On Vanishing of Gromov--Witten Invariants
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912547476602880 |
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| author | Pak, Igor Robichaux, Colleen Xu, Weihong |
| author_facet | Pak, Igor Robichaux, Colleen Xu, Weihong |
| contents | We consider the decision problem of whether a particular Gromov--Witten invariant on a partial flag variety is zero. We prove that for the $3$-pointed, genus zero invariants, this problem is in the complexity class ${\sf AM}$ assuming the Generalized Riemann Hypothesis (GRH), and therefore lies in the second level of polynomial hierarchy ${\sf PH}$.
For the proof, we construct an explicit system of polynomial equations through a translation of the defining equations. We also need to prove an extension of the Parametric Hilbert's Nullstellensatz to obtain our central reduction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_15715 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Vanishing of Gromov--Witten Invariants Pak, Igor Robichaux, Colleen Xu, Weihong Algebraic Geometry Discrete Mathematics Combinatorics Primary: 14N35, Secondary: 05E14, 14M15, 68Q17, 68Q25 We consider the decision problem of whether a particular Gromov--Witten invariant on a partial flag variety is zero. We prove that for the $3$-pointed, genus zero invariants, this problem is in the complexity class ${\sf AM}$ assuming the Generalized Riemann Hypothesis (GRH), and therefore lies in the second level of polynomial hierarchy ${\sf PH}$. For the proof, we construct an explicit system of polynomial equations through a translation of the defining equations. We also need to prove an extension of the Parametric Hilbert's Nullstellensatz to obtain our central reduction. |
| title | On Vanishing of Gromov--Witten Invariants |
| topic | Algebraic Geometry Discrete Mathematics Combinatorics Primary: 14N35, Secondary: 05E14, 14M15, 68Q17, 68Q25 |
| url | https://arxiv.org/abs/2508.15715 |