Vanishing of Schubert coefficients via the effective Hilbert nullstellensatz

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 ${{\mathsf {PH}}}$ . We prove this problem is in ${{\mathsf {AM}}} \cap {{\mathsf {coAM}}}$ assuming the Generalized Riemann Hypothesis ( $\mathrm{GRH}$ ), that is, relatively low in ${{\mathsf {PH}}}$ . 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.