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 +1−2πC q1/2 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 + πC2 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(Fq) 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 6

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