In this article, we formulate necessary and sufficient polynomial equations for the existence of a symmetry plane or an order-two axial symmetry for a totally symmetric tensor of order n ≥ 1 . These conditions are effective and of degree n (the tensor’s order) in the components of the normal to the plane (or the direction of the axial symmetry). These results are then extended to obtain necessary and sufficient polynomial conditions for the existence of such symmetries for an elasticity tensor, a piezo-electricity tensor or a piezo-magnetism pseudo-tensor.

中文翻译：

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减少平面或轴张量对称的代数条件

在本文中，我们为 n ≥ 1 阶完全对称张量的对称平面或二阶轴对称的存在性制定了必要且充分的多项式方程。这些条件是有效的，并且在平面法线（或轴对称方向）的分量中的阶数为 n（张量级）。然后将这些结果扩展以获得必要且充分的多项式条件，用于弹性张量、压电张量或压电磁伪张量的此类对称性的存在。