Filters








6,195 Hits in 5.7 sec

Functional decomposition of polynomials: The wild case

Joachim von zur Gathen
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.  ... 
doi:10.1016/s0747-7171(08)80054-5 fatcat:6xlsmeh2wfaztcty2hp65xocrq

Functional Decomposition of Symbolic Polynomials

Stephen M. Watt
2008 2008 International Conference on Computational Sciences and Its Applications  
This article extends 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  ... 
doi:10.1109/iccsa.2008.71 dblp:conf/iccsa/Watt08 fatcat:iqyabbnyqrcc7he5t4im3rindm

Some Results on the Functional Decomposition of Polynomials [article]

Mark Giesbrecht
2010 arXiv   pre-print
We consider 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.  ... 
arXiv:1004.5433v1 fatcat:zjrzqqlmkbhbdb7fopnl562gee

Tame rational functions: Decompositions of iterates and orbit intersections [article]

Fedor Pakovich
2022 arXiv   pre-print
We also show that for a tame rational 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.  ... 
arXiv:2001.05818v4 fatcat:rercneo7grc3biogg5rhngzfei

Functional decomposition ofpolynomials: the tame case

Joachim von zur Gathen
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  ... 
doi:10.1016/s0747-7171(08)80014-4 fatcat:td25whsh6rgsvoocwu3dsloxem

Functional Decomposition Using Principal Subfields

Luiz E. Allem, Juliane G. Capaverde, Mark van Hoeij, Jonas Szutkoski
2017 Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation - ISSAC '17  
In this paper we use 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  ... 
doi:10.1145/3087604.3087608 dblp:conf/issac/AllemCHS17 fatcat:b5y2uycufnhabog4dtz45aeaem

On a compactification of a Hurwitz space in the wild case [article]

Sylvain Maugeais
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.  ... 
arXiv:math/0509118v1 fatcat:4ihsy7pz4jbcla6ed2gbhbb76u

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

Naoki Imai, Takahiro Tsushima
2020 arXiv   pre-print
We construct a family 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.  ... 
arXiv:1603.04693v3 fatcat:kveb6g2glfgxlkgbxa7wgvgf6u

Algebras whose Coxeter polynomials are products of cyclotomic polynomials [article]

Jose-Antonio de la Peña
2013 arXiv   pre-print
in 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.  ... 
arXiv:1310.1557v1 fatcat:nogzxj77dvcefeo2aupwa3ggwi

Composition collisions and projective polynomials

Joachim von zur Gathen, Mark Giesbrecht, Konstantin Ziegler
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) .  ... 
doi:10.1145/1837934.1837962 dblp:conf/issac/GathenGZ10 fatcat:ynccr4tm65aubbrpu47t5jujjq

Diophantine equations with Euler polynomials [article]

D. Kreso, Cs. Rakaczki
2013 arXiv   pre-print
In this paper we determine possible 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  ... 
arXiv:1312.3907v1 fatcat:efpptyaxibdwjiypp5lbw2yele

Hall polynomials for affine quivers [article]

Andrew Hubery
2007 arXiv   pre-print
In 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.  ... 
arXiv:math/0703178v2 fatcat:zvouckiaxbd55ft2vyzwrmdgje

Giesbrecht's algorithm, the HFE cryptosystem and Ore's p^s-polynomials [article]

Robert S. Coulter, George Havas, Marie Henderson
2016 arXiv   pre-print
We end with some observations on 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.  ... 
arXiv:1611.04479v1 fatcat:3c23gi6vsbhhxbhoxu5dxzfg4a

Composition collisions and projective polynomials [article]

Joachim von zur Gathen, Mark Giesbrecht, Konstantin Ziegler
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).  ... 
arXiv:1005.1087v1 fatcat:ijund3lxkzgcdl4pmwiexru5hi

Decomposition of ordinary difference polynomials

Mingbo Zhang, Xiao-Shan Gao
2009 Journal of symbolic computation  
Keywords 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] .  ... 
doi:10.1016/j.jsc.2009.04.001 fatcat:5rpquxk4mvdpxhmga5xs26alla
« Previous Showing results 1 — 15 out of 6,195 results