A Real Generalized Trisecant Trichotomy

Kristian Ranestad, Anna Seigal, Kexin Wang
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Abstract

The classical trisecant lemma says that a general chord of a non-degenerate space curve is not a trisecant; that is, the chord only meets the curve in two points. The generalized trisecant lemma extends the result to higher-dimensional varieties. It states that the linear space spanned by general points on a projective variety intersects the variety in exactly these points, provided the dimension of the linear space is smaller than the codimension of the variety and that the variety is irreducible, reduced, and non-degenerate. We prove a real analogue of the generalized trisecant lemma, which takes the form of a trichotomy. Along the way, we characterize the possible numbers of real intersection points between a real projective variety and a complimentary dimension real linear space. We show that any integer of correct parity between a minimum and a maximum number can be achieved. We then specialize to Segre-Veronese varieties, where our results apply to the identifiability of independent component analysis, tensor decomposition and to typical tensor ranks.
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真正的广义三十进制三分法
经典的十三次方程(trisecant lemma)说的是,非退化空间曲线的一般弦不是十三次方程;也就是说,弦只在两个点上与曲线相交。广义十等分两点定理将这一结果推广到了更高维度的品种上。它指出,只要线性空间的维数小于广域的维数,并且广域是不可还原的、还原的和非退化的,那么由投影广域上的一般点所跨的线性空间正好在这些点上与广域相交。我们证明了广义三等分lemma 的实数类比,它采用了三分法的形式。同时,我们还描述了实射影变与有余维实线性空间之间实交点的可能数目。我们证明了最小值和最大值之间任何正确奇偶性的整数都可以实现。然后,我们专门研究了 Segre-Veronese 变体,我们的结果适用于独立分量分析的可识别性、张量分解和典型张量等级。
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