How did Seki Takakazu find the real roots of higher-degree polynomial equations?

Seki Takakazu, the prominent figure of the wasan tradition, advanced the study of higher-degree polynomial equations by building upon ancient Chinese mathematical techniques. In his work Kaiindai no hō (1685; “Method for Solving Hidden Problems”), he described an ancient Chinese method for obtaining a root and successfully extended it to identify all the real roots of such equations.1

Seki’s approach involved a sophisticated classification of problems that accounted for the existence of both negative and multiple roots, a phenomenon previously noted by his contemporary Sawaguchi Kazuyuki. To facilitate these complex computations, Seki developed a "literal" written algebra using characters, which liberated mathematicians from the limitations of traditional counting rods. By substituting a tabular notational system for the cumbersome rod-based methods, he simplified the handling of equations involving multiple unknowns. These innovations allowed him and his disciples to refine 13th-century Chinese algorithms, ultimately enabling a more comprehensive and generalized approach to solving polynomial equations.21