关于波利亚和普京纳的精确表示
On Exact Polya and Putinar's Representations
摘要：
我们考虑的问题，找到确切的平方和（SOS）分解为某些类的非负多元多项式，依赖于半定规划（SDP）求解器。首先，我们提供了一个混合的数字-符号算法计算精确的合理SOS分解多项式躺在内部的SOS锥。它使用任意精度的SDP求解器计算输入多项式扰动的近似SOS分解。由于扰动项的存在，得到了精确的SOS分解。我们证明了输出大小和运行时间的比特复杂度估计在输入多项式的次数上都是多项式的，在变量的数量上都是指数的。其次，我们应用该算法分别计算了基本紧半代数集上正定形式和正多项式的精确Polya和Putinar表示。我们也比较了我们的算法与现有的计算机代数方法，包括柱形代数分解和临界点方法的实现。
英文摘要：
We consider the problem of finding exact sums of squares (SOS) decompositions for certain classes of non-negative multivariate polynomials, relying on semidefinite programming (SDP) solvers. We start by providing a hybrid numeric-symbolic algorithm computing exact rational SOS decompositions for polynomials lying in the interior of the SOS cone. It computes an approximate SOS decomposition for a perturbation of the input polynomial with an arbitrary-precision SDP solver. An exact SOS decomposition is obtained thanks to the perturbation terms. We prove that bit complexity estimates on output size and runtime are both polynomial in the degree of the input polynomial and simply exponential in the number of variables. Next, we apply this algorithm to compute exact Polya and Putinar's representations respectively for positive definite forms and positive polynomials over basic compact semi-algebraic sets. We also compare the implementation of our algorithms with existing methods in computer algebra including cylindrical algebraic decomposition and critical point method.
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