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Reduction of bivariate polynomials from convex-dense to dense, with application to factorizations

2012
*
Mathematics of Computation
*

This

doi:10.1090/s0025-5718-2011-02562-7
fatcat:l5dum3l7srhgxdkjjwsi754xvi
*reduction*simply consists in computing an invertible monomial transformation that produces a*polynomial**with*a*dense*size*of*the same order*of*magnitude as the size*of*the integral*convex*hull*of*... In this article we present a new algorithm for reducing the usual sparse*bivariate**factorization*problems*to*the*dense*case. ... We would like*to*thank the anonymous referees for their helpful comments. ...##
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Theoretical Properties
[chapter]

2015
*
The Fence Methods
*

Two major ingredients are the

doi:10.1142/9789814596077_0009
fatcat:7rmqxc74afagha2moms4jjz35u
*reduction**from*the*bivariate*case*to*the univariate one, and the*reduction**from*any number*to*two variables. ...*Convex*-*dense**bivariate**factorization*11.5.53 Remark In the worst case, the size*of*the irreducible*factorization*is exponential in the sparse size*of*the*polynomial*f*to*be*factored*. ... Minkowski sum, 9 Newton polytope, 9 NP-hardness, 11 Ostrowski theorem, 9 separable*factorization*, 3 sparse*factorization*, 9 sparse*polynomial*representation, 9 straight-line program, 11, 12 straight-line ...##
###
Theoretical Properties
[chapter]

2011
*
Chapman & Hall/CRC Biostatistics Series
*

Two major ingredients are the

doi:10.1201/b10783-7
fatcat:zqyyjtuzsjf7zpekrozupsqwye
*reduction**from*the*bivariate*case*to*the univariate one, and the*reduction**from*any number*to*two variables. ...*Convex*-*dense**bivariate**factorization*11.5.53 Remark In the worst case, the size*of*the irreducible*factorization*is exponential in the sparse size*of*the*polynomial*f*to*be*factored*. ... Minkowski sum, 9 Newton polytope, 9 NP-hardness, 11 Ostrowski theorem, 9 separable*factorization*, 3 sparse*factorization*, 9 sparse*polynomial*representation, 9 straight-line program, 11, 12 straight-line ...##
###
Theoretical Properties
[chapter]

2013
*
Handbook of Finite Fields
*

Two major ingredients are the

doi:10.1201/b15006-5
fatcat:cubpnr7y3fbfpivjinvw2dqmvy
*reduction**from*the*bivariate*case*to*the univariate one, and the*reduction**from*any number*to*two variables. ...*Convex*-*dense**bivariate**factorization*11.5.53 Remark In the worst case, the size*of*the irreducible*factorization*is exponential in the sparse size*of*the*polynomial*f*to*be*factored*. ... Minkowski sum, 9 Newton polytope, 9 NP-hardness, 11 Ostrowski theorem, 9 separable*factorization*, 3 sparse*factorization*, 9 sparse*polynomial*representation, 9 straight-line program, 11, 12 straight-line ...##
###
Index to Volumes 37 and 38

2004
*
Journal of symbolic computation
*

-I., Characterization

doi:10.1016/s0747-7171(04)00109-9
fatcat:q3cckydpknhjhinygacsvlj52y
*of*Pythagorean curves and Pythagoreanization using a rational transform, 377 An effective decision method for semidefinite*polynomials*, 83 An objective representation*of*the Gaussian ... , 101 EBERLY, W. and GIESBRECHT, M., Efficient decomposition*of*separable algebras, 35 Efficient decomposition*of*separable algebras, 35 EGNER, S. and PÜSCHEL, M., Symmetry-based matrix*factorization*, ... families*of*singularities, 1191 FUKUDA, K.,*From*the zonotope construction*to*the Minkowski addition*of**convex*polytopes, 1261 GAO, S., KALTOFEN, E. and LAUDER, A.G.B., Deterministic distinct-degree*factorization*...##
###
Indecomposability of polynomials via Jacobian matrix

2010
*
Journal of Algebra
*

Indecomposable

doi:10.1016/j.jalgebra.2010.01.007
fatcat:szb566a6ivhytmivniyku3yym4
*polynomials*are a special class*of*absolutely irreducible*polynomials*. ... Some improvements*of*important effective results on absolute irreducibility have recently appeared using Ruppert's matrix. ... The second author was supported by the "Abdus Salam" center, ICTP, Trieste (Italy); for this he wishes*to*thank all staff*of*this center. ...##
###
An efficient sparse adaptation of the polytope method over Fp and a record-high binary bivariate factorisation

2008
*
Journal of symbolic computation
*

*of*terms

*of*the input

*bivariate*

*polynomial*. ... terms t satisfies t < d 3/4 , and which is known

*to*be the product

*of*two sparse

*factors*. ... for allowing the use

*of*their facilities

*to*generate the reported experiments. ...

##
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Decomposition of polytopes and polynomials
[article]

2000
*
arXiv
*
pre-print

Motivated by a connection

arXiv:math/0012099v1
fatcat:jtr3cfk3ynflzjls2hmlt7hisq
*with*the*factorization**of*multivariate*polynomials*, we study integral*convex*polytopes and their integral decompositions in the sense*of*the Minkowski sum. ...*Applications**of*our algorithm include absolute irreducibility testing and*factorization**of**polynomials*via their Newton polytopes. ... In Section 5, we describe*applications**of*our algorithms*to**polynomials**with*respect*to*their irreducibility and*factorization*. ...##
###
Factoring polynomials via polytopes

2004
*
Proceedings of the 2004 international symposium on Symbolic and algebraic computation - ISSAC '04
*

Our main contribution is

doi:10.1145/1005285.1005289
dblp:conf/issac/SalemGL04
fatcat:hgqj77cawvfh3ka6qzfvjd5roq
*to*present an algorithm for*factoring**bivariate**polynomials*which is able*to*exploit*to*some extent the sparsity*of**polynomials*. ... We give details*of*an implementation which we used*to**factor*randomly chosen sparse and composite*polynomials**of*high degree over the binary field. ... These two papers reduce sparse*polynomials**with*more than two variables*to**bivariate*or univariate*polynomials*which are then treated as*dense**polynomials*. ...##
###
Page 7094 of Mathematical Reviews Vol. , Issue 97K
[page]

1997
*
Mathematical Reviews
*

polyhedron,

*with**application**to*mold design (109-120); David Dobkin and Dimitrios Gunopulos, Geometric problems in machine learning (121-132); Ronen Basri and David Jacobs, Matching*convex*polygons and ...*of*the Galois groups*of*the resolvent*factors*for the direct and inverse Galois problems (456-468); Jacques-Arthur Weil, First integrals and Darboux*polynomials**of*homogeneous linear differential systems ...##
###
Essentially Reductive Hilbert Modules II
[article]

2006
*
arXiv
*
pre-print

Many Hilbert modules over the

arXiv:math/0607722v1
fatcat:alckxh32m5e3xko3meaodbkvve
*polynomial*ring in m variables are essentially*reductive*, that is, have commutators which are compact. ... Arveson has raised the question*of*whether the closure*of*homogeneous ideals inherit this property and provided motivation*to*seek an affirmative answer. ... Note that the fact that I is radical forces the generating*polynomials**to*be prime*factors*having the form z t i − αz u j for α = 0. ...##
###
Decomposition of Polytopes and Polynomials

2001
*
Discrete & Computational Geometry
*

Motivated by a connection

doi:10.1007/s00454-001-0024-0
fatcat:e32o3ai6vrbhrh5vndy7pnrhdi
*with*the*factorization**of*multivariable*polynomials*, we study integral*convex*polytopes and their integral decompositions in the sense*of*the Minkowski sum. ...*Applications**of*our algorithms include absolute irreducibility testing and*factorization**of**polynomials*via their Newton polytopes. ... In Section 5, we describe*applications**of*our algorithms*to**polynomials**with*respect*to*their irreducibility and*factorization*. ...##
###
On the bit-complexity of sparse polynomial and series multiplication

2013
*
Journal of symbolic computation
*

As an

doi:10.1016/j.jsc.2012.06.004
fatcat:cxvzdjkkzncxvh7tvhe4gca36y
*application*, we are able*to*count the number*of*the absolutely irreducible*factors**of*a multivariate*polynomial**with*a cost that is essentially quadratic in the number*of*the integral points in the ... This new complexity bound is*to*be compared*to*a recent algorithm by Weimann that computes the irreducible*factorization**of*a*bivariate**polynomial*within a cost that grows*with*|S | 3 [Wei09a, Wei09b]. ... The absolute*factorization**of*F mainly reduces*to*linear algebra by considering the following map: where K[x, y] S y represents the subset*of*the*polynomials**with*support in S y (and similarly for S x ...##
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On the Complexity of Diophantine Geometry in Low Dimensions
[article]

1998
*
arXiv
*
pre-print

, well within the second level

arXiv:math/9811088v1
fatcat:2p7qxmzr6ra5hjimaajaocqspu
*of*the*polynomial*hierarchy. ... Better still, we show that the truth*of*the Generalized Riemann Hypothesis implies that detecting roots in Q^n for the*polynomial*systems in (I) can be done via a two-round Arthur-Merlin protocol, i.e. ... We also note that while Main Theorem 2 deals*with*using*reduction*mod p*to*count roots over Q, other results, such as [Koi96, Thm. 8] and [Bür99, Thm. 4 .1], use*reduction*mod p*to*determine the existence ...##
###
Intent Preference Decoupling for User Representation on Online Recommender System

2020
*
Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence
*

Accurately characterizing the user's current interest is the core

doi:10.24963/ijcai.2020/353
dblp:conf/ijcai/WangL20a
fatcat:3zig2w4pc5bd7iinrcxfaxjaiq
*of*recommender systems. However, users' interests are dynamic and affected by intent*factors*and preference*factors*. ... The learning*of*the intent is considered as a meta-learning task and fast adaptive*to*the current browsing; the learning*of*the preference is based on the calibrated user intent and constantly updated ... Acknowledgments The work*of*the first author was supported by NSFC (71571163) and Zhejiang NSF (LY19G010001). Proceedings*of*the Twenty-Ninth International Joint Conference on Artificial Intelligence ...
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