Compositions and collisions at degree
            <i>p</i>
            <sup>2</sup>

Blankertz, Raoul; Gathen, Joachim von zur; Ziegler, Konstantin

doi:10.1145/2442829.2442846

Cited by 3 publications

(4 citation statements)

References 18 publications

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“…We have seen that ν q,n tends to 1 unless p 2 n. Blankertz, von zur Gathen and Ziegler [1] show that lim e−→∞ ν p e ,p 2 = 2/3 if p = 2, 1 otherwise.…”

Section: Proof (I)mentioning

confidence: 72%

“…If n = p 2 , the methods of this paper do not yield satisfactory bounds. However, this has been completely resolved by Blankertz, von zur Gathen and Ziegler [1], as follows.…”

Section: Conditions Bounds On Tmentioning

confidence: 99%

“…One of the two difficult cases without a lower bound 1 − ε is n = p 2 (I.B). This has been completely resolved by Blankertz, von zur Gathen and Ziegler [1], who determined the exact size of D p 2 . We include the weaker result that follows in that case from the present method, but forego a proof.…”

Section: Counting General Decomposable Polynomialsmentioning

confidence: 99%

“…The precise upper bound is always valid, by (ii). But in the lower bound of (iii), when p is the smallest prime divisor of n and divides n exactly twice, then ε is only in O(1), but Blankertz, von zur Gathen and Ziegler [1] give an exact count for n = p 2 , using function field theoretic methods. In all other cases, ε is in O(q −1 ) over F q .…”

Section: Introductionmentioning

confidence: 99%

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Counting Decomposable Univariate Polynomials

Gathen¹

2014

Combinator. Probab. Comp.

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A univariate polynomial f over a field is decomposable if it is the composition f = g • h of two polynomials g and h whose degree is at least 2. We determine an approximation to the number of decomposables over a finite field. The tame case, where the field characteristic p does not divide the degree n of f, is reasonably well understood, and we obtain exponentially decreasing relative error bounds. The wild case, where p divides n, is more challenging and our error bounds are weaker.

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“…We have seen that ν q,n tends to 1 unless p 2 n. Blankertz, von zur Gathen and Ziegler [1] show that lim e−→∞ ν p e ,p 2 = 2/3 if p = 2, 1 otherwise.…”

Section: Proof (I)mentioning

confidence: 72%

“…If n = p 2 , the methods of this paper do not yield satisfactory bounds. However, this has been completely resolved by Blankertz, von zur Gathen and Ziegler [1], as follows.…”

Section: Conditions Bounds On Tmentioning

confidence: 99%

Section: Counting General Decomposable Polynomialsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Counting Decomposable Univariate Polynomials

Gathen¹

2014

Combinator. Probab. Comp.

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Tame decompositions and collisions

Ziegler¹

2014

Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation

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A univariate polynomial f over a field is decomposable if f = g • h = g(h) for nonlinear polynomials g and h. It is intuitively clear that the decomposable polynomials form a small minority among all polynomials over a finite field. The tame case, where the characteristic p of F q does not divide n = deg f , is fairly well-understood, and we have reasonable bounds on the number of decomposables of degree n. Nevertheless, no exact formula is known if n has more than two prime factors. In order to count the decomposables, one wants to know, under a suitable normalization, the number of collisions, where essentially different (g, h) yield the same f . In the tame case, Ritt's Second Theorem classifies all 2-collisions.We introduce a normal form for multi-collisions of decompositions of arbitrary length with exact description of the (non)uniqueness of the parameters. We obtain an efficiently computable formula for the exact number of such collisions at degree n over a finite field of characteristic coprime to p. This leads to an algorithm for the exact number of decomposable polynomials at degree n over a finite field F q in the tame case.

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A polynomial time algorithm for computing all minimal decompositions of a polynomial

Blankertz

2014

ACM Commun. Comput. Algebra

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The composition of two polynomials g(h) = g • h is a polynomial. For a given polynomial f we are interested in finding a functional decomposition f = g • h. In this paper an algorithm is described, which computes all minimal decompositions in polynomial time. In contrast to many previous decomposition algorithms this algorithm works without restrictions on the degree of the polynomial and the characteristic of the ground field. The algorithm can be iteratively applied to compute all decompositions. It is based on ideas of Landau & Miller (1985) and Zippel (1991). Additionally, an upper bound on the number of minimal decompositions is given.

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Compositions and collisions at degree p ²

Cited by 3 publications

References 18 publications

Counting Decomposable Univariate Polynomials

Counting Decomposable Univariate Polynomials

Tame decompositions and collisions

A polynomial time algorithm for computing all minimal decompositions of a polynomial

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Compositions and collisions at degree p 2

Cited by 3 publications

References 18 publications

Counting Decomposable Univariate Polynomials

Counting Decomposable Univariate Polynomials

Tame decompositions and collisions

A polynomial time algorithm for computing all minimal decompositions of a polynomial

Contact Info

Product

Resources

About

Compositions and collisions at degree p ²