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<p>Let <i>X</i> be a (real or complex) infinite dimensional linear space. We establish conditions on a homogeneous polynomial <i>P</i> on <i>X</i> so that, if <i>W</i> is any finite dimensional subspace of <i>X</i> on which <i>P</i> vanishes, then <i>P</i> vanishes on an infinite dimensional subspace of <i>X</i> containing <i>W</i>. In the complex case, this is a step beyond the classical result due to Plichko and Zagorodnyuk. Applications to the real case are also provided.</p>

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Zero sets of homogeneous polynomials containing infinite dimensional spaces

  • 期刊论文

  • 发表日期：

    2025年5月27日

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  • Mikaela Aires,
  • Geraldo Botelho

摘要

Let X be a (real or complex) infinite dimensional linear space. We establish conditions on a homogeneous polynomial P on X so that, if W is any finite dimensional subspace of X on which P vanishes, then P vanishes on an infinite dimensional subspace of X containing W. In the complex case, this is a step beyond the classical result due to Plichko and Zagorodnyuk. Applications to the real case are also provided.