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Asympotic bounds for Bombieri's inequality on products of homogeneous polynomials

J. M. Aldaz, H. Render

Abstract

Let $P$ be a fixed homogeneous polynomial. We present a sharp condition on $P$ guaranteeing the existence of asymptotically larger bounds in Bombieri's inequality, so for every homogeneous polynomial $q_m$ of degree $m$ we have \begin{equation*} \left\Vert P q_{m}\right\Vert _{a}\geq C_{P} m^{l\left( P\right) /2}\left\Vert q_{m}\right\Vert _{a}, \end{equation*} where $\| \cdot \| _{a}$ denotes the apolar norm. Explicit estimates for $C_P > 0$ and $l(P) > 0$ are given.

Asympotic bounds for Bombieri's inequality on products of homogeneous polynomials

Abstract

Let be a fixed homogeneous polynomial. We present a sharp condition on guaranteeing the existence of asymptotically larger bounds in Bombieri's inequality, so for every homogeneous polynomial of degree we have \begin{equation*} \left\Vert P q_{m}\right\Vert _{a}\geq C_{P} m^{l\left( P\right) /2}\left\Vert q_{m}\right\Vert _{a}, \end{equation*} where denotes the apolar norm. Explicit estimates for and are given.
Paper Structure (4 sections, 12 theorems, 74 equations)

This paper contains 4 sections, 12 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Background definitions and results
  3. Proof of the sufficiency part of Theorem \ref{['ThmMain1']}
  4. Proof of the necessity part of Theorem \ref{['ThmMain1']}

Key Result

Theorem 1

Let $P_k: \mathbb{C}^d \to \mathbb{C}$ be a non-zero homogeneous polynomial of degree $k \ge 2$, and let $1 \le \rho < k$ be a natural number. If the polynomials contained in do not have a common zero $c\in \mathbb{C}^d \setminus \{0\},$ then taking we have that for all $m\geq 0$ and all $f_{m}\in\mathcal{P}_{m}\left( \mathbb{C}^{d}\right)$, On the other hand, if there is a $c\in\mathbb{C}^{d}

Theorems & Definitions (21)

  • Theorem 1
  • Corollary 2
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • Proposition 5
  • proof
  • Proposition 6
  • proof
  • ...and 11 more