A slice decomposition is an expression of a homogeneous polynomial as a sum of forms with a linear factor. A strength decomposition is an expression of a homogeneous polynomial as a sum of reducible forms. The slice rank and strength of a polynomial are the minimal lengths of such decompositions, respectively. The slice rank is an upper bound for the strength and the gap between these two values can be arbitrary large. However, in line with a conjecture by Catalisano et al. on the dimensions of secant varieties of the varieties of reducible forms, we conjecture that equality holds for general forms. By using a weaker version of Fröberg's Conjecture on the Hilbert series of ideals generated by general forms, we show that our conjecture holds up to degree 7 and in degree 9.

切片分解是将齐次多项式表示为具有线性因子的形式之和。强度分解是将齐次多项式表示为可归约形式的总和。多项式的切片秩和强度分别是此类分解的最小长度。切片等级是强度的上限，这两个值之间的差距可以任意大。然而，与 Catalisano 等人的猜想一致。关于可约形式簇的割线簇的维数，我们推测等式对一般形式成立。通过使用 Fröberg 对由一般形式生成的希尔伯特理想系列的猜想的较弱版本，我们证明了我们的猜想在 7 次和 9 次时成立。