Values of ℓ ₀ Norms for Solutions of Linear systems Arising From Monomial Sphere Maps

We consider gaps that arise in the possible numbers of non-vanishing components for certain linear systems that arise when studying rational sphere maps. When the source dimension is [Formula: see text], we find an explicit formula for an integer [Formula: see text] such that every number exceeding [Formula: see text] is the value for the minimum embedding dimension of some polynomial (hence rational) sphere map. We conjecture, but do not prove, that [Formula: see text] is sharp. This number is asymptotic to [Formula: see text]; this asymptotic value improves what has been suggested to be quadratic in [Formula: see text]. Our methods are combinatorial.