A new symbolic algorithm to compute sums of squares multipliers (certificates) to witness the membership of non-negative univariate polynomials in a saturated univariate quadratic module is presented. Certificates are first computed in terms of natural generators introduced by Kuhlmann and Marshall for an Archimedean saturated quadratic module; natural generators can be easily read-off from a semialgebraic set. In the univariate case, an Archimedean quadratic module is also a preordering since it is closed under multiplication; certificates have different representations when a polynomial is viewed as a member in a quadratic module versus in a preordering An algorithm is given to compute certificates of natural generators in terms of the original generators; it uses a construction introduced by Kuhlmann, Marshall, and Schwartz known as the ``Basic Lemma'', which splits the non-negative factors of generators. To compute a quadratic module certificate, certificates of products of natural generators are computed using a detailed case analysis based on the types of natural generators. An implementation of the algorithms proposed in Maple is also discussed. The certificates obtained using this implementation are compared with those generated by RealCertify. We discuss examples where RealCertify is unable to find certificates while the proposed method is successful.

翻译：提出了一种新的符号算法，用于计算平方和乘子（证书），以验证非负单变量多项式属于饱和单变量二次模。首先，基于Kuhlmann和 Marshall 为阿基米德饱和二次模引入的自然生成元计算证书；自然生成元可轻松从半代数集中读出。在单变量情形下，阿基米德二次模也是预序模，因其对乘法封闭；当多项式被视为二次模或预序模中的元素时，证书具有不同的表示形式。给出了一个算法，用于计算原始生成元下自然生成元的证书；该算法利用了由Kuhlmann、Marshall和Schwartz引入的“基本引理”构造，该引理将生成元的非负因子分解。为计算二次模证书，基于自然生成元的类型，通过详细的案例分析法计算自然生成元乘积的证书。还讨论了在Maple中提出算法的实现。将此实现获取的证书与RealCertify生成的证书进行了比较。我们讨论了RealCertify无法找到证书而所提方法成功的实例。