论文标题
Reznick的Potitivstellensatz的稀疏版本
A sparse version of Reznick's Positivstellensatz
论文作者
论文摘要
如果$ f $是一种积极的确定形式，则Reznick的Potitivstellensatz [Mathematische Zeitschrift。 220（1995），第75--97页说，存在$ k \ in \ mathbf {n} $，以便$ {\ | x \ |^{2K} _2} f $是多项式的正方形之和。假设$ f $可以写为表格的总和$ \ sum_ {l = 1}^p f_l $，其中每个$ f_l $都取决于初始变量的一个子集，并且假设这些子集满足所谓的运行交叉点，那么我们提供了Reznick portivivstellensatz的稀疏版本。 Namely, there exists $k \in \mathbf{N}$ such that $f=\sum_{l = 1}^p {{σ_l}/{H_l^{k}}}$, where $σ_l$ is a sum of squares of polynomials, $H_l$ is a uniform polynomial denominator, and both polynomials $σ_l，h_l $涉及与$ f_l $相同的变量，对于每个$ l = 1，\ dots，p $。换句话说，$ f $的稀疏模式也反映在此稀疏版本的Reznick的积极证书中。接下来，我们使用此结果还获得了（i）多项式在整个空间上的多项式的阳性证书，并且（ii）在（可能是非紧密的）基本的半分布集中的多项式非负值，假设输入数据满足了运行的交点。两者都是由于Putinar和vasilescu引起的阳性证书的稀疏版本。
If $f$ is a positive definite form, Reznick's Positivstellensatz [Mathematische Zeitschrift. 220 (1995), pp. 75--97] states that there exists $k\in\mathbf{N}$ such that ${\| x \|^{2k}_2}f$ is a sum of squares of polynomials. Assuming that $f$ can be written as a sum of forms $\sum_{l=1}^p f_l$, where each $f_l$ depends on a subset of the initial variables, and assuming that these subsets satisfy the so-called running intersection property, we provide a sparse version of Reznick's Positivstellensatz. Namely, there exists $k \in \mathbf{N}$ such that $f=\sum_{l = 1}^p {{σ_l}/{H_l^{k}}}$, where $σ_l$ is a sum of squares of polynomials, $H_l$ is a uniform polynomial denominator, and both polynomials $σ_l,H_l$ involve the same variables as $f_l$, for each $l=1,\dots,p$. In other words, the sparsity pattern of $f$ is also reflected in this sparse version of Reznick's certificate of positivity. We next use this result to also obtain positivity certificates for (i) polynomials nonnegative on the whole space and (ii) polynomials nonnegative on a (possibly non-compact) basic semialgebraic set, assuming that the input data satisfy the running intersection property. Both are sparse versions of a positivity certificate due to Putinar and Vasilescu.
