We obtain an estimate on the average cardinality of the value set of any family of monic polynomials of F q [T ] of degree d for which s consecutive coefficients a d−1 , . . . , a d−s are fixed. Our estimate holds without restrictions on the characteristic of F q and asserts thatis such an average cardinality, µ d := d r=1 (−1) r−1 /r! and a := (a d−1 , . . . , d d−s ). We provide an explicit upper bound for the constant underlying the O-notation in terms of d and s with "good" behavior. Our approach reduces the question to estimate the number of F q -rational points with pairwise-distinct coordinates of a certain family of complete intersections defined over F q . We show that the polynomials defining such complete intersections are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning the singular locus of the varieties under consideration, from which a suitable estimate on the number of F q -rational points is established.
We obtain estimates on the number |A λ | of elements on a linear family A of monic polynomials of Fq[T ] of degree n having factorization pattern λ := 1 λ 1 2 λ 2 · · · n λn . We show that |A λ | = T (λ) q n−m +O(q n−m−1/2 ), where T (λ) is the proportion of elements of the symmetric group of n elements with cycle pattern λ and m is the codimension of A. Furthermore, if the family A under consideration is "sparse", then |A λ | = T (λ) q n−m + O(q n−m−1 ). Our estimates hold for fields Fq of characteristic greater than 2. We provide explicit upper bounds for the constants underlying the O-notation in terms of λ and A with "good" behavior. Our approach reduces the question to estimate the number of Fq-rational points of certain families of complete intersections defined over Fq. Such complete intersections are defined by polynomials which are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning their singular locus, from which precise estimates on their number of Fq-rational points are established.
Numbers whose continued fraction expansion contains only small digits have been extensively studied. In the real case, the Hausdorff dimension σ M of the reals with digits in their continued fraction expansion bounded by M was considered, and estimates of σ M for M → ∞ were provided by Hensley (the case of a fixed bound M when the denominator N tends to ∞. Later, Hensley (Pac. J. Math. 151(2):237-255, 1991) dealt with the case of a bound M which may depend on the denominator N, and obtained a precise estimate on the cardinality of rational numbers of denominator less than N whose digits (in the continued fraction expansion) are less than M(N), provided the bound M(N) is large enough with respect to N. This paper improves this last result of Hensley towards four directions. First, it considers various continued fraction expansions; second, it deals with various probability settings (and not only the uniform probability); third, it studies the case of all possible sequences M(N), with the only restriction that M(N) is at least equal to a given constant M
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