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In a system of interacting f=1 bosons (in the subspace where the total spinin the z-direction is vanishing), we prove inequalities for the ground stateexpectation value of the density of spin-0 bosons. The inequalities imply thatthe ground state possesses "polar" or "antiferromagnetic" order when thequadratic Zeeman term q is large enough. In the low density limit, theinequalities establish the existence of a sharp transition at q=0 when q isvaried.

"Polar" and "antiferromagnetic" order in f=1 many-boson systems