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arXiv · 2403.10400

A Fischer type decomposition theorem from the apolar inner product

J. M. Aldaz·H. Render
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Abstract

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.

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J. M. Aldaz, H. Render. 2024-03-15. A Fischer type decomposition theorem from the apolar inner product. https://doi.org/10.1007/s13324-023-00844-4

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