Let$P_n(y_1,\dots ,y_n):=\prod _{1\le i<j\le n}(1-\frac{y_i}{y_j})$ and$P_n:=\underset{(y_1,\dots ,y_n)}{\sup }{P}_n(y_1,\dots ,y_n)$ where the supremum is taken over the n-ples $(y_1,\dots ,y_n)$ of real numbers satisfying $0<|y_1|<|y_2|<\cdots <|y_n|$. We prove that $P_n\le 2^{\lfloor n/2\rfloor }$ for every n, i.e., we extend to all n the bound that Pohst proved for $n\le 11$. As a consequence, the bound for the absolute discriminant of a totally real field in terms of its regulator is now proved for every degree of the field.

中文翻译：

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Pohst 不等式的推广

让${磷}_n({是}_1,\dots ,{是}_n)：=\prod _{1\le \mathrm{一世}<j\le n}(1-\frac{{是}_{\mathrm{一世}}}{{是}_j})$ 和${磷}_n：=\underset{({是}_1,\dots ,{是}_n)}{\mathrm{超}}{磷}_n({是}_1,\dots ,{是}_n)$其中supremum接管了n个$({是}_1,\dots ,{是}_n)$ 实数满足 $0<|{是}_1|<|{是}_2|<\cdots <|{是}_n|$. 我们证明${磷}_n\le 2^{\lfloor n/2\rfloor }$对于每个n，即，我们将Pohst 证明的界限扩展到所有n$n\le 11$. 因此，现在证明了一个完全实域的绝对判别式在其调节器方面的界限，适用于该域的每一级。