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：

---

论 Exact Reznick、Hilbert-Artin 和 Putinar 的表述

我们考虑依赖于半定规划 (SDP) 求解器为某些类别的非负多元多项式计算精确平方和 (SOS) 分解的问题。我们提供了一种混合数字符号算法，该算法使用位于 SOS 锥体内部的多项式的有理系数计算精确的有理 SOS 分解。该算法的第一步是使用任意精度 SDP 求解器计算输入多项式扰动的近似 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都是输入多项式次数的多项式和变量数量的单指数。下一个，我们应用该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。我们还报告了使用这些算法和现有替代方法（例如临界点方法和圆柱代数分解）进行的实际实验。