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On exact Reznick, Hilbert-Artin and Putinar's representations
Journal of Symbolic Computation ( IF 0.7 ) Pub Date : 2021-04-01 , DOI: 10.1016/j.jsc.2021.03.005
Victor Magron , Mohab Safey El Din

We consider the problem of computing exact sums of squares (SOS) decompositions for certain classes of non-negative multivariate polynomials, relying on semidefinite programming (SDP) solvers.

We provide a hybrid numeric-symbolic algorithm computing exact rational SOS decompositions with rational coefficients for polynomials lying in the interior of the SOS cone. The first step of this algorithm computes an approximate SOS decomposition for a perturbation of the input polynomial with an arbitrary-precision SDP solver. Next, an exact SOS decomposition is obtained thanks to the perturbation terms and a compensation phenomenon. We prove that bit complexity estimates on output size and runtime are both polynomial in the degree of the input polynomial and singly exponential in the number of variables. Next, we apply this algorithm to compute exact Reznick, Hilbert-Artin's representation and Putinar's representations respectively for positive definite forms and positive polynomials over basic compact semi-algebraic sets. We also report on practical experiments done with the implementation of these algorithms and existing alternatives such as the critical point method and cylindrical algebraic decomposition.



中文翻译:

关于确切的雷兹尼克,希尔伯特·阿廷和普蒂纳尔的表述

我们考虑依赖半定规划(SDP)求解器来计算某些类非负多元多项式的精确平方和(SOS)分解的问题。

我们提供了一种混合数值符号算法,计算精确位于SOS圆锥内部的多项式的有理系数的有理SOS分解。该算法的第一步是使用任意精度的SDP求解器为输入多项式的扰动计算近似的SOS分解。接下来,由于扰动项和补偿现象,获得了精确的SOS分解。我们证明,对输出大小和运行时间的位复杂度估计在输入多项式的程度上都是多项式,而在变量数量上则是单指数。接下来,我们应用此算法分别计算基本紧致半代数集上的正定形式和正多项式的精确Reznick,Hilbert-Artin表示和Putinar表示。

更新日期:2021-04-09
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