张量的E-行列式
E-Determinants of Tensors
AI总结：
将对称张量的对称超行列式概念推广到一般张量的E-行列式，证明其继承矩阵行列式的许多性质，并应用于多项式系统可解性及张量特征多项式。
摘要：
我们将对称张量的对称超行列式概念推广到一般张量的E-行列式。我们证明了E-行列式继承了矩阵行列式的许多性质。这些性质包括：多项式系统的可解性、张量复合的E-行列式、块张量的E-行列式的乘积公式、Hadamard不等式、Gersgrin不等式和Minikowski不等式。作为一个简单应用，我们证明如果多项式系统的首项系数张量是对角线元素非零的三角张量，则该系统一定有解。我们通过E-行列式研究张量的特征多项式。当维数为二时，给出了特征多项式系数的显式公式。
英文摘要：
We generalize the concept of the symmetric hyperdeterminants for symmetric tensors to the E-determinants for general tensors. We show that the E-determinant inherits many properties of the determinant of a matrix. These properties include: solvability of polynomial systems, the E-determinat of the composition of tensors, product formula for the E-determinant of a block tensor, Hadamard's inequality, Gersgrin's inequality and Minikowski's inequality. As a simple application, we show that if the leading coefficient tensor of a polynomial system is a triangular tensor with nonzero diagonal elements, then the system definitely has a solution. We investigate the characteristic polynomial of a tensor through the E-determinant. Explicit formulae for the coefficients of the characteristic polynomial are given when the dimension is two.
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