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Recently, Berge theta hypergraphs have received special attention due to thesimilarity 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 ofendpoints. In this work, we determine the Tur\'{a}n number of $3$-uniform Bergetheta hypergraph when $\ell=3$ and $t$ is relatively small. More precisely, weprovide 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 constantfactor. The construction is algebraic, which is based on some equations overfinite fields, and the parameter $t$ in our construction is much smaller thanthat in random algebraic construction. Our main technique is using theresultant of polynomials, which appears to be a powerful technique to eliminatevariables.

3-uniform hypergraphs with few Berge paths of length three between any two vertices