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Polynomial-Time Power-Sum Decomposition of Polynomials [article]

Mitali Bafna, Jun-Ting Hsieh, Pravesh K. Kothari, Jeff Xu
2022 arXiv   pre-print
We give efficient algorithms for finding power-sum decomposition of an input polynomial P(x)= ∑_i≤ m p_i(x)^d with component p_is.  ...  Specifically, our algorithm succeeds in decomposing a sum of m ∼Õ(n) generic quadratic p_is for d=3 and more generally the dth power-sum of m ∼ n^2d/15 generic degree-K polynomials for any K ≥ 2.  ...  Acknowledgements P.K. thanks Ankit Garg for preliminary discussions on power-sum decomposition.  ... 
arXiv:2208.00122v1 fatcat:tar2nncesrbz5mxci6oyicpf6a

Efficient evaluation of noncommutative polynomials using tensor and noncommutative Waring decompositions [article]

Eric Evert, J. William Helton, Shiyuan Huang, Jiawang Nie
2021 arXiv   pre-print
For example, we consider a "Waring decomposition" in which each product of linear terms is actually a power of a single linear NC polynomial or more generally a power of a homogeneous NC polynomial.  ...  This paper analyses a Waring type decomposition of a noncommuting (NC) polynomial p with respect to the goal of evaluating p efficiently on tuples of matrices.  ...  The classical commutative Waring problem can be generalized from representation by sums of powers of linear functions to representation by sums of powers of homogeneous polynomials.  ... 
arXiv:1903.05910v2 fatcat:j3gel7w6nvg4lpvk2ghhia2vea

Approximating the maximum of a polynomial over a polytope: Handelman decomposition and continuous generating functions [article]

Jesús De Loera and Brandon Dutra and Matthias Köppe
2016 arXiv   pre-print
We investigate a way to approximate the maximum of a polynomial over a polytopal region by using Handelman's polynomial decomposition and continuous multivariate generating functions.  ...  The maximization problem is NP-hard, but our approximation methods will run in polynomial time when the dimension is fixed.  ...  Acknowledgments The authors are grateful to the excellent and very detailed comments of the referees.  ... 
arXiv:1601.04118v2 fatcat:amwydd6eqvff3lxpdt7adb7rfi


Andrés Marcos, Declan G. Bates, Ian Postlethwaite
2005 IFAC Proceedings Volumes  
In this paper an algorithm that provides an equivalent, but of reduced order, representation for multivariate polynomial matrices is given.  ...  The algorithm is applied to the problem of finding minimal linear fractional transformation models.  ...  The "presence" degree, σ(δ i ), is defined as the number of times, including powers, a variable δ i appears in an expression (monomial, polynomial or matrix).  ... 
doi:10.3182/20050703-6-cz-1902.00999 fatcat:cl3n3ifot5brjci3ptlmi57hgq

Derandomization and absolute reconstruction for sums of powers of linear forms [article]

Pascal Koiran, Mateusz Skomra
2021 arXiv   pre-print
We study the decomposition of multivariate polynomials as sums of powers of linear forms.  ...  (ii) For an input polynomial with rational coefficients, the algorithm runs in polynomial time when implemented in the bit model of computation.  ...  The anonymous referee suggested several improvements in the presentation of the paper, and a significant simplification in the hitting set constructions of Section 6.  ... 
arXiv:1912.02021v5 fatcat:axmheh4t45eyhalgzkcquzboli

Learning sums of powers of low-degree polynomials in the non-degenerate case [article]

Ankit Garg, Neeraj Kayal, Chandan Saha
2020 arXiv   pre-print
We develop algorithms for writing a polynomial as sums of powers of low degree polynomials.  ...  Consider an n-variate degree-d polynomial f which can be written as f = c_1Q_1^m + ... + c_s Q_s^m, where each c_i∈F^×, Q_i is a homogeneous polynomial of degree t, and t m = d.  ...  Institute for the Theory of Computing in December 2018.  ... 
arXiv:2004.06898v2 fatcat:tpebdeaqwnh55h26iob3yy2s6u

Prime and composite polynomials

J. F. Ritt
1922 Transactions of the American Mathematical Society  
Our task in the present paper will be, after showing the constancy of the number of prime polynomials in the distinct decompositions of a given polynomial into prime polynomials, to determine those polynomials  ...  which have two or more distinct decompositions into prime polynomials.  ...  which is not a power of a prime, then to every permutation of the prime factors of the exponent there corresponds a separate decomposition of F(z) into prime powers of z.  ... 
doi:10.1090/s0002-9947-1922-1501189-9 fatcat:n2hywal6svhxxid3i3o2b7srpm

Prime and Composite Polynomials

J. F. Ritt
1922 Transactions of the American Mathematical Society  
If F(z) is a power of z with an exponent which is not a power of a prime, then to every permutation of the prime factors of the exponent there corresponds a separate decomposition of F(z) into prime powers  ...  Then X 1a assumes the value infinity n times for z = so-, and is therefore a polynomial of degree n. Similarly pX is a polynomial of degree m.  ... 
doi:10.2307/1988911 fatcat:fjxteh5nhvdzrmc3f2xdadehka

Symmetric polynomials for 2D shape representation

Renato M. P. Negrinho, Pedro M. Q. Aguiar
2014 2014 IEEE International Conference on Image Processing (ICIP)  
We single out elementary symmetric polynomials and power sums as particular families of polynomials that further enable obtaining completeness with respect to point labeling.  ...  We discuss the usage of symmetric polynomials for representing 2D shapes in their most general form, i.e., arbitrary sets of unlabeled points in the plane.  ...  POWER SUMS Power sums are the most widely known symmetric polynomials.  ... 
doi:10.1109/icip.2014.7025959 dblp:conf/icip/NegrinhoA14 fatcat:mvomz5ghwjcalclifjwnf7v5re

Duality of sum of nonnegative circuit polynomials and optimal SONC bounds [article]

Dávid Papp
2019 arXiv   pre-print
sum-of-squares polynomials.  ...  Seidler and de Wolff recently showed that sums of nonnegative circuit (SONC) polynomials can be used to compute global lower bounds (called SONC bounds) for polynomials in this manner in polynomial time  ...  The author is grateful to Mareike Dressler (UCSD) for pointing out the reference to Jie Wang's recent work [37] on the support of SONC polynomials.  ... 
arXiv:1912.04718v1 fatcat:fes5gx4p6vfwbkmrgocgtrobli

Function Evaluation on FPGAs using On-Line Arithmetic Polynomial Approximation

Rachid Beguenane, Stephane Simard, Arnaud Tisserand
2006 2006 IEEE North-East Workshop on Circuits and Systems  
It presents on-line arithmetic operators for the polynomial approximation of some functions (e.g., reciprocal, square-root, sine, cosine, exponential, logarithm).  ...  The proposed method is based on polynomial approximations with sparse coefficients well suited for FPGA implementation.  ...  In a future work, we will provide an optimized power stage with a its cost and on-line delay estimations. The sum stage uses the decomposition characteristics.  ... 
doi:10.1109/newcas.2006.250959 fatcat:hyero3iehjfwhofvm5rcn56luq

Algorithms for Weighted Sums of Squares Decomposition of Non-negative Univariate Polynomials [article]

Victor Magron and Mohab Safey El Din and Markus Schweighofer
2017 arXiv   pre-print
In this article, we describe, analyze and compare both from the theoretical and practical points of view, two algorithms computing such a weighted sums of squares decomposition for univariate polynomials  ...  It is well-known that every non-negative univariate real polynomial can be written as the sum of two polynomial squares with real coefficients.  ...  Preliminary experiments yield very promising results when the bitsize of the polynomials is small, e.g. for power sums of degree up to 1000.  ... 
arXiv:1706.03941v1 fatcat:ipo4v3w2q5f7nihzafeozcq52m

Determinantal Representations and the Hermite Matrix [article]

Tim Netzer, Daniel Plaumann, Andreas Thom
2011 arXiv   pre-print
If some power of a polynomial admits a definite determinantal representation, then its Hermite matrix is a sum of squares.  ...  We relate the question to sums of squares decompositions of a certain Hermite matrix.  ...  We first show in Section 1 that a definite determinantal representation of some power of p of the correct size always yields a sum-of-squares decomposition of H(p) (Thm. 1.6).  ... 
arXiv:1108.4380v1 fatcat:r2qifbuhy5h7pigugprglh2s4m

Algorithms for weighted sum of squares decomposition of non-negative univariate polynomials

Victor Magron, Mohab Safey El Din, Markus Schweighofer
2018 Journal of symbolic computation  
Preliminary experiments yield very promising results when the bitsize of the polynomials is small, e.g. for power sums of degree up to 1000.  ...  Table 2 : 2 Comparison results of output size and performance between Algorithm univsos1 and Algorithm univsos2 for non-negative power sums of increasing degrees.  ...  We obtain an approximate rational sums of squares decomposition of the polynomial p ε with the auxiliary procedure sum two squares (Step 8), relying on an arbitrary precision complex root finder.  ... 
doi:10.1016/j.jsc.2018.06.005 fatcat:tuydv76grrcgjc4vdrt3pzp2ge

Attack Based on Direct Sum Decomposition against the Nonlinear Filter Generator [chapter]

Jingjing Wang, Xiangxue Li, Kefei Chen, Wenzheng Zhang
2012 Lecture Notes in Computer Science  
In this paper, we present the direct sum decomposition of the NLFG output sequence that leads to a system of linear equations in the initial state of the NLFG and further to an efficient algebraic attack  ...  The nonlinear filter generator (NLFG) is a powerful building block commonly used in stream ciphers.  ...  Then we identify the direct sum decomposition of the NLFG keystream corresponding to that particular couple of co-prime polynomials.  ... 
doi:10.1007/978-3-642-31410-0_4 fatcat:5s2p4u4ovze5lcpshsr3wkjomy
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