A polynomial resultant approach to algebraic constructions of extremal graphs

The Tur & aacute;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 of the most important problems in extremal graph theory. However, the order of magnitude of ex(n, H) for bipartite graphs is known only in a handful of cases. In particular, giving explicit constructions of extremal graphs is very challenging in this field. In this paper, we develop a polynomial resultant approach to the algebraic construction of explicit extremal graphs, which can efficiently decide whether a specified structure exists. A key insight in our approach is the multipolynomial resultant, which is a fundamental tool of computational algebraic geometry. Our main results include the matched lower bounds on the Tur & aacute;n number of 1-subdivision of K3,t1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{3,t_{1}}$$\end{document} and the linear Tur & aacute;n number of the Berge theta hypergraph Theta 3,t2B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta_{3,t_{2}}<^>{B}$$\end{document}, where t1 = 25 and t2 = 217. Moreover, the constant t1 improves the random algebraic construction of Bukh and Conlon (2018) and makes the known estimation better on the smallest value of t1 concerning a problem posed by Conlon et al. (2021) by reducing the value from a magnitude of 1056 to the number 25, while the constant t2 improves a result of He and Tait (2019).