Vanishing of Schubert coefficients via the effective Hilbert nullstellensatz

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.