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<p>In this paper we study the symmetric rank of products of linear forms and an irreducible quadratic form. The main result presents a new, non-trivial lower bound for the rank, and the arguments rely on the apolarity lemma. In the special case of degree 4 we give a list of normal forms for such quartics, and we apply our general result to compute the rank of almost all of them. These families of quartics provide examples of polynomials of generic, supergeneric, and even maximal rank.</p>

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Symmetric rank of some reducible quartics

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年1月2日

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  • Liena Colarte-Gómez,
  • Francesco Galuppi

摘要

In this paper we study the symmetric rank of products of linear forms and an irreducible quadratic form. The main result presents a new, non-trivial lower bound for the rank, and the arguments rely on the apolarity lemma. In the special case of degree 4 we give a list of normal forms for such quartics, and we apply our general result to compute the rank of almost all of them. These families of quartics provide examples of polynomials of generic, supergeneric, and even maximal rank.