6 YVES AUBRY AND FRANC ¸OIS RODIER Hence #C (Fq) q + 1 2πC q1/2. The maximum number of rational points on the curve C on the surface V is 7(d−3) by Proposition 7. If q +1−2πCq1/2 7(d−3), then C (Fq) V , therefore X(Fq) V , and δ(f) 4 by Theorem 3. But this condition is equivalent to q 2πC q1/2 7(d 3) + 1 0. The condition is satisfied when q1/2 πC + 7(d 3) 1 + π2 C hence when q d4 18d3 + 121d2 348d + 362 or 5 d q1/4 + 4.6. 4. Polynomials functions with δ 4 If the function f is a polynomial of one variable with coefficients in Fq of degree d 3, we consider again as in section 3 the intersection X of S1 and S2, which are now cylinders in the affine space A4(Fq) with equations respectively Pf(x, y, z) = 0 and Pf(x, y, t) = 0 and which are of dimension 3 as affine varieties. Lemma 10. The algebraic set X has dimension 2, i.e. it is an affine surface. Moreover, it has degree (d 3)2. Proof. We have to show that the hypersurfaces S1 and S2 do not have a common irreducible component. Since these hypersurfaces are two cylinders, it is enough to prove that the polynomial defining S1 does not vanish on the whole of a straight line (x0, y0, z, t0) where x0, y0, t0 are fixed and satisfy Pf(x0, y0, t0) = 0. Indeed, S1 is defined by the polynomial Pf(x, y, z), which takes the value Pf(x0, y0, z) = f(x0) + f(y0) + f(z) + f(x0 + y0 + z) (x0 + y0)(x0 + z)(y0 + z) at the point (x0, y0, z, t0). If we set x0 + y0 = s0, the homogeneous term of degree di in Pf(x, y, z) becomes di(xdi−1 0 + zdi−1) + s0Qi(x0, z) (z + s0 + x0)(z + x0) where Qi is a polynomial in x0 and z of degree di 2. If di is odd, the numerator of this term is of degree di 2, and hence does not vanish, so it is the same for the polynomial Pf(x0, y0, z). Hence, X has dimension 2. Moreover, X is the intersection of two hypersurfaces of degree d 3, thus it has degree (d 3)2. The surface X is reducible. Let X = i Xi be its decomposition in absolutely irreducible components. We embed the affine surface X into a projective space P4(F q ) with homogeneous coordinates (x : y : z : t : u). Consider the hyperplane at infinity H∞ defined by the equation u = 0 and let X∞ be the intersection of the projective closure X of X with H∞. Then X∞ is the intersection of two surfaces in this hyperplane, which
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