New results on computing certificates of strictly positive polynomials in Archimedean quadratic modules are presented. The results build upon (i) Averkov's method for generating a strictly positive polynomial for which a membership certificate can be more easily computed than the input polynomial whose certificate is being sought, and (ii) Lasserre's method for generating a certificate by successively approximating a nonnegative polynomial by sums of squares. First, a fully constructive method based on Averkov's result is given by providing details about the parameters; further, his result is extended to work on arbitrary subsets, in particular, the whole Euclidean space $\mathbb{R}^n$, producing globally strictly positive polynomials. Second, Lasserre's method is integrated with the extended Averkov construction to generate certificates. Third, the methods have been implemented and their effectiveness is illustrated. Examples are given on which the existing software package RealCertify appears to struggle, whereas the proposed method succeeds in generating certificates. Several situations are identified where an Archimedean polynomial does not have to be explicitly included in a set of generators of an Archimedean quadratic module. Unlike other approaches for addressing the problem of computing certificates, the methods/approach presented is easier to understand as well as implement.

翻译：本文提出了关于计算阿基米德二次模中严格正多项式证书的新结果。这些结果基于（i）Averkov 的方法，用于生成一个严格正多项式，其成员资格证书比待求证书的输入多项式更易计算；以及（ii）Lasserre 的方法，通过逐次逼近非负多项式为平方和来生成证书。首先，基于 Averkov 的结果，通过详细说明参数，给出了一种完全构造性方法；进一步，将其结果推广至任意子集，特别是整个欧几里得空间 $\\mathbb{R}^n$，从而生成全局严格正多项式。其次，将 Lasserre 的方法与扩展的 Averkov 构造相结合以生成证书。第三，这些方法已实现，并展示了其有效性。给出了现有软件包 RealCertify 似乎难以处理，而所提方法成功生成证书的示例。识别了若干情形，其中阿基米德多项式无需显式包含在阿基米德二次模的生成元集中。与解决证书计算问题的其他方法相比，所提出的方法/途径更易于理解和实现。