Explicit estimates for the number of rational points of singular complete intersections over a finite field

Abstract

Abstract. Let V ⊂ P n (Fq) be a complete intersection defined over a finite field Fq of dimension r and singular locus of dimension at most 0 ≤ s ≤ r − 2. We obtain an explicit version of the Hooley-Katz estimate ||V (Fq)| − pr| = O(q (r+s+1)/2 ), where |V (Fq)| denotes the number of Fq-rational points of V and pr := |P r (Fq)|. Our estimate improves all the previous estimates in several important cases. Our approach relies on tools of classical algebraic geometry. A crucial ingredient is a new effective version of the Bertini smoothness theorem, namely an explicit upper bound of the degree of a proper Zariski closed subset of (P n ) s+1 (Fq) which contains all the singular linear sections of V of codimension s + 1.

Estimates on the number ofFq–rational solutions of variants of diagonal equations over finite fields

“…By means of such study we were able to prove that, in all the cases, the set of common zeros in F q of the involved polynomials is a complete intersection whose singular locus has a "controlled" dimension. This allowed us to apply certain explicit estimates on the number of F q -rational zeros of projective complete intersections defined over F q to obtain a conclusion for the problem under consideration (see, e.g., [24], [7], [9] or [32]).…”

On the value set of small families of polynomials over a finite field, III

“…, Y s have full rank in A s . We remark that varieties defined by polynomials of this type arise in several combinatorial problems over finite fields (see, e.g., [CMP12], [CMPP14], [MPP14], [CMP15b] and [MPP15]). Finally, let A N := T d + a d−1 T d−1 + · · · + a 0 ∈ F q [T ] : G i (a d−1 , .…”

“…Hypotheses (H 1 ), (H 2 ) and (H 3 ) hold due to general facts of varieties defined by symmetric polynomials (see [MPP15] for details). Further, it can be shown that (H 4 ) holds by a generalization of the arguments proving the validity of (H 4 ) for the linear family A L .…”

“…An estimate for S A r . We shall rely on the following estimate ([CMP15a, Theorem 1.3]; see also [GL02], [CM07] or [MPP16] for similar estimates): if W ⊂ P n is a normal complete intersection defined over F q of dimension l ≥ 2 and multidegree (e 1 , . .…”

Explicit estimates for polynomial systems defining irreducible smooth complete intersections

“…, f s in the ndimensional projective space P n F over an algebraic closure F of F q . Indeed, if V is known to be a nonsingular or an absolutely irreducible complete intersection, then estimates on the deviation from the expected number of points of V in P n (F q ) are obtained in Deligne (1974), Hooley (1991), Ghorpade & Lachaud (2002), Cafure et al (2015), Matera et al (2016). This motivates the study of the "frequency" with which such geometric properties arise.…”