Odd-order obstruction for scalar elliptic operators in at least three dimensions ($n\ge 3\Rightarrow N\text{ even}$)

The principal symbol of an odd-order scalar differential operator is an odd map from the frequency sphere to the complex plane. The Borsuk-Ulam theorem forces a zero on a two-dimensional sphere, contradicting ellipticity. Restriction to three variables proves the same obstruction in every greater dimension.