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The Tur\'{a}n problem asks for the largest number of edges ex$(n,H)$ in an$n$-vertex graph not containing a fixed forbidden subgraph $H$, which is one ofthe most important problems in extremal graph theory. However the order ofmagnitude of ex$(n,H)$ for bipartite graphs is known only in a handful ofcases. In particular, giving explicit constructions of extremal graphs is verychallenging in this field. In this paper, we develop a polynomail resultantapproach to algebraic construction of explicit extremal graphs, which canefficiently decide whether a specified structure exists. A key insight in ourapproach is the multipolynomial resultant, which is a fundamental tool ofcomputational algebraic geometry. Our main results include the matched lowersbounds for Tur\'{a}n number of $1$-subdivision of $K_{3,t_{1}}$ and linearTur\'{a}n number of Berge theta hyerpgraph $\Theta_{3,t_{2}}^{B}$ with$t_{1}=25$ and $t_{2}=217$. Moreover, the constant $t_{1}$ improves the randomalgebraic construction of Bukh and Conlon~[Rational exponents in extremal graphtheory, J. Eur. Math. Soc. 20 (2018), 1747-1757] and makes progress on theknown estimation for the smallest value of $t_{1}$ concerning a problem posedby Conlon, Janzer and Lee ~[More on the extremal number of subdivisions,Combinatorica, to appear], while the constant $t_{2}$ improves a result of Heand Tait~[Hypergraphs with few berge paths of fixed length between vertices,SIAM J. Discrete Math., 33(3), 1472-1481].

A polynomial resultant approach to algebraic construction of extremal graphs