given for a set with at most three points on a line to be a cubic curve. The case of an elliptic curve is of particular interest.

Early explicit examples of non-desarguesian plane geometries

2011 ◽  
Vol 102 (1-2) ◽  
pp. 179-188 ◽  
Author(s):  
Markus Stroppel
Keyword(s):  
Desarguesian Plane

A Class of Non-Desarguesian Projective Planes

1957 ◽  
Vol 9 ◽  
pp. 378-388 ◽  
Author(s):  
D. R. Hughes
Keyword(s):  
Positive Integer ◽  
Projective Plane ◽  
Collineation Group ◽  
Difference Sets ◽  
Projective Planes ◽  
Desarguesian Projective Plane ◽  
Desarguesian Plane ◽  
Desarguesian Projective Planes

In (7), Veblen and Wedclerburn gave an example of a non-Desarguesian projective plane of order 9; we shall show that this plane is self-dual and can be characterized by a collineation group of order 78, somewhat like the planes associated with difference sets. Furthermore, the technique used in (7) will be generalized and we will construct a new non-Desarguesian plane of order p2n for every positive integer n and every odd prime p.

Veblen–Wedderburn hybrid planes

1969 ◽  
Vol 309 (1497) ◽  
pp. 157-170 ◽  
Keyword(s):  
Projective Plane ◽  
Galois Field ◽  
Real Field ◽  
Type I ◽  
Type Ii ◽  
Desarguesian Plane ◽  
The Real ◽  
Desarguesian Planes ◽  
Hughes Plane

The paper describes a method of redistributing the points of the collinear sets in a Desarguesian plane so as to produce a (hybrid) projective plane which is non-Desarguesian. The method is applied to the construction: (i) of a plane over a prescribed subfield of the real field, and (ii) of a plane (over a Galois field) which is proved to be identical with the Hughes plane. On the basis of this construction algebraic relations in the field can be interpreted as incidence relations in the hybrid plane. In order to verify that the planes of type (i) are not isomorphic with Desarguesian planes, some conditions are established which show that all planes of this type (as well as of type (ii)) contain Fano subplanes.

scholarly journals Blocking and Double Blocking Sets in Finite Planes

10.37236/5717 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Jan De Beule ◽  
Tamás Héger ◽  
Tamás Szőnyi ◽  
Geertrui Van de Voorde
Keyword(s):  
Projective Planes ◽  
Blocking Set ◽  
Blocking Sets ◽  
Desarguesian Plane ◽  
Baer Subplanes ◽  
Desarguesian Projective Planes ◽  
Finite Planes ◽  
Value Sets ◽  
Minimal Blocking ◽  
Desarguesian Affine Plane

In this paper, by using properties of Baer subplanes, we describe the construction of a minimal blocking set in the Hall plane of order $q^2$ of size $q^2+2q+2$ admitting $1-$, $2-$, $3-$, $4-$, $(q+1)-$ and $(q+2)-$secants. As a corollary, we obtain the existence of a minimal blocking set of a non-Desarguesian affine plane of order $q^2$ of size at most $4q^2/3+5q/3$, which is considerably smaller than $2q^2-1$, the Jamison bound for the size of a minimal blocking set in an affine Desarguesian plane of order $q^2$.We also consider particular André planes of order $q$, where $q$ is a power of the prime $p$, and give a construction of a small minimal blocking set which admits a secant line not meeting the blocking set in $1$ mod $p$ points. Furthermore, we elaborate on the connection of this problem with the study of value sets of certain polynomials and with the construction of small double blocking sets in Desarguesian projective planes; in both topics we provide some new results.

Moulton Planes

1961 ◽  
Vol 13 ◽  
pp. 427-436 ◽  
Author(s):  
William A. Pierce
Keyword(s):  
Translation Plane ◽  
Affine Plane ◽  
Negative Slope ◽  
Arbitrary Field ◽  
Desarguesian Plane ◽  
Real Numbers ◽  
Classical Plane ◽  
Real Affine ◽  
Desargues Theorem ◽  
Ordered Pairs

In 1902, F. R. Moulton (12) gave an early example of a non-Desarguesian plane. Its ‘points” are ordered pairs (x, y) of real numbers. Its “lines” coincide with lines of the real affine plane except that lines of negative slope are “bent” on the x-axis, line {y = b + mx}, for negative m, being replaced by {y = b + mx if y ≤ 0, y = [m/2]. [x + (b/m)] if y > 0}. A certain Desarguesian configuration in the classical plane is shifted just enough to vitiate Desargues’ Theorem for Moulton's geometry. The plane is neither a translation plane (“Veblen-Wedderburn” in the sense of Hall (7), p. 364) nor even the dual of one (Veblen and Wedderburn (17). It is natural to ask if the same construction is feasible when real numbers are replaced by elements from an arbitrary field.

A Class of Non-Desarguesian Plane Geometries

1950 ◽  
Vol 57 (6) ◽  
pp. 381 ◽  
Author(s):  
Kenneth Levenberg
Keyword(s):  
Desarguesian Plane

A Real Non-Desarguesian Plane

1963 ◽  
Vol 70 (5) ◽  
pp. 522 ◽  
Author(s):  
K. Sitaram
Keyword(s):  
Desarguesian Plane

scholarly journals One-Factorisations of Complete Graphs Constructed in Desarguesian Planes of Certain Odd Square Orders

10.37236/8378 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Nicola Pace ◽  
Angelo Sonnino
Keyword(s):  
Complete Graph ◽  
Affine Group ◽  
Geometric Construction ◽  
Complete Graphs ◽  
Desarguesian Plane ◽  
Desarguesian Planes

A geometric construction of one-factorisations of complete graphs $K_{q(q-1)}$ is provided for the case when either $q=2^d +1$ is a Fermat prime, or $q=9$. This construction uses the affine group $\mathrm{AGL}(1,q)$, points and ovals in the Desarguesian plane $\mathrm{PG}(2,q^{2})$ to produce one-factorisations of the complete graph $K_{q(q-1)}$.

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