On real typical ranks
On real typical ranks
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在真实的典型队伍中
DOI：
10.1007/s40574-017-0134-0
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
G. Ottaviani
中科院分区：
文献类型：
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作者：
A. Bernardi;Grigoriy Blekherman;G. Ottaviani
我们研究关于真实的簇X的典型秩。例如张量秩（X是塞格雷簇）和对称张量秩（X是Veronese簇）。我们表明，任何之间的最小典型秩和最大典型秩的秩也是典型的。我们研究典型的秩n元对称张量的顺序d，或等价的齐次多项式的次数d在n个变量，为小值的n和d。我们证明了4是真实的三元三次体唯一的典型秩，而四元三次体只有5和6的典型秩。对于三元四次数，我们表明6和7是典型的秩，并且所有典型的秩都在6和8之间。对于三元五次，我们表明，典型的秩是7和13之间。
We study typical ranks with respect to a real variety X. Examples of such are tensor rank (X is the Segre variety) and symmetric tensor rank (X is the Veronese variety). We show that any rank between the minimal typical rank and the maximal typical rank is also typical. We investigate typical ranks of n-variate symmetric tensors of order d, or equivalently homogeneous polynomials of degree d in n variables, for small values of n and d. We show that 4 is the unique typical rank of real ternary cubics, and quaternary cubics have typical ranks 5 and 6 only. For ternary quartics we show that 6 and 7 are typical ranks and that all typical ranks are between 6 and 8. For ternary quintics we show that the typical ranks are between 7 and 13.