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 { {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.