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 singly exponential in the cardinality of the Newton polytope (or doubly 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.

翻译：本文研究针对特定非负多元多项式计算精确平方和分解的问题，该方法依赖于半定规划求解器。我们提出一种混合数值-符号算法，可为位于平方和锥内部的、具有有理系数的多项式计算精确的有理平方和分解。该算法的第一步通过任意精度半定规划求解器，为输入多项式的扰动形式计算近似平方和分解。随后，借助扰动项与补偿机制获得精确的平方和分解。我们证明输出规模与运行时间的比特复杂度估计在牛顿多胞形基数上均为单指数级（或在变量数量上为双指数级）。进一步，我们将此算法分别应用于计算正定齐式的精确Reznick表示、Hilbert-Artin表示，以及基本紧半代数集上正多项式的Putinar表示。文中同时报告了这些算法的实现与现有替代方法（如临界点法与柱形代数分解）的实际实验对比。