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Combinatorica

, Volume 37, Issue 5, pp 805–836 | Cite as

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

  • Eda Cesaratto
  • Guillermo MateraEmail author
  • Mariana Pérez
Original Paper

Abstract

We estimate the number |A λ| of elements on a linear family A of monic polynomials of F q [T] of degree n having factorization pattern \(\lambda : = {1^{{\lambda _1}}}{2^{{\lambda _2}}} \cdot \cdot \cdot {n^{{\lambda _n}}}\). 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 F q 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 F q -rational points of certain families of complete intersections defined over F q . 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 F q -rational points are established.

Mathematics Subject Classification (2000)

11T06 05E05 05E40 12C05 14G05 14G15 

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Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Eda Cesaratto
    • 1
    • 2
  • Guillermo Matera
    • 1
    • 2
    Email author
  • Mariana Pérez
    • 1
  1. 1.Instituto del Desarrollo HumanoUniversidad Nacional de General SarmientoBuenos AiresArgentina
  2. 2.National Council of Science and Technology (CONICET)Buenos AiresArgentina

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