西 南 交 通 大 学 学 报
第 55 卷 第 2 期
2020 年 4 月
JOURNAL OF SOUTHWEST JIAOTONG UNIVERSITY
Vol. 55 No. 2
Apr. 2020
ISSN: 0258-2724 DOI：10.35741/issn.0258-2724.55.2.46
Research articleMathematics
R
ESULTS ON THE
P
ROJECTIVE
P
LANE OVER A
F
INITE
F
IELD OF
O
RDER
S
EVENTEEN
有限阶七阶上射影平面的结果
Najm A. M. AL-Seraji, Hussam. H. Jawad
Department of Mathematics, College of Science, University of Mstansiriyah Baghdad, Iraq, [email protected], [email protected]
Received: January 15, 2020 ▪ Review: April 7, 2020 ▪ Accepted: April 24, 2020
This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
Abstract
The main goal of this research is to find the projective mapping that transforms a geometric formation called an -set onto an arc such that the domain of the mapping is a subset of the projective line
PG , such that a -set is called a pentad, a -set is a hexad, a -set is a heptad, a -set is an octad, and a -set is a nonad, mapped onto a conic . The research also aims to find the stabilizer group of points on a non-singular cubic curve, with or without rational inflection points, on the projective plane over a finite field of order seventeen, and to give some examples.
Keywords: Cubic Curve, Arc, Stabilizer Group
摘要 这项研究的主要目标是找到一种射影映射，该映射将称为-组的几何形式转换为弧，以使映 射的域是射影线 PG 的子集，因此-组被称为五元组， 一个-组是一个六角，一个-组是一个庚烷， 一个-组是一个八度，一个-组是一个非广告，映射到一个圆锥上。 该研究的目的还在于在有限级 的 17 阶场上的投影平面上找到非奇异三次曲线上具有或不具有有理拐点的稳定点组，并给出一些 例子。 关键词: 三次曲线，圆弧，稳定器组
I. I
NTRODUCTION In 𝑃𝐺 , a -arc or -set is a special case of a -arc that is a set of 𝑘 points, no three of which are collinear. Al-Seraji[1] has classified 𝑘-sets in 𝑃𝐺(1,𝑞) for 𝑞=17. In the first part of this research, we have relied on these classifications to find a projective mapping between -arcs, where is , and a parameter for each transformation, such that if
is any -set, then a -secant of is a line such that . The previous researcher recommended such method to classify certain geometric structures, specifically the projective line of order sixteen PG . Hirschfeld and Al-Seraji [3] also classified k-sets on the projective line of order seventeen, and Al-Zangana and Shehab [4] computed the inequivalent -sets in PG .
The second part of this research aims to find the stabilizer group of points on a cubic curve on the projective plane over a finite field of order seventeen. We have presented some examples, and have explained the modality of finding the inflection points for some cubic curves. The effect of stabilizers on the points of some cubic curves has also been described.
There have been previous studies of this subject, some of which have been recalled by researchers. Al-Seraji and Ajaj [5] found the stabilizer groups of cubic curves on the projective plane over a finite field of order four, and they also studied the properties of their groups. Al-Zangana [6] has also shown the cubic curves on the projective plane over a finite field of order nineteen. The computing tool used in the current study is the mathematical programming language (GAP) [7], [8], [9], [10], [11], [12], [13].
II. R
ESEARCHA
IMThe main goal of this research is to find the projective mapping that transforms a geometric formation called an -set onto an arc such that the
domain of the mapping is a subset of the projective line PG . The aim is also to improve its efficiency so that some related problems can be solved.
III. R
ESEARCHM
ETHODS ANDR
ESULTSA. The Points on the Conic
Let : PG ∈
PG be a projective mapping defined as follows:
= and .
The subsets of the projective line PG are pentads, hexads, heptads, octads, and nonads. The distinct -arc, in , are be taken from Al-Seraji [1], but we aim to present it in a more comprehensive way by writing it in terms of the primitive element , in order to find an inverse projective mapping.
Equations conics used are:
(1)
(2)
(3)
(4) These equations are taken from [1]. We show the points corresponding to each equation in Table 1 as follows:
Table 1.
The points on coincs
No. Conic The points
1
2
3
B. The Transformations between the 5-Arcs
According to the fundamental theory of projective geometry, in PG(2,17), there is a unique projectivity transforming any four points, of which three are on a line, into four points of which no three are on a line, but in -arcs, it will be dependent on the external points of -arcs, which is , and in the same manner for the next
-arcs.
The distinct 5-arcs on a conic , are given in Table 2 as follows:
Table 2.
The distinct 5-arcs on a conic
No. Symbol The distinct 5-arcs 1 , , , , 2 , , , , 3 , , , , 4 , , , ,
The distinct pentads under the effect of projective mapping are given in Table 3 as follows:
Table 3.
The distinct pentads
No. Symbol The distinct pentads 1 , , , , 2 , , , , 3 , , , , 4 , , , ,
The projective mapping is given in Table 4 as follows:
Table 4.
The projective mapping between the -arcs [ Projective mapping Transform [ , , ] , , ] , , ] , , ]
These transformations are not unique, meaning that there are more transformers for every set, for example:
, ,
.
C. The Transformations between the -Arcs
The distinct 6-arcs on a conic , are given in Table 5 as follows:
Table 5.
The distinct 6-arcs on a conic
No. Symbol The distinct 6-arcs 1 { } 2 { } 3 { } 4 } 5 { } 6 { } 7 { } 8 { } 9 { } 10 , , , , ,
In this study, the distinct hexads under the effect of projective mapping are given in Table 6 as follows:
Table 6.
The distinct hexads
No. Symbol The distinct hexads 1 { } 2 { } 3 { } 4 { } 5 { } 6 } 7 { } 8 { }
9 { } 10 { }
The projective mapping is given in Table 7 as follows:
Table 7.
Projective mapping on the -arcs
[ Projective mapping Transformations
The transformation is
unique, but the rest of the transformations are not unique; for example:
, .
D. The Transformations between the -Arcs
The inequivalent -arcs on a conic are given in Table 8 as follows:
Table 8.
The inequivalents -arcs on a conics
No. Symbol The inequivalents -arcs 1 2 3 4 5 6 7 8 9 10
The distinct heptads under the effect of projective mapping are given in Table 9 as follows:
Table 9.
The distinct heptads
No. Symbol The distinct heptads 1 2 3 4 5 6 7 8 9 10
The projective mapping between -arcs is given in Table 10 as follows:
Table 10.
The projective mapping between the -arcs
Projective mapping Transformations
Some transformations may be unique, for example , & , whereas other transformations may be not unique, for example:
, ,
.
E. The Transformations between the -Arcs
The inequivalent -arcs on a conic are given in Table 11 as follows:
Table 11.
The inequivalents -arcs on a conics
No. Symbol The inequivalents -arcs 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
The distinct octads under the effect of projective mapping are given in Table 12 as follows:
Table 12.
The distinct octads
No. Symbol The distinct octads 1
3 4 5 6 7 8 9 10 11 12 13 , 14 15 16 17
The projective mapping are given in the Table 13 as follows:
Table 13.
The projective mapping between the -arcs
Projective mapping Transformations
Some transformations may be unique for
example , ,
& and others
transformations may be not unique for example:
, .
F. The Transformations between the -Arcs
The inequivalents -arcs on a conics are given in the Table 14 as follows:
Table 14.
The inequivalents -arcs on a conics
No. Symbol The inequivalents -arcs 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
The distinct nonads under the effect of projective mapping are given in Table 15 as follows:
Table 15.
The distinct nonads
No. Symbol The distinct nonads 1 2 , 3 4 = 5 6 = 7 , 8 , 9 10 , 11 12 13 14 , 15 16 { 17
The projective mapping are given in Table 16 as follows:
Table 16.
Projective mapping between the -arcs
Projective mapping Transformations
Some transformations may be unique for
example , , , ,
& and others transformations may be not unique for example:
, .
G. The Stabilizer Group of Points on Cubics
We will work to findThe stabilizers group of points on the non-singular cubics, firstly we will be find the stabilizers of the non-singular cubic
with three rational inflexions points, secondly will be find the stabilizers of the non-singular cubic with one rational inflexion point, finally will be find the stabilizers of the non-singular with no rational inflexions, all these equations have been taken from Al- Seraji [1].
1) The Stabilizers of the Non-Singular Cubics with Three Rational Inflections
Initially, we will display the stabilizers of the non-singular cubics with three rational inflections in Table 17 as follows:
Table 17.
The stabilizers of cubics with three rational inflections point
No. Canonical form Size The stabilizers
1 18
2 12
3 21
4 21
6 15 7 12 8 24 9 12 10 18 11 15 12 18 13 21 14 18 15 24 16 24
The points on the cubic curve are given in Table 18.
Table 18. Points on the cubic
The value of
The generator matrix G of the previous cubic is
(5) The values of parameters =
, such that ,
where in and represents the rows of Moreover,
= =
(6)
Since , ,
. Therefore, , , are inflection points of .
Let be the stabilizer of points on the cubic curve.
Then we can be study the effect of on the points of cubic curve on Table 19, as follows:
Table 19.
The effect on the points of cubic curve
No. Effect on the points of 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
So fixed of the two points and shifted the remaining.
But fixed three points. We will explain this in Table 20, as follows:
Table 20.
Effect on points of cubic curve
No. Effect on points of cubic curve 1 2 3 4 5 6
The same way for the remaining cubics.
2) The Stabilizers of the Non-Singular Cubic with One Rational Inflection
The stabilizers of the non-singular cubic with one rational inflection are given in Table 21, as follows:
Table 21.
The stabilizers of the non-singular cubic with one rational inflexion No. Canonical form Size The stabilizers
1 18 2 18 3 24 4 24 5 17 6 17 7 14 8 14
9 15 10 15 11 20 12 20 13 12 14 12 15 25 16 25 17 16 18 20 19 26 20 10
3) The Stabilizers of the Non-Singular Cubic with No Rational Inflection
The stabilizers of the non-singular cubic with no rational inflections are given in Table 22, as follows:
Table 22.
The stabilizers of the non-singular cubic with no rational inflexions
No. Canonical form Size The stabilizers
1 15 2 24 3 15 4 21 5 18 6 24
7 12 8 24 9 21 10 15 11 18 12 18 13 21 14 12 15 12 16 18
IV. C
ONCLUSIONSBased on the mentioned methodological steps, the research aim has been achieved successfully as planned. The projective mapping which transforms a geometric formation called -sets has been obtained as previously mentioned. Also, the stabilizers group of points on the non-singular cubic curve on a projective plane over a finite field of order seventeen has been determined and proved by examples. The results showed an efficient method to solve such complicated examples.
A
CKNOWLEDGEMENTPraise be to Allaah, Lord of the Worlds, for completing this research, I would like to thank my supervisor Dr. Najm Abdulzahra Makhrib Al-Seraji, I also thank department of mathematics, college of science, University of Mstansiriyah, and everyone who contributed to the completion of this research.
