Generating series for irreducible polynomials over finite fields

Bodin, Arnaud

doi:10.1016/j.ffa.2009.11.002

Cited by 3 publications

(2 citation statements)

References 14 publications

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“…Von zur Gathen, Viola & Ziegler (2013) extend those results to multivariate polynomials and give further information such as exact formulas and generating functions. Bodin (2008) gives a recursive formula for the number of irreducible bivariate polynomials and remarks on a generalization for more than two variables; he follows up with Bodin (2010).…”

Section: Introductionmentioning

confidence: 99%

Survey on Counting Special Types of Polynomials

Gathen¹,

Ziegler²

2015

Lecture Notes in Computer Science

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Most integers are composite and most univariate polynomials over a finite field are reducible. The Prime Number Theorem and a classical result of Gauß count the remaining ones, approximately and exactly.For polynomials in two or more variables, the situation changes dramatically. Most multivariate polynomials are irreducible. This survey presents counting results for some special classes of multivariate polynomials over a finite field, namely the the reducible ones, the s-powerful ones (divisible by the s-th power of a nonconstant polynomial), the relatively irreducible ones (irreducible but reducible over an extension field), the decomposable ones, and also for reducible space curves. These come as exact formulas and as approximations with relative errors that essentially decrease exponentially in the input size.Furthermore, a univariate polynomial f is decomposable if f = g • h for some nonlinear polynomials g and h. It is intuitively clear that the decomposable polynomials form a small minority among all polynomials. The tame case, where the characteristic p of F q does not divide n = deg f , is fairly well-understood, and we obtain closely matching upper and lower bounds on the number of decomposable polynomials. In the wild case, where p does divide n, the bounds are less satisfactory, in particular when p is the smallest prime divisor of n and divides n exactly twice. The crux of the matter is to count the number of collisions, where essentially different (g, h) yield the same f . We present a classification of all collisions at degree n = p 2 which yields an exact count of those decomposable polynomials.

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Section: Introductionmentioning

confidence: 99%

Survey on Counting Special Types of Polynomials

Gathen¹,

Ziegler²

2015

Lecture Notes in Computer Science

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“…In the bivariate case, von zur Gathen (2008) proves precise approximations with an exponentially decreasing relative error. Bodin (2008) gives a recursive formula for the number of irreducible bivariate polynomials and remarks on a generalization for more than two variables; he follows up with Bodin (2010). Some further types of multivariate polynomials are examined from a counting perspective: decomposable ones (von zur Gathen (2010), Bodin, Dèbes & Najib (2009)), singular ones (von zur Gathen ( 2008)), and pairs of coprime polynomials (Hou & Mullen (2009)).…”

Section: Introductionmentioning

confidence: 99%

Counting Reducible, Powerful, and Relatively Irreducible Multivariate Polynomials over Finite Fields

Gathen¹,

Viola²,

Ziegler³

2013

SIAM J. Discrete Math.

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We present counting methods for some special classes of multivariate polynomials over a finite field, namely, the reducible ones, the s-powerful ones (divisible by the sth power of a nonconstant polynomial), and the relatively irreducible ones (irreducible but reducible over an extension field). One approach employs generating functions, and another one uses a combinatorial method. They yield exact formulas and approximations with relative errors that essentially decrease exponentially in the input size.

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Cryptography

2013

Handbook of Finite Fields

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Generating series for irreducible polynomials over finite fields

Cited by 3 publications

References 14 publications

Survey on Counting Special Types of Polynomials

Survey on Counting Special Types of Polynomials

Counting Reducible, Powerful, and Relatively Irreducible Multivariate Polynomials over Finite Fields

Cryptography

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