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

Gathen, Joachim von zur; Viola, Alfredo; Ziegler, Konstantin

doi:10.1137/110854680

Cited by 7 publications

(6 citation statements)

References 20 publications

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“…For r ≥ 3 and d ≥ 4r − 8, the following upper bound holds: Proof. Let c k,r = (2ek) r(r+1)(k 2 +1)+4rg k,r for k ∈ N. According to (9) and Theorem 13, we have the inequality…”

Section: The Number Of F Q -Reducible Curvesmentioning

confidence: 99%

“…The inequality (9) shows that an upper bound on the number of F q -reducible cycles in P r of dimension 1 and degree d can be deduced from an upper bound on the degree of the Chow variety C d,r of curves over F q of degree d in P r . In order to obtain an upper bound on the latter, we consider a suitable variant of the approach of Kollár [18, Exercise I.…”

Section: An Upper Bound On the Degree Of The Restricted Chow Variety mentioning

confidence: 99%

“…This question was recently taken up by Bodin [2] and Hou and Mullen [16]. The sharpest bounds are in von zur Gathen [8] for bivariate and von zur Gathen et al [9] for multivariate polynomials.…”

Section: Introductionmentioning

confidence: 99%

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The number of reducible space curves over a finite field

Cesaratto

Gathen²,

Matera

2013

Journal of Number Theory

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“…For r ≥ 3 and d ≥ 4r − 8, the following upper bound holds: Proof. Let c k,r = (2ek) r(r+1)(k 2 +1)+4rg k,r for k ∈ N. According to (9) and Theorem 13, we have the inequality…”

Section: The Number Of F Q -Reducible Curvesmentioning

confidence: 99%

Section: An Upper Bound On the Degree Of The Restricted Chow Variety mentioning

confidence: 99%

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The number of reducible space curves over a finite field

Cesaratto

Gathen²,

Matera

2013

Journal of Number Theory

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“…In the bivariate case, von zur Gathen (2008) proves precise approximations with an exponentially decreasing relative error. 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: 79%

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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“…This work was subsequently refined and extended in [1,7,8,15,16]. Our Theorem 1.1 may be interpreted as a determination of the q-adic asymptotics of M d,n (q) as n → ∞.…”

Section: 2mentioning

confidence: 91%

Liminal reciprocity and factorization statistics

Hyde¹

2019

Algebraic Combinatorics

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Let M d,n (q) denote the number of monic irreducible polynomials in Fq[x1, x2, . . . , xn] of degree d. We show that for a fixed degree d, the sequence M d,n (q) converges coefficientwise to an explicitly determined rational function M d,∞ (q). The limit M d,∞ (q) is related to the classic necklace polynomial M d,1 (q) by an involutive functional equation we call liminal reciprocity. The limiting first moments of factorization statistics for squarefree polynomials are expressed in terms of symmetric group characters as a consequence of liminal reciprocity, giving a liminal analog of the twisted Grothendieck-Lefschetz formula of Church, Ellenberg, and Farb.

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

Cited by 7 publications

References 20 publications

The number of reducible space curves over a finite field

The number of reducible space curves over a finite field

Survey on Counting Special Types of Polynomials

Liminal reciprocity and factorization statistics

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