一类循环对称张量的正定性
Positive definiteness of a class of cyclic symmetric tensors
AI总结：
本文研究一类循环对称张量的正定性，建立了四阶三维循环对称张量半正定及四阶n维对称张量正定的充要条件，并据此给出循环与非循环情形的正定性判定，进而得到三元四次齐次多项式的若干不等式。
摘要：
针对四阶三维循环对称张量，建立了其半正定性的一个充分必要条件。对于四阶n维对称张量，给出了正定性的一个充分必要条件。借助该条件，给出了一类四阶三维循环对称张量的正定性判定。此外，还给出了一类非循环对称张量的正定性判定。通过应用这些结论，为三元四次齐次多项式建立了若干（严格）不等式。
英文摘要：
For a 4th order 3-dimensional cyclic symmetric tensor, a sufficient and necessary condition is bulit for its positive semi-definiteness. A sufficient and necessary condition of positive definiteness is showed for a 4th order $n$-dimensional symmetric tensor. With the help of such a condition, the positive definiteness of a class of 4th order 3-dimensional cyclic symmetric tensors is given. Moreover, the positive definiteness of a class of non-cyclic symmetric tensors is showed also. By applying these conclusions, several (strict) inequalities are erected for ternary quartic homogeneous polynomials.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
