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In $PG(3,q^2)$, with $q$ odd, we determine the possible intersection sizes ofa Hermitian surface $\mathcal{H}$ and an irreducible quadric $\mathcal{Q}$having the same tangent plane $\pi$ at a common point $P\in{\mathcalQ}\cap{\mathcal H}$.

Intersections of the Hermitian surface with irreducible quadrics in $PG(3,q^2)$, $q$ odd