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Factorization patterns on nonlinear families of univariate polynomials over a finite field

  • Guillermo MateraEmail author
  • Mariana Pérez
  • Melina Privitelli
Article
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Abstract

We estimate the number \(|\mathcal {A}_{{\varvec{\lambda }}}|\) of elements on a nonlinear family \(\mathcal {A}\) of monic polynomials of \(\mathbb {F}_{q}[T]\) of degree r having factorization pattern \({\varvec{\lambda }}:=1^{\lambda _1}2^{\lambda _2}\ldots r^{\lambda _r}\). We show that \(|\mathcal {A}_{{\varvec{\lambda }}}|= \mathcal {T}({\varvec{\lambda }})\,q^{r-m}+\mathcal {O}(q^{r-m-{1}/{2}})\), where \(\mathcal {T}({\varvec{\lambda }})\) is the proportion of elements of the symmetric group of r elements with cycle pattern \({\varvec{\lambda }}\) and m is the codimension of \(\mathcal {A}\). We provide explicit upper bounds for the constants underlying the \(\mathcal {O}\)-notation in terms of \({\varvec{\lambda }}\) and \(\mathcal {A}\) with “good” behavior. We also apply these results to analyze the average-case complexity of the classical factorization algorithm restricted to \(\mathcal {A}\), showing that it behaves as good as in the general case.

Keywords

Finite fields Factorization patterns Symmetric polynomials Complete intersections Singular locus Classical factorization algorithm Average-case complexity 

Notes

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Guillermo Matera
    • 1
    • 2
    Email author
  • Mariana Pérez
    • 1
    • 3
  • Melina Privitelli
    • 2
    • 4
  1. 1.Instituto del Desarrollo HumanoUniversidad Nacional de General SarmientoLos PolvorinesArgentina
  2. 2.National Council of Science and Technology (CONICET)Buenos AiresArgentina
  3. 3.Universidad Nacional de HurlinghamHurlinghamArgentina
  4. 4.Instituto de CienciasUniversidad Nacional de General SarmientoLos PolvorinesArgentina

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