We continue the study initiated by H. S. Shapiro on Fischer decompositions of entire functions, showing that such decomposition exist in a weak sense (we do not prove uniqueness) under hypotheses regarding the order of the entire function f to be expressed as \(f= P\cdot q+r\), the polynomial P, and bounds on the apolar norm of homogeneous polynomials of degree m. These bounds, previously used by Khavinson and Shapiro, and by Ebenfelt and Shapiro, can be interpreted as a quantitative, asymptotic strengthening of Bombieri’s inequality. In the special case where both the dimension of the space and the degree of P are two, we characterize for which polynomials P such bounds hold.

我们继续 HS Shapiro 发起的关于整个函数的 Fischer 分解的研究，表明在整个函数f的阶表示为\(f= P 的假设下，这种分解以弱意义存在（我们不证明唯一性）\cdot q+r\)，多项式P ，以及m 次齐次多项式的非极范数的界限。这些界限以前被哈文森和夏皮罗以及埃本菲尔特和夏皮罗使用过，可以解释为邦别里不平等的定量、渐近强化。在空间维数和P次数均为 2 的特殊情况下，我们描述了哪些多项式P 的边界成立。