Algebraic envelopes of plane arcs in Galois planes.

Massimo Giulietti

Dipartimento di Matematica, Università di Perugia, 06123 Perugia, Italy

              (giuliet@dipmat.unipg.it)

Abstract A method to calculate equations of algebriac envelopes associated to a k-arc in P G(2, q) is given. It is based on Gröbner Bases of ideals in GF (q)[x_{0}, x_{1}, x_{2}].

1 Introduction

We recall a fundamental theorem by B. Segre.

Theorem 1.1 Let K be a k-arc of P G(2, q). If q is odd, then there exists a plane curve Γ′ in the dual plane of P G(2, F), where F denotes the algebraic closure of GF (q), satisfying the properties below. Such a curve is called algebraic envelope of K. For P ∈ P G(2, F), let ℓ_{P} denote the corresponding line in the dual plane.

1. Γ′ is defined over GF (q).

2. The degree of Γ′ is 2t, with t := q − k + 2 being the number of 1-secants through a point of K.

3. The kt 1-secants of K belong to Γ′.

4. Each 1-secant ℓ of K through a point P ∈ K is counted twice in the intersection of Γ′ with ℓ_{P}, i.e. I(ℓ, Γ′ ∩ ℓ_{P}) = 2.

5. The curve Γ′ contains no 2-secant of K.

6. The irreducible components of Γ′ have multiplicity at most 2, and Γ′ has at least one component of multiplicity 1.

7. a) If k > \frac{2}{3}(q + 2), then K is not complete if and only if Γ′ 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 Γ′ does not admit a quadratic component over GF (q). c) If k > 4, then Γ′ does not admit any component of degree d < 3 which is not defined over GF (q).

8. If k > \frac{2}{3}(q + 2), then there exists a unique curve in the dual plane of P G(2, F) satisfying properties 2,3,4,5.

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 Gröbner Basis for multivariate polynomial rings.

We fix the following notation:

• K = {P^{1}, …, P^{k}}, P^{j} = [P_{0}^{j}, P_{1}^{j}, P_{2}^{j}], is a k-arc in P G(2, q) with q odd;

• t = q − k + 2;

• M′ = \frac{M}{c}, where M(x_{0}, x_{1}, x_{2}) = Π_{j=1}^{k}(P_{0}^{j}x_{0} + P_{1}^{j}x_{1} + P_{2}^{j}x_{2}) and c is the leading coefficient of M with respect to the monomial order grlex ;

• l^{j,1} = [l_{0}^{j,1}, l_{1}^{j,1}, l_{2}^{j,1}], …, l^{j,t} = [l_{0}^{j,t}, l_{1}^{j,t}, l_{2}^{j,t}] are points of the dual plane of P G(2, q) corresponding to the t = q − k + 2 1-secants of K through P^{j}.

• for P_{0}^{j} ≠ 0 let L_{P^{j}} the following ideal of GF (q)[x_{0}, x_{1}, x_{2}]: L_{P^{j}} =< Π_{ı=1}^{t}(l_{2}^{ı}x_{1}−l_{1}^{ı}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);

• L_{K} = ⋂_{P ∈K} L_{P}.

We recall the following proposition ([2, Prop. 2.3]):

Proposition 1.1 Let K be a k-arc in P G(2, q), with q odd, containing both E_{1} = [0, 1, 0] and E_{2} = [0, 0, 1]. Let G = {g_{1}, …, g_{s}} the reduced Gröbner Basis of L_{K} with respect to the monomial order grlex. Then there exists one and only one element g ∈ G such that g ≠ M′ and deg g = 2t. Moreover, if I is the set of all homogeneous polynomials representing equations of the algebraic envelopes of K, we have that:

I = \{ \begin{matrix}\{ag + bM′ | a ∈ GF (q)^{⋆}, deg(b) ≤ 2t − k\}, & k ≤ 2t \\ \{ag | a ∈ GF (q)^{⋆}\}, & k > 2t. \end{matrix}

2 MAGMA instructions

In this section we provide the translation of Prop. 1.1 in MAGMA language instructions. In particular, the algorithm below works for any arc satisfying condition 8 of Thm. 1.1.

//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;

References

[1] D. COX - J. LITTLE - D. O’SHEA, Ideals, Varieties, and Algorithms, Springer-Verlag, 1991.

[2] M. GIULIETTI, Inviluppi di k-archi in piani proiettivi sopra campi finiti e Basi di Gröbner, Rendiconti del Circolo Matematico di Palermo serie II, 48, 1998, 191-200.

[3] J.W.P. HIRSCHFELD, Projective Geometries over Finite Fields, Oxford, Clarendon Press, 1979.

[4] R.J. WALKER, Algebriac curves, Springer-Verlag, 1978.