\documentstyle[12pt,fleqn]{article}
\language0 
\title{Algebraic envelopes of plane arcs in Galois planes.}

\author{Massimo Giulietti}
\date{Dipartimento di Matematica,  Universit\`a di Perugia,\\
      06123 Perugia, Italy\\
                 ({\tt giuliet@dipmat.unipg.it})}

\newtheorem{deff}{Definition}[section]
\newtheorem{prop}{Proposition}[section]
\newtheorem{propo}{Claim}[section]
\newtheorem{lem}{Lemma}[section]
\newtheorem{oss}{Observation}[section]
\newtheorem{theorem}{Theorem}[section]

\begin{document}
\baselineskip =18pt
\maketitle
\noindent 

\begin{abstract}
A method to calculate equations of algebriac envelopes associated to 
a $k$-arc in $PG(2,q)$ is given. It is based on Gr{\"o}bner Bases
of ideals in $GF(q)[x_{0},x_{1},x_{2}]$. 
%This method is then applied
%to the complete 24-arc in $PG(2,29)$ by Chao and Kaneta.
\end{abstract}
\section{Introduction}
We recall a fundamental theorem by B. Segre.
\begin{theorem}\label{seg2}
Let $K$ be a $k$-arc of $PG(2,q)$. If $q$ is odd, then there exists
a plane curve $\Gamma '$ in the dual plane of
$PG(2,F)$, where $F$ denotes the 
algebraic closure of $GF(q)$, satisfying the properties below. Such 
a curve is called {\em algebraic  envelope} of $K$. 
For $P\in PG(2,F)$, let $\ell_P$ denote the corresponding 
line in the dual plane. 
    \begin{enumerate}

\item $\Gamma '$ is defined over $GF(q)$.

\item The degree of $\Gamma '$ is $2t$, with $t:=q-k+2$ being the number 
of 1-secants through a point of $K$.

\item The $kt$ 1-secants of $K$ belong to $\Gamma '$.

\item Each 1-secant $\ell$ of $K$ through a point $P\in K$ is counted
twice in the intersection of $\Gamma '$ with $\ell_P$, i.e. 
$I(\ell, \Gamma '\cap \ell_P)=2$.

\item The curve $\Gamma '$ contains no 2-secant of $K$.

\item The irreducible components of $\Gamma '$ have multiplicity at most
2, and $\Gamma '$ has at least one component of multiplicity 1.

\item a) If $k>\frac{2}{3}(q+2)$, then $K$ is not complete if and only if 
$\Gamma '$ admits a linear component defined over $GF(Q)$. 
b) If $K$ is not contained in a conic and $k>\frac{1}{4}(3q+5)$,
then $\Gamma '$ does not admit a quadratic component over $GF(q)$. 
c) If $k>4$, then $\Gamma '$ does not admit any component of degree
$d<3$ which is not defined over $GF(q)$.
\item If $k> \frac{2}{3}(q+2)$, then there exists a unique curve in the
dual plane of $PG(2,F)$ satisfying properties 2,3,4,5. 

\end{enumerate}

\end{theorem}
In this paper we provide an algorithm to compute equations for the algebraic
envelopes associated to a fixed arc $K$; the main tool is the theory
of Gr{\"o}bner Basis for multivariate polynomial rings.

We fix the following notation:
\begin{itemize}
\item $K=\{P^1, \ldots, P^k\}$, $P^j=[P_0^j,P_1^j,P_2^j]$, is a $k$-arc in 
$PG(2,q)$ with $q$ odd;
\item $t=q-k+2$;
\item $M'=\frac{M}{c}$, where $M(x_0,x_1,x_2)=\Pi_{j=1}^{k}(P_0^jx_0
+P_1^jx_1+P_2^jx_2)$ and $c$ is the leading coefficient of $M$ with respect
to the monomial order {\em grlex };
\item $l^{j,1}=[l^{j,1}_{0},l^{j,1}_{1},l^{j,1}_{2}], \ldots , 
l^{j,t}=[l^{j,t}_{0},l^{j,t}_{1},l^{j,t}_{2}]$ are points of the dual plane of $PG(2,q)$ corresponding to the $t=q-k+2$ 1-secants of $K$ through $P^j$.
\item for $P^j_0\neq 0$ let $L_{P^j}$ the following ideal of 
$GF(q)[x_0,x_1,x_2]$:
$
L_{P^j}=
<\Pi_{\imath =1}^{t}(l^{\imath}_{2}x_{1}-l^{\imath}_{1}x_{2})^{2},
P_{0}x_{0}+P_{1}x_{1}+P_{2}x_{2}>.
$
(If $P_{0}^j=0$ the above definition extends in a natural way);
\item $L_{K}=\bigcap_{P \in K}L_{P}$.
\end{itemize}
We recall the following proposition (\cite[Prop. 2.3]{giul}):
\begin{prop}\label{chiave}
Let $K$ be a $k$-arc in $PG(2,q)$, with $q$ odd, containing both 
$E_{1}=[0,1,0]$ and $E_{2}=[0,0,1]$. Let ${\cal G}=\{g_{1}, \ldots , 
g_{s}\}$ the reduced Gr\"obner Basis of $L_{K}$ with respect to the monomial order {\rm grlex}. 
Then there exists one and only one element $g\in {\cal G}$ such that $g\neq M'$ and $\deg{g}=2t$. 
Moreover, if ${\cal I}$ is the set of all homogeneous polynomials 
representing equations of the algebraic envelopes of $K$, we have that:
\[
{\cal I}=
\left\{ \begin{array}{ll}
\{ag+bM' \mid a \in GF(q)^{\star}, \:\:
deg(b) \leq 2t - k\}, & \:\:\: k \leq 2t \\
\{ag \mid a \in GF(q)^{\star}\}, & \:\:\: k> 2t.
\end{array} \right. 
\]
\end{prop}

\section{MAGMA instructions}
In this section we provide the translation of Prop. \ref{chiave} in 
MAGMA language instructions. 
%For the sake of simplicity we refer to a fixed arc
%, the one exhibited by Chao and Kaneta in \cite{CH}, 
%the same 
In particular, the algorithm below works for any arc satisfying condition 8 of Thm. \ref{seg2}.

\begin{verbatim}
//definition of the environment (the value  q is supposed
//to contain an odd prime power)

F:=GF(q);
P,V,L:=ProjectivePlane(q);
R<x0,x1,x2>:=PolynomialRing(F,3,"glex");

//definition of the arc and of its tangents; the value K is 
//supposed to contain a subset of V - the set of the points 
//of the plane - having k<q+2 
//elements and containing both [0,1,0] and [0,0,1]

KK:=SetToSequence(K);
k:=#KK;
t:=q-k+2;
if IsArc(K) and (k gt 2*t) then
  tg1:=AllTangents(K);
  tg:=SetToSequence(tg1);
  
//construction of the ideal LK

  LK:=ideal<R,1>;
  Ret:={R|};
  for P in KK do
    //for all P, we construct the ideal LP
    caso:=0;
    if P[1] ne 0 then
      caso:=1;
      r:=x0+(1/P[1])*(P[2]*x1+P[3]*x_2);
    end if;
    if (P[1] eq 0) and (P[2] ne 0) then
      caso:=2;
      r:=x1+(1/P[2])*(P[1]*x0+P[3]*x_2);
    end if;
    if (P[1] eq 0) and (P[2] eq 0)  then
      caso:=3;
      r:=x2+(1/P[3])*(P[2]*x1+P[1]*x_0);
    end if;  
    fattore:=1;
    case caso:
      when 1:
        for l in tg do
          if P in l then
            a:=(l[3]*x1-l[2]*x2)*(l[3]*x1-l[2]*x2);
            fattore:=fattore*a;
          end if;
        end for;
      when 2:
        for l in tg do
          if P in l then
            a:=(l[3]*x0-l[1]*x2)*(l[3]*x0-l[1]*x2);
            fattore:=fattore*a;
          end if;
        end for;
      when 3:
        for l in tg do
          if P in l then
            a:=(l[1]*x1-l[2]*x0)*(l[1]*x1-l[2]*x0);
            fattore:=fattore*a;
          end if;
        end for;
    end case;
    LP:=ideal<R|fattore,r>;
    Groebner(LP);
    LK:=LK meet LP;
  end for;

//we find the only element in the reduced Groebner 
//basis of LK of degree less than or equal
//to 2*t
  Groebner(LK);
  GB:=GroebnerBasis(LK);
  EB:={R|};
  for G in GB do
    if TotalDegree(G) le 2*t then
      EB:=EB join {G};
    end if;
  end for;
  EB1:=SetToSequence(EB);
  INV:=EB1(1);
end if;  


\end{verbatim}


\begin{thebibliography}{9999}
%\bibitem{CH} J.M. CHAO - H. KANETA, {\em A complete 24-arc in $PG(2,29)$ %with the
%automorphism group $PSL(2,7)$}, Roma, Rendiconti di Matematica, serie VII,
%vol. 16, 1996, 537-544.
\bibitem{COX} D. COX - J. LITTLE - D. O'SHEA, {\em Ideals, Varieties, and
Algorithms}, Springer-Verlag, 1991.
\bibitem{giul}M. GIULIETTI, {\em Inviluppi di $k$-archi in piani proiettivi
sopra campi finiti e Basi di Gr\"obner}, Rendiconti del Circolo Matematico
di Palermo serie II, {\bf 48}, 1998, 191-200. 
\bibitem{HIR} J.W.P. HIRSCHFELD, {\em Projective Geometries over Finite Fields}, Oxford, Clarendon Press, 1979.
\bibitem{WA} R.J. WALKER, {\em Algebriac curves}, Springer-Verlag, 1978. 
\end{thebibliography}
%\tableofcontents
\end{document}
