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Functional decomposition of polynomials: The wild case

1990
*
Journal of symbolic computation
*

*The*

*functional*

*decomposition*problem is: given f

*of*degree n = rs, determine whether such g and h exist, and, in

*the*affirmative

*case*, compute them. ... An apparently difficult

*case*is when

*the*characteristic p

*of*

*the*ground field divides r. This paper presents a

*polynomial*-time partial solution for this "

*wild*"

*case*; it works, e.g., when p2 t r. ... Dickerson (1989) applies

*polynomial*

*decomposition*to

*the*inversion

*of*automorphisms

*of*

*polynomial*rings. ) is a (

*functional*)

*decomposition*

*of*f. ...

##
###
Functional Decomposition of Symbolic Polynomials

2008
*
2008 International Conference on Computational Sciences and Its Applications
*

This article extends

doi:10.1109/iccsa.2008.71
dblp:conf/iccsa/Watt08
fatcat:iqyabbnyqrcc7he5t4im3rindm
*the*notion*of*univariate*polynomial**decomposition*to symbolic*polynomials*and presents an algorithm to compute these*decompositions*. ... Earlier work has presented algorithms to factor and compute GCDs*of*symbolic Laurent*polynomials*, that is multivariate*polynomials*whose exponents are themselves integer-valued*polynomials*. ... Conclusions We have extended*the*notion*of**functional**decomposition**of**polynomials*to*the*domain*of*symbolic*polynomials*and have shown that if such a*decomposition*exists either*the*inner*decomposition*...##
###
Some Results on the Functional Decomposition of Polynomials
[article]

2010
*
arXiv
*
pre-print

We consider

arXiv:1004.5433v1
fatcat:zjrzqqlmkbhbdb7fopnl562gee
*the**wild**case*in some depth, and deal with those*polynomials*whose*decompositions*are in some sense*the*"wildest":*the*additive*polynomials*. ... We determine*the*maximum number*of**decompositions*and show some*polynomial*time algorithms for certain classes*of**polynomials*with*wild**decompositions*. ... Many open questions remain in*the**wild**case*for*polynomial**decomposition*. ...##
###
Tame rational functions: Decompositions of iterates and orbit intersections
[article]

2022
*
arXiv
*
pre-print

We also show that for a tame rational

arXiv:2001.05818v4
fatcat:rercneo7grc3biogg5rhngzfei
*function*A*decompositions**of*its iterates A^∘ d, d≥ 1, into compositions*of*rational*functions*can be obtained from*decompositions**of*a single iterate A^∘ N for N ... Let A be a rational*function**of*degree at least two on*the*Riemann sphere. We say that A is tame if*the*algebraic curve A(x)-A(y)=0 has no factors*of*genus zero or one distinct from*the*diagonal. ...*The**polynomial**case*6.1.*Polynomial**decompositions*. ...##
###
Functional decomposition ofpolynomials: the tame case

1990
*
Journal of symbolic computation
*

*The*

*functional*

*decomposition*problem is: given f

*of*degree n = rs, determine whether such g and h exist, and, in

*the*affirmative

*case*, compute them. ... If g and h are

*polynomials*

*of*degrees r and s over a field, their

*functional*composition f = g(h) has degree n = rs. ... Besides

*the*open questions mentioned in

*the*text (and

*the*more difficult "

*wild*

*case*"), a next goal would be to elucidate

*the*structure

*of*(and find algorithms for) rational

*decompositions*f = gob with ...

##
###
Functional Decomposition Using Principal Subfields

2017
*
Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation - ISSAC '17
*

In this paper we use

doi:10.1145/3087604.3087608
dblp:conf/issac/AllemCHS17
fatcat:b5y2uycufnhabog4dtz45aeaem
*the*idea*of*principal subfields and fast subfield-intersection techniques to compute*the*subfield lattice*of*K(t)/K(f(t)). ... Let f∈ K(t) be a univariate rational*function*. It is well known that any non-trivial*decomposition*g ∘ h, with g,h∈ K(t), corresponds to a non-trivial subfield K(f(t))⊊ L ⊊ K(t) and vice-versa. ... complexity for*polynomial**decomposition*(specially in*the**wild**case*): given f (t) ∈ Fq[t], we can find all minimal*decompositions**of*f with an expected number*of*Õ(rn 2 dp) operations in Fq plus*the*cost ...##
###
On a compactification of a Hurwitz space in the wild case
[article]

2005
*
arXiv
*
pre-print

*The*purpose

*of*this article is to construct and describe a modular compactification

*of*this space. ... Denote by H_g, g', p^c

*the*moduli space

*of*morphisms

*of*degree p between smooth curves

*of*genus g and g' and with constant ramification. ... Introduction

*The*aim

*of*this work is to construct a compactification

*of*one Hurwitz space in

*the*

*wild*

*case*. ...

##
###
Affinoids in the Lubin-Tate perfectoid space and simple supercuspidal representations II: wild case
[article]

2020
*
arXiv
*
pre-print

We construct a family

arXiv:1603.04693v3
fatcat:kveb6g2glfgxlkgbxa7wgvgf6u
*of*affinoids in*the*Lubin-Tate perfectoid space and their formal models such that*the*middle cohomology*of*their reductions realizes*the*local Langlands correspondence and*the*local ...*The*reductions*of**the*formal models are isomorphic to*the*perfections*of*some Artin-Schreier varieties, whose cohomology realizes primitive Galois representations. ... [Wei10] for some special*case*at a finite level). It generalizes a part*of**the*result in [Yos10] to higher conductor*cases*. ...##
###
Algebras whose Coxeter polynomials are products of cyclotomic polynomials
[article]

2013
*
arXiv
*
pre-print

in

arXiv:1310.1557v1
fatcat:nogzxj77dvcefeo2aupwa3ggwi
*the*derived category Der(modA)*of**the*module category modA*of*finite dimensional left A-modules. We say that A is an algebra*of*cyclotomic type if*the*characteristic*polynomial*?A*of*? ... A is a product*of*cyclotomic*polynomials*. ...*The*remaining equivalences follow from (3.2). 4. On*the**decomposition**of**the*Coxeter*polynomial**of*an algebra*of*cyclotomic type 4.1. ...##
###
Composition collisions and projective polynomials

2010
*
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation - ISSAC '10
*

*The*

*functional*

*decomposition*

*of*

*polynomials*has been a topic

*of*great interest and importance in pure and computer algebra and their applications. ... This work investigates

*the*

*decomposition*

*of*

*polynomials*whose degree is a power

*of*p. ...

*The*authors thank Toni Bluher for telling us about

*the*applications

*of*projective

*polynomials*, and an anonymous referee for pointing us to Helleseth & Kholosha (2010) . ...

##
###
Diophantine equations with Euler polynomials
[article]

2013
*
arXiv
*
pre-print

In this paper we determine possible

arXiv:1312.3907v1
fatcat:efpptyaxibdwjiypp5lbw2yele
*decompositions**of*Euler*polynomials*E_k(x), i.e. possible ways*of*writing Euler*polynomials*as a*functional*composition*of**polynomials**of*lower degree. ... Using this result together with*the*well-known criterion*of*Bilu and Tichy, we prove that*the*Diophantine equation -1^k +2 ^k - ... + (-1)^x x^k=g(y), with g∈Q[X]*of*degree at least 2 and k≥ 7, has only ... Dijana Kreso supported by*the*Austrian Science Fund (FWF): W1230-N13 and NAWI Graz. Csaba Rakaczki was supported, in part, by*the*Hungarian Academy*of*Sciences under ...##
###
Hall polynomials for affine quivers
[article]

2007
*
arXiv
*
pre-print

In

arXiv:math/0703178v2
fatcat:zvouckiaxbd55ft2vyzwrmdgje
*the*finite and cyclic*cases*, this approach provides a new and simple proof*of**the*existence*of*Hall*polynomials*. ... In general, these*polynomials*are defined with respect to*the**decomposition*classes*of*Bongartz and Dudek, a generalisation*of**the*Segre classes for square matrices. ... This completes*the*proof*of**the*Main Theorem.*Wild*quivers We finish by offering a definition*of*Hall*polynomials*for*wild*quivers. Let Q be an arbitrary quiver. ...##
###
Giesbrecht's algorithm, the HFE cryptosystem and Ore's p^s-polynomials
[article]

2016
*
arXiv
*
pre-print

We end with some observations on

arXiv:1611.04479v1
fatcat:3c23gi6vsbhhxbhoxu5dxzfg4a
*the*security*of**the*Hidden Field Equation (HFE) cryptosystem, where p-*polynomials*play a central role. ... We report on a recent implementation*of*Giesbrecht's algorithm for factoring*polynomials*in a skew-*polynomial*ring. ... Acknowledgments This work was supported by*the*Australian Research Council. ...##
###
Composition collisions and projective polynomials
[article]

2010
*
arXiv
*
pre-print

*The*

*functional*

*decomposition*

*of*

*polynomials*has been a topic

*of*great interest and importance in pure and computer algebra and their applications. ... This work investigates

*the*

*decomposition*

*of*

*polynomials*whose degree is a power

*of*p. ... Acknowledgments

*The*authors thank Toni Bluher for telling us about

*the*applications

*of*projective

*polynomials*, and an anonymous referee for pointing us to Helleseth & Kholosha (2010). ...

##
###
Decomposition of ordinary difference polynomials

2009
*
Journal of symbolic computation
*

Keywords

doi:10.1016/j.jsc.2009.04.001
fatcat:5rpquxk4mvdpxhmga5xs26alla
*Decomposition*· Differential*polynomial*· Pseudo linear differential*polynomial*· Differential degree Introduction*The*study on*functional**decomposition*algorithms started with*the**decomposition*...*The*algorithm is implemented in Maple for*the*constant field*case*.*The*program can be used to decompose differential*polynomials*with thousands*of*terms effectively. ... Gathen proposed algorithms to find univariate*decompositions*in both*of**the*tame and*the**wild**cases*[31, 32] . ...
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