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

Jérémy Berthomieu, Grégoire Lecerf
2012 Mathematics of Computation  
This 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.  ... 
doi:10.1090/s0025-5718-2011-02562-7 fatcat:l5dum3l7srhgxdkjjwsi754xvi

Theoretical Properties [chapter]

2015 The Fence Methods  
Two major ingredients are the 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  ... 
doi:10.1142/9789814596077_0009 fatcat:7rmqxc74afagha2moms4jjz35u

Theoretical Properties [chapter]

2011 Chapman & Hall/CRC Biostatistics Series  
Two major ingredients are the 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  ... 
doi:10.1201/b10783-7 fatcat:zqyyjtuzsjf7zpekrozupsqwye

Theoretical Properties [chapter]

2013 Handbook of Finite Fields  
Two major ingredients are the 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  ... 
doi:10.1201/b15006-5 fatcat:cubpnr7y3fbfpivjinvw2dqmvy

Index to Volumes 37 and 38

2004 Journal of symbolic computation  
-I., Characterization 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  ... 
doi:10.1016/s0747-7171(04)00109-9 fatcat:q3cckydpknhjhinygacsvlj52y

Indecomposability of polynomials via Jacobian matrix

G. Chèze, S. Najib
2010 Journal of Algebra  
Indecomposable 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.  ... 
doi:10.1016/j.jalgebra.2010.01.007 fatcat:szb566a6ivhytmivniyku3yym4

An efficient sparse adaptation of the polytope method over Fp and a record-high binary bivariate factorisation

Fatima K. Abu Salem
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.  ... 
doi:10.1016/j.jsc.2007.10.011 fatcat:rji56jrosfalrbgczqd2jszdhe

Decomposition of polytopes and polynomials [article]

S. Gao, A.G.B. Lauder
2000 arXiv   pre-print
Motivated by a connection 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.  ... 
arXiv:math/0012099v1 fatcat:jtr3cfk3ynflzjls2hmlt7hisq

Factoring polynomials via polytopes

Fatima Abu Salem, Shuhong Gao, Alan G. B. Lauder
2004 Proceedings of the 2004 international symposium on Symbolic and algebraic computation - ISSAC '04  
Our main contribution is 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.  ... 
doi:10.1145/1005285.1005289 dblp:conf/issac/SalemGL04 fatcat:hgqj77cawvfh3ka6qzfvjd5roq

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]

Ronald G. Douglas
2006 arXiv   pre-print
Many Hilbert modules over the 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.  ... 
arXiv:math/0607722v1 fatcat:alckxh32m5e3xko3meaodbkvve

Decomposition of Polytopes and Polynomials

S. Gao, A. G. B. Lauder
2001 Discrete & Computational Geometry  
Motivated by a connection 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.  ... 
doi:10.1007/s00454-001-0024-0 fatcat:e32o3ai6vrbhrh5vndy7pnrhdi

On the bit-complexity of sparse polynomial and series multiplication

Joris van der Hoeven, Grégoire Lecerf
2013 Journal of symbolic computation  
As an 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  ... 
doi:10.1016/j.jsc.2012.06.004 fatcat:cxvzdjkkzncxvh7tvhe4gca36y

On the Complexity of Diophantine Geometry in Low Dimensions [article]

J. Maurice Rojas
1998 arXiv   pre-print
, well within the second level 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  ... 
arXiv:math/9811088v1 fatcat:2p7qxmzr6ra5hjimaajaocqspu

Intent Preference Decoupling for User Representation on Online Recommender System

Zhaoyang Liu, Haokun Chen, Fei Sun, Xu Xie, Jinyang Gao, Bolin Ding, Yanyan Shen
2020 Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence  
Accurately characterizing the user's current interest is the core 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  ... 
doi:10.24963/ijcai.2020/353 dblp:conf/ijcai/WangL20a fatcat:3zig2w4pc5bd7iinrcxfaxjaiq
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