Schubert coefficients are nonnegative integers $c^w_{u,v}$ that arise in Algebraic Geometry and play a central role in Algebraic Combinatorics. It is a major open problem whether they have a combinatorial interpretation, i.e, whether $c^w_{u,v} \in \#{\sf P}$. We study the closely related vanishing problem of Schubert coefficients: $\{c^w_{u,v}=^?0\}$. Until this work it was open whether this problem is in the polynomial hierarchy ${\sf PH}$. We prove that $\{c^w_{u,v}=^?0\}$ in ${\sf coAM}$ assuming the GRH. In particular, the vanishing problem is in ${\Sigma_2^{{\text{p}}}}$. Our approach is based on constructions lifted formulations, which give polynomial systems of equations for the problem. The result follows from a reduction to Parametric Hilbert's Nullstellensatz, recently studied in arXiv:2408.13027. We apply our construction to show that the vanishing problem is in ${\sf NP}_{\mathbb{C}} \cap {\sf P}_{\mathbb{R}}$ in the Blum--Shub--Smale (BSS) model of computation over complex and real numbers respectively. Similarly, we prove that computing Schubert coefficients is in $\#{\sf P}_{\mathbb{C}}$, a counting version of the BSS model. We also extend our results to classical types. With one notable exception of the vanishing problem in type $D$, all our results extend to all types.

翻译：舒伯特系数是代数几何中出现的非负整数$c^w_{u,v}$，在代数组合学中扮演着核心角色。它们是否具有组合解释（即是否满足$c^w_{u,v} \in \#{\sf P}$）是一个重要的开放性问题。我们研究与之密切相关的舒伯特系数消失问题：$\{c^w_{u,v}=^?0\}$。在本工作之前，该问题是否属于多项式层级${\sf PH}$尚未解决。我们在假设广义黎曼猜想（GRH）成立的条件下，证明$\{c^w_{u,v}=^?0\}$属于${\sf coAM}$。特别地，该消失问题属于${\Sigma_2^{{\text{p}}}}$。我们的方法基于构造提升公式，为该问题提供多项式方程组系统。该结果源于对参数化希尔伯特零点定理的归约，该定理近期在arXiv:2408.13027中得到了研究。我们应用该构造证明，在分别处理复数和实数的布卢姆-舒布-斯梅尔（BSS）计算模型中，消失问题属于${\sf NP}_{\mathbb{C}} \cap {\sf P}_{\mathbb{R}}$。类似地，我们证明舒伯特系数的计算属于$\#{\sf P}_{\mathbb{C}}$，即BSS模型的计数版本。我们还将结果推广到经典类型。除$D$型中的消失问题这一显著例外外，我们的所有结果均适用于所有类型。