Counting Reducible, Powerful, and Relatively Irreducible Multivariate Polynomials over Finite Fields

The number of reducible space curves over a finite field

“…For r ≥ 3 and d ≥ 4r − 8, the following upper bound holds: Proof. Let c k,r = (2ek) r(r+1)(k 2 +1)+4rg k,r for k ∈ N. According to (9) and Theorem 13, we have the inequality #R…”

“…The inequality (9) shows that an upper bound on the number of F q -reducible cycles in P r of dimension 1 and degree d can be deduced from an upper bound on the degree of the Chow variety C d,r of curves over F q of degree d in P r . In order to obtain an upper bound on the latter, we consider a suitable variant of the approach of Kollár [18, Exercise I.3.28] (see also [12]).…”

“…In order to obtain bounds on the probability that an F q -curve in P r is F q -reducible, we take as a lower bound on the number of all F q -curves of P r the number #P (d, r)(F q ) of plane F q -curves in P r . Bounds on the number #R(d, r)(F q ) of plane F q -reducible curves of P r are provided by the estimates for irreducible bivariate and multivariate polynomials of [8] and [9]. For the homogeneous case, these estimates imply that the #R(d, 2) of F q -reducible curves in P (d, 2) is bounded as…”

“…This question was recently taken up by Bodin [2] and Hou and Mullen [16]. The sharpest bounds are in von zur Gathen [8] for bivariate and von zur Gathen et al [9] for multivariate polynomials.…”

Generating series for irreducible polynomials over finite fields

“…Our proof of (3) appears to be simple compare with these references. Finally, in the forthcoming [10], the formula I(z) is also obtained and applied to get an approximation: I n = N n −q bn+ν−1 (1 + 2/q + O(1/q 2 )), where ν and n 5 are fixed, while q grows to infinity. The end of this section is devoted to the proof of Theorem 1.…”