八边形扇形上二次型的经典多项式不等式
Classical polynomial inequalities for quadratic forms on an octagonal sector
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中文总结 AI 辅助
该研究针对正八边形第一象限扇形上的实二次型建立一组尖锐不等式，确定了多个全局常数，得到q=4/3时的Bohnenblust–Hille型系数不等式。
中文摘要 AI 辅助
我们针对正八边形第一象限扇形上的实二次型建立了一组尖锐不等式。从相关多项式单位球的完整极点描述出发，计算了欧氏梯度的精确逐点伯恩斯坦函数。一条极端曲线控制着该问题：其端点在斜率为1/2时起作用，之后最大化器遵循显式卡尔达诺分支。我们得到了尖锐马尔可夫常数2√5、精确相对二次极化常数2、规范无条件常数3以及体相对玻尔半径1/√3。还确定了所有1≤q≤∞的最优系数ℓ_q比较，所有这些全局常数的极值均为同一个范数1多项式，在q=4/3时结果是尖锐的固定空间Bohnenblust–Hille型系数不等式。
英文摘要
We establish a collection of sharp inequalities for real quadratic forms on the first-quadrant sector of a regular octagon. Starting from the complete extreme-point description of the associated polynomial unit ball, we compute the exact pointwise Bernstein function for the Euclidean gradient. One extreme curve controls the problem: its endpoint is active up to slope $1/2$, after which the maximizer follows an explicit Cardano branch. We obtain the sharp Markov constant $2\sqrt5$, the exact relative quadratic polarization constant $2$, the canonical unconditional constant $3$, and the body-relative Bohr radius $1/\sqrt3$. We also determine the optimal coefficient $\ell_q$-comparison for every $1\le q\le\infty$. The same norm-one polynomial is extremal for all these global constants. At $q=4/3$ the result is a sharp fixed-space coefficient inequality of \textit{Bohnenblust--Hille type}.
