We give lower bounds for the numbers of real solutions in problems appearing in Schubert calculus in the Grassmannian $${\mathop{\rm Gr}(n,d)}$$Gr(n,d) related to osculating flags. It is known that such solutions are related to Bethe vectors in the Gaudin model associated to $${\mathop{\rm gl}_n}$$gln. The Gaudin Hamiltonians are self-adjoint with respect to a non-degenerate indefinite Hermitian form. Our bound comes from the computation of the signature of that form.

中文翻译：

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舒伯特微积分问题中实数解数的下界

我们给出了舒伯特微积分中出现的问题的实数下界，与密切标志相关的格拉斯曼学 $${\mathop{\rm Gr}(n,d)}$$Gr(n,d)。众所周知，此类解与与 $${\mathop{\rm gl}_n}$$gln 相关联的 Gaudin 模型中的 Bethe 向量有关。Gaudin 哈密顿量相对于非退化不定 Hermitian 形式是自伴的。我们的界限来自于对该表格签名的计算。