Explicit polynomial sequences with maximal spaces of partial derivatives and a question of K. Mulmuley
Explicit polynomial sequences with maximal spaces of partial derivatives and a question of K. Mulmuley
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DOI：
10.4086/toc.2019.v015a003
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发表时间：
2017-05
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通讯作者：
Fulvio Gesmundo;J. Landsberg
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作者：
Fulvio Gesmundo;J. Landsberg
我们回答 K. Mulmuley 的问题：在 [Efremenko-Landsberg-Schenck-Weyman] 中表明，平移偏导数的方法不能用于将填充永久物与行列式分开。 Mulmuley 询问这种“不行”的结果是否可以扩展到没有填充的模型。我们使用迭代矩阵乘法多项式证明情况确实如此。我们还提供了几个具有最大偏导数空间的多项式示例，包括完全对称多项式。我们对这些多项式应用 Koszul 平坦化，以获得第一个显式多项式序列，其对称边界等级下界高于通过偏导数可达到的界限。
We answer a question of K. Mulmuley: In [Efremenko-Landsberg-Schenck-Weyman] it was shown that the method of shifted partial derivatives cannot be used to separate the padded permanent from the determinant. Mulmuley asked if this "no-go" result could be extended to a model without padding. We prove this is indeed the case using the iterated matrix multiplication polynomial. We also provide several examples of polynomials with maximal space of partial derivatives, including the complete symmetric polynomials. We apply Koszul flattenings to these polynomials to have the first explicit sequence of polynomials with symmetric border rank lower bounds higher than the bounds attainable via partial derivatives.