Consider a homogeneous additive polynomial of the form $F(\mathbf{x})=a_1{x}_1^k+a_2{x}_2^k+\cdots +a_s{x}_s^k$, where the coefficients are integers. The function ${\mathrm{\Gamma }}^{⁎}(k)$ is defined as the smallest number s of variables such that F is guaranteed to have a nontrivial zero in every p-adic field ${\mathbb{Q}}_p$ regardless of the coefficients. In this article, we calculate the value of ${\mathrm{\Gamma }}^{⁎}(k)$ for all $33\le k\le 64$ for which this value is not already known.

中文翻译：

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Γ⁎(k) 函数的更精确值

考虑以下形式的齐次加法多项式 $F(\mathbf{X})={\mathrm{一个}}_1{X}_1^{克}+{\mathrm{一个}}_2{X}_2^{克}+\cdots +{\mathrm{一个}}_{秒}{X}_{秒}^{克}$，其中系数是整数。功能${\mathrm{\Gamma }}^{⁎}(克)$被定义为变量的最小数量s，使得F保证在每个p- adic 域中都有一个非平凡的零${\mathbb{问}}_p$与系数无关。在本文中，我们计算${\mathrm{\Gamma }}^{⁎}(克)$ 对所有人 $33\le 克\le 64$ 此值未知。