标题： 边界的幂单项式的秩 显示英文标题

标题： Border Ranks of Monomials

摘要： 年轻展开式，由Landsberg和Ottaviani引入，为割线簇提供了行列式方程，其非零性为张量和特别是多项式的边界秩提供了下界。 我们研究了适用于年轻展开式的单项式最优形状，这些形状展示了年轻展开式方法的极限。 特别是，它们为大量单项式提供了最佳可能的下界，包括所有次数不超过6的单项式、3个变量的单项式以及变量乘积的任何次幂。 另一方面，对于次数为7及更高的单项式，存在一些单项式，使得任何年轻展开式都无法给出与Landsberg和Teitler推测的紧上界相匹配的下界。 显示英文摘要

摘要： Young flattenings, introduced by Landsberg and Ottaviani, give determinantal equations for secant varieties and their non-vanishing provides lower bounds for border ranks of tensors and in particular polynomials. We study monomial-optimal shapes for Young flattenings, which exhibit the limits of the Young flattening method. In particular, they provide the best possible lower bound for large classes of monomials including all monomials up to degree 6, monomials in 3 variables, and any power of the product of variables. On the other hand, for degree 7 and higher there are monomials for which no Young flattening can give a lower bound that matches the conjecturally tight upper bound of Landsberg and Teitler.