The symmetric subrank of homogeneous polynomial is the largest number of terms in a diagonal form to which it can be specialized by a (typically non-invertible) linear variable substitution. Building on earlier work by Derksen-Makam-Zuiddam and Biaggi-Chang-Draisma-Rupniewski for ordinary tensors, we determine the asymptotic behavior of symmetric subrank and symmetric border subrank of degree-d forms as the number of variables tends to infinity. Furthermore, by using results from geometric invariant theory we show that for cubic (resp. quartic) forms the symmetric subrank and symmetric border subrank coincide if the latter is at most three (resp. two).

翻译：齐次多项式的对称子秩是指在（通常非可逆的）线性变量代换下，其能化为对角形式后所含非零项的最大数目。基于Derksen-Makam-Zuiddam以及Biaggi-Chang-Draisma-Rupniewski先前关于普通张量的工作，我们确定了当变量个数趋于无穷时，d次形式的对称子秩与对称边界子秩的渐近行为。此外，利用几何不变量理论的结果，我们证明对于三次（或四次）形式，若对称边界子秩至多为3（或2），则对称子秩与对称边界子秩相等。