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October 2, 2025Open Access

Vanishing of Schubert coefficients via the effective Hilbert nullstellensatz

Key Points

The study proves that deciding Schubert coefficients is in AM ∩ coAM under assumptions of the Generalized Riemann Hypothesis.
Key metrics include classification of Schubert Vanishing within the polynomial hierarchy, showing it is relatively low in PH.
Our method employs Purbhoo’s criterion to develop explicit polynomial systems, leading to significant findings in complexity.
Finding indicates potential extensions to all classical types, enhancing understanding of Schubert coefficients' behaviors.

Abstract

Abstract Schubert Vanishing is a problem of deciding whether Schubert coefficients are zero. Until this work it was open whether this problem is in the polynomial hierarchy { {PH}}. We prove this problem is in { {AM}} { {coAM}} assuming the Generalized Riemann Hypothesis (GRH), that is, relatively low in { {PH}}. Our approach uses Purbhoo’s criterion 57 to construct explicit polynomial systems for the problem. The result follows from a reduction to Parametric Hilbert’s Nullstellensatz, recently analyzed in 2. We extend our results to all classical types.

Vanishing of Schubert coefficients via the effective Hilbert nullstellensatz

Key Points

Abstract

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