scholarly journals The Degree of the Splitting Field of a Random Polynomial over a Finite Field

10.37236/1823 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
John D. Dixon ◽  
Daniel Panario
Keyword(s):  
Finite Field ◽  
Symmetric Group ◽  
Random Permutation ◽  
Splitting Field ◽  
Random Polynomial ◽  
The Mean ◽  
Order Of An Element

The asymptotics of the order of a random permutation have been widely studied. P. Erdös and P. Turán proved that asymptotically the distribution of the logarithm of the order of an element in the symmetric group $S_{n}$ is normal with mean ${1\over2}(\log n)^{2}$ and variance ${1\over3}(\log n)^{3}$. More recently R. Stong has shown that the mean of the order is asymptotically $\exp(C\sqrt{n/\log n}+O(\sqrt{n}\log\log n/\log n))$ where $C=2.99047\ldots$. We prove similar results for the asymptotics of the degree of the splitting field of a random polynomial of degree $n$ over a finite field.

Generation of the symmetric group Sn2

2021 ◽  
pp. 2150107
Author(s):  
Carlos Zequeira Sánchez ◽  
Evaristo José Madarro Capó ◽  
Guillermo Sosa-Gómez
Keyword(s):  
Symmetric Group ◽  
Random Element ◽  
Matrix Representation ◽  
Random Permutation ◽  
Random Permutations ◽  
Dimension Formula ◽  
Indispensable Tool

In various scenarios today, the generation of random permutations has become an indispensable tool. Since random permutation of dimension [Formula: see text] is a random element of the symmetric group [Formula: see text], it is necessary to have algorithms capable of generating any permutation. This work demonstrates that it is possible to generate the symmetric group [Formula: see text] by shifting the components of a particular matrix representation of each permutation.

scholarly journals Random permutations and their discrepancy process

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Guillaume Chapuy
Keyword(s):  
Computational Biology ◽  
Symmetric Group ◽  
Gaussian Process ◽  
Brownian Bridge ◽  
Random Permutation ◽  
Partial Sums ◽  
Random Permutations ◽  
New Process ◽  
International Audience

International audience Let $\sigma$ be a random permutation chosen uniformly over the symmetric group $\mathfrak{S}_n$. We study a new "process-valued" statistic of $\sigma$, which appears in the domain of computational biology to construct tests of similarity between ordered lists of genes. More precisely, we consider the following "partial sums": $Y^{(n)}_{p,q} = \mathrm{card} \{1 \leq i \leq p : \sigma_i \leq q \}$ for $0 \leq p,q \leq n$. We show that a suitable normalization of $Y^{(n)}$ converges weakly to a bivariate tied down brownian bridge on $[0,1]^2$, i.e. a continuous centered gaussian process $X^{\infty}_{s,t}$ of covariance: $\mathbb{E}[X^{\infty}_{s,t}X^{\infty}_{s',t'}] = (min(s,s')-ss')(min(t,t')-tt')$.

scholarly journals Effective bounds for the maximal order of an element in the symmetric group

1989 ◽  
Vol 53 (188) ◽  
pp. 665-665 ◽  
Author(s):  
Jean-Pierre Massias ◽  
Jean-Louis Nicolas ◽  
Guy Robin
Keyword(s):  
Symmetric Group ◽  
Maximal Order ◽  
Order Of An Element

On Random Generation of the Symmetric Group

1993 ◽  
Vol 2 (4) ◽  
pp. 505-512 ◽  
Author(s):  
Tomasz Łuczak ◽  
László Pyber
Keyword(s):  
Symmetric Group ◽  
Random Permutation ◽  
Invariant Subset ◽  
Random Generation ◽  
Element Set

We prove that the probability i(n, k) that a random permutation of an n element set has an invariant subset of precisely k elements decreases as a power of k, for k ≤ n/2. Using this fact, we prove that the fraction of elements of Sn belong to transitive subgroups other than Sn or An tends to 0 when n → ∞, as conjectured by Cameron. Finally, we show that for every ∈ > 0 there exists a constant C such that C elements of the symmetric group Sn, chosen randomly and independently, generate invariably Sn with probability at least 1 − ∈. This confirms a conjecture of McKay.

An 𝔽 p2 -maximal Wiman sextic and its automorphisms

2021 ◽  
Vol 21 (4) ◽  
pp. 451-461
Author(s):  
Massimo Giulietti ◽  
Motoko Kawakita ◽  
Stefano Lia ◽  
Maria Montanucci
Keyword(s):  
Riemann Surface ◽  
Automorphism Group ◽  
Finite Field ◽  
Symmetric Group ◽  
Closed Field ◽  
Complex Field ◽  
Hermitian Curve ◽  
Full Automorphism Group ◽  
Algebraically Closed Field

Abstract In 1895 Wiman introduced the Riemann surface 𝒲 of genus 6 over the complex field ℂ defined by the equation X 6+Y 6+ℨ 6+(X 2+Y 2+ℨ 2)(X 4+Y 4+ℨ 4)−12X 2 Y 2 ℨ 2 = 0, and showed that its full automorphism group is isomorphic to the symmetric group S 5. We show that this holds also over every algebraically closed field 𝕂 of characteristic p ≥ 7. For p = 2, 3 the above polynomial is reducible over 𝕂, and for p = 5 the curve 𝒲 is rational and Aut(𝒲) ≅ PGL(2,𝕂). We also show that Wiman’s 𝔽192 -maximal sextic 𝒲 is not Galois covered by the Hermitian curve H19 over the finite field 𝔽192 .

scholarly journals Zeros of an algebraic polynomial with nonequal means random coefficients

2004 ◽  
Vol 2004 (63) ◽  
pp. 3389-3395
Author(s):  
K. Farahmand ◽  
P. Flood
Keyword(s):  
Algebraic Polynomial ◽  
Asymptotic Estimate ◽  
Random Variables ◽  
Random Coefficients ◽  
Random Polynomial ◽  
Expected Number ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
The Mean

This paper provides an asymptotic estimate for the expected number of real zeros of a random algebraic polynomiala0+a1x+a2x2+⋯+an−1xn−1. The coefficientsaj(j=0,1,2,…,n−1)are assumed to be independent normal random variables with nonidentical means. Previous results are mainly for identically distributed coefficients. Our result remains valid when the means of the coefficients are divided into many groups of equal sizes. We show that the behaviour of the random polynomial is dictated by the mean of the first group of the coefficients in the interval(−1,1)and the mean of the last group in(−∞,−1)∪(1,∞).

scholarly journals A Characteristic Map for the Symmetric Space of Symplectic Forms Over a Finite Field

2020 ◽  
Author(s):  
Jimmy He
Keyword(s):  
Symmetric Space ◽  
Finite Field ◽  
Symmetric Group ◽  
Spherical Function ◽  
Symmetric Functions ◽  
Macdonald Polynomials ◽  
Symplectic Forms ◽  
Parabolic Induction ◽  
Characteristic Map ◽  
Combinatorial Formulas

Abstract The characteristic map for the symmetric group is an isomorphism relating the representation theory of the symmetric group to symmetric functions. An analogous isomorphism is constructed for the symmetric space of symplectic forms over a finite field, with the spherical functions being sent to Macdonald polynomials with parameters $(q,q^2)$. An analogue of parabolic induction is interpreted as a certain multiplication of symmetric functions. Applications are given to Schur positivity of skew Macdonald polynomials with parameters $(q,q^2)$ as well as combinatorial formulas for spherical function values.

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