Counting decomposable multivariate polynomials

Gathen, Joachim von zur

doi:10.1007/s00200-011-0141-9

Cited by 10 publications

(1 citation statement)

References 14 publications

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“…The item (3) has already been obtained in [2] and [9], where some explicit constants are computed. For example in two variables, and when n is the product of at least three prime numbers, Theorem 5.1 of [2] yields the same first order estimate of J n : N n − J n − q ℓ−1N n ℓ 1−q −n−1 1−q −n/ℓ−1 n q ℓ−1 q n/ℓN n ℓ 1−q −n−1 1−q −n/ℓ−1 .…”

Section: Indecomposable Polynomialsmentioning

confidence: 99%

Generating series for irreducible polynomials over finite fields

Bodin

2010

Finite Fields and Their Applications

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Section: Indecomposable Polynomialsmentioning

confidence: 99%

Generating series for irreducible polynomials over finite fields

Bodin

2010

Finite Fields and Their Applications

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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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On Fast Division Algorithm for Polynomials Using Newton Iteration

Cao

2012

Information Computing and Applications

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Counting decomposable multivariate polynomials

Cited by 10 publications

References 14 publications

Generating series for irreducible polynomials over finite fields

Generating series for irreducible polynomials over finite fields

Survey on Counting Special Types of Polynomials

On Fast Division Algorithm for Polynomials Using Newton Iteration

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