We study the symmetric tensor rank of multiplication over finite field extensions using linearized polynomials. Via field trace, symmetric linearized polynomials are identified with symmetric bilinear forms and symmetric matrices, allowing symmetric tensor decompositions to be reformulated as spanning problems by rank-one symmetric linearized polynomials. We translate these spanning conditions into explicit linear systems over finite fields and use the Frobenius automorphism to obtain computationally effective criteria. As applications, we recover known values of the symmetric bilinear complexity for small extension degrees and obtain explicit symmetric decompositions for several parameters. We also introduce the symmetric tensor-rank of a symmetric rank-metric code and show that, for the natural one-dimensional Gabidulin code associated with finite field multiplication, this invariant coincides with the symmetric tensor rank of the multiplication map.

翻译：我们使用线性化多项式研究有限域扩张上乘法映射的对称张量秩。通过域迹算子，对称线性化多项式可与对称双线性形式及对称矩阵一一对应，从而将对称张量分解问题重新表述为秩一对称线性化多项式的张成问题。我们将这些张成条件转化为有限域上的显式线性方程组，并利用Frobenius自同构获得计算有效的判据。作为应用，我们恢复了小扩张次数下对称双线性复杂度的已知值，并给出了若干参数下的显式对称分解。此外，我们引入了对称秩度量码的对称张量秩概念，并证明：对于与有限域乘法相关联的自然一维Gabidulin码，该不变量恰好等于乘法映射的对称张量秩。