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

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.

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Correspondence to Guillermo Matera.

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The authors were partially supported by the grants PIP CONICET 11220130100598, PIO CONICET-UNGS 14420140100027, UNGS 30/3180 and STIC-AmSud 13STIC-02 “Dynalco”.

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Cesaratto, E., Matera, G. & Pérez, M. The distribution of factorization patterns on linear families of polynomials over a finite field. Combinatorica 37, 805–836 (2017). https://doi.org/10.1007/s00493-015-3330-5

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Mathematics Subject Classification (2000)

  • 11T06
  • 05E05
  • 05E40
  • 12C05
  • 14G05
  • 14G15