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<p>In this paper, we study the product of <i>n</i> linear forms over function fields. We calculate the maximum value of the minima of the forms with determinant one when <i>n</i> is small. The value is equal to the natural bound given by algebraic number theory. Our proof is based on a reduction theory of diagonal group orbits on homogeneous spaces. We also show that the forms defined algebraically correspond to periodic orbits with respect to the diagonal group actions.</p>

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The product of linear forms over function fields

  • 期刊论文

  • 发表日期：

    2026年5月6日

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  • Wenyu Guo,
  • Xuan Liu,
  • Ronggang Shi

摘要

In this paper, we study the product of n linear forms over function fields. We calculate the maximum value of the minima of the forms with determinant one when n is small. The value is equal to the natural bound given by algebraic number theory. Our proof is based on a reduction theory of diagonal group orbits on homogeneous spaces. We also show that the forms defined algebraically correspond to periodic orbits with respect to the diagonal group actions.