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

2022
*
arXiv
*
pre-print

We give efficient algorithms for finding

arXiv:2208.00122v1
fatcat:tar2nncesrbz5mxci6oyicpf6a
*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*. ...##
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Efficient evaluation of noncommutative polynomials using tensor and noncommutative Waring decompositions
[article]

2021
*
arXiv
*
pre-print

For example, we consider a "Waring

arXiv:1903.05910v2
fatcat:j3gel7w6nvg4lpvk2ghhia2vea
*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*. ...##
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Approximating the maximum of a polynomial over a polytope: Handelman decomposition and continuous generating functions
[article]

2016
*
arXiv
*
pre-print

We investigate a way to approximate the maximum

arXiv:1601.04118v2
fatcat:amwydd6eqvff3lxpdt7adb7rfi
*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. ...##
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A MULTIVARIATE POLYNOMIAL MATRIX ORDER-REDUCTION ALGORITHM FOR LINEAR FRACTIONAL TRANSFORMATION MODELLING

2005
*
IFAC Proceedings Volumes
*

In this paper an algorithm that provides an equivalent, but

doi:10.3182/20050703-6-cz-1902.00999
fatcat:cl3n3ifot5brjci3ptlmi57hgq
*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). ...##
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Derandomization and absolute reconstruction for sums of powers of linear forms
[article]

2021
*
arXiv
*
pre-print

We study the

arXiv:1912.02021v5
fatcat:axmheh4t45eyhalgzkcquzboli
*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. ...##
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Learning sums of powers of low-degree polynomials in the non-degenerate case
[article]

2020
*
arXiv
*
pre-print

We develop algorithms for writing a

arXiv:2004.06898v2
fatcat:tpebdeaqwnh55h26iob3yy2s6u
*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. ...##
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Prime and composite polynomials

1922
*
Transactions of the American Mathematical Society
*

Our task in the present paper will be, after showing the constancy

doi:10.1090/s0002-9947-1922-1501189-9
fatcat:n2hywal6svhxxid3i3o2b7srpm
*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. ...##
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Prime and Composite Polynomials

1922
*
Transactions of the American Mathematical Society
*

If F(z) is a

doi:10.2307/1988911
fatcat:fjxteh5nhvdzrmc3f2xdadehka
*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. ...##
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Symmetric polynomials for 2D shape representation

2014
*
2014 IEEE International Conference on Image Processing (ICIP)
*

We single out elementary symmetric

doi:10.1109/icip.2014.7025959
dblp:conf/icip/NegrinhoA14
fatcat:mvomz5ghwjcalclifjwnf7v5re
*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*. ...##
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Duality of sum of nonnegative circuit polynomials and optimal SONC bounds
[article]

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*. ...

##
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Function Evaluation on FPGAs using On-Line Arithmetic Polynomial Approximation

2006
*
2006 IEEE North-East Workshop on Circuits and Systems
*

It presents on-line arithmetic operators for the

doi:10.1109/newcas.2006.250959
fatcat:hyero3iehjfwhofvm5rcn56luq
*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. ...##
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Algorithms for Weighted Sums of Squares Decomposition of Non-negative Univariate Polynomials
[article]

2017
*
arXiv
*
pre-print

In this article, we describe, analyze and compare both from the theoretical and practical points

arXiv:1706.03941v1
fatcat:ipo4v3w2q5f7nihzafeozcq52m
*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. ...##
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Determinantal Representations and the Hermite Matrix
[article]

2011
*
arXiv
*
pre-print

If some

arXiv:1108.4380v1
fatcat:r2qifbuhy5h7pigugprglh2s4m
*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). ...##
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Algorithms for weighted sum of squares decomposition of non-negative univariate polynomials

2018
*
Journal of symbolic computation
*

Preliminary experiments yield very promising results when the bitsize

doi:10.1016/j.jsc.2018.06.005
fatcat:tuydv76grrcgjc4vdrt3pzp2ge
*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. ...##
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Attack Based on Direct Sum Decomposition against the Nonlinear Filter Generator
[chapter]

2012
*
Lecture Notes in Computer Science
*

In this paper, we present the direct

doi:10.1007/978-3-642-31410-0_4
fatcat:5s2p4u4ovze5lcpshsr3wkjomy
*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*. ...
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