  Recently, Berge theta hypergraphs have received special attention due to the
similarity with Berge even cycles. Let $r$-uniform Berge theta hypergraph
$\Theta_{\ell,t}^{B}$ be the $r$-uniform hypergraph consisting of $t$
internally disjoint Berge paths of length $\ell$ with the same pair of
endpoints. In this work, we determine the Tur\'{a}n number of $3$-uniform Berge
theta hypergraph when $\ell=3$ and $t$ is relatively small. More precisely, we
provide an explicit construction giving \begin{align*}
\textup{ex}_{3}(n,\Theta_{3,217}^{B})=\Omega(n^{\frac{4}{3}}). \end{align*}
This matches an earlier upper bound by He and Tait up to an absolute constant
factor. The construction is algebraic, which is based on some equations over
finite fields, and the parameter $t$ in our construction is much smaller than
that in random algebraic construction. Our main technique is using the
resultant of polynomials, which appears to be a powerful technique to eliminate
variables.
