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

Matera, Guillermo; Pérez, Mariana; Privitelli, Melina

doi:10.48550/arxiv.1310.3293

Cited by 1 publication

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“…To this end, observe that a polynomial f ∈ A is not square-free if and only if its discriminant is equal to zero. In [MPP13] we study the so-called discriminant locus of A, namely the set A nsq formed by the elements of A whose discriminant is equal to zero (see also [FS84] for further results on discriminant loci). According to [MPP13, Theorem A.3], the discriminant locus A nsq is the set of F q -rational points of a hypersurface of degree n(n−1) of a suitable (n − m)-dimensional affine space.…”

Section: The Number Of Polynomials In a λmentioning

confidence: 99%

“…In [CMP12b], [CMPP14] and [MPP13] a methodology to deal with combinatorial problems over finite fields is developed. It is based on the fact that many combinatorial problems can be described by means of symmetric polynomials, and varieties defined by symmetric polynomials have particular features that can be exploited in order to obtain "good" estimates on their number of F q -rational points.…”

Section: Factorization Patterns Of Polynomials With Prescribed Coeffi...mentioning

confidence: 99%

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The distribution of factorization patterns on linear families of polynomials over a finite field

2016

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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.

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Section: The Number Of Polynomials In a λmentioning

confidence: 99%

Section: Factorization Patterns Of Polynomials With Prescribed Coeffi...mentioning

confidence: 99%