We show that positivity (≥0) on $\mathbb{R}_{+}^{n}$ and on $\mathbb{R}^{n}$ of real symmetric polynomials of degree at most p in $n\ge 2$ variables is solvable by algorithms running in polynomial time in the number n of variables. For real symmetric quartics, we find discriminants which lead to the efficient algorithms QE4+ and QE4 running in $O(n)$ time. We describe the Maple implementation of both algorithms, which are then used not only for testing concrete inequalities (with given numerical coefficients and number of variables), but also for proving symbolic inequalities.

中文翻译：

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关于实对称多项式的正性检验算法

我们证明了 $\mathbb{R}_{+}^{n}$ 和 $\mathbb{R}^{n}$ 的正数（≥0）在 $n\ 中最多为 p 次的实对称多项式ge 2$ 变量可通过在多项式时间内以 n 个变量运行的算法求解。对于实对称四次方程，我们找到了导致在 $O(n)$ 时间内运行的高效算法 QE4+ 和 QE4 的判别式。我们描述了这两种算法的 Maple 实现，然后它们不仅用于测试具体的不等式（具有给定的数值系数和变量数量），还用于证明符号不等式。