[4] He created a new algebraic notation system and, motivated by astronomical computations, did work on infinitesimal calculus and Diophantine equations.

His successors later developed a school dominant in Japanese mathematics until the end of the Edo period.

While it is not clear how much of the achievements of wasan are Seki's, since many of them appear only in writings of his pupils, some of the results parallel or anticipate those discovered in Europe.

While in Ko-shu han, he was involved in a surveying project to produce a reliable map of his employer's land.

[8] The material in these works consisted of algebra with numerical methods, polynomial interpolation and its applications, and indeterminate integer equations.

Accordingly, a target of Seki and his contemporary Japanese mathematicians was the development of general multivariable algebraic equations and elimination theory.

In the Chinese approach to polynomial interpolation, the motivation was to predict the motion of celestial bodies from observed data.

In 1671, Sawaguchi Kazuyuki [ja] (沢口 一之), a pupil of Hashimoto Masakazu (橋本 正数) in Osaka, published Kokon Sanpō Ki (古今算法記), in which he gave the first comprehensive account of Chinese algebra in Japan.

In the end of the book, he challenged other mathematicians with 15 new problems, which require multi-variable algebraic equations.

In his book of 1674, however, Seki gave only single-variable equations resulting from elimination, but no account of the process at all, nor his new system of algebraic symbols.

In 1678, Tanaka Yoshizane (田中 由真), who was from Hashimoto's school and was active in Kyoto, authored Sanpō Meiki (算法明記), and gave new solutions to Sawaguchi's 15 problems, using his version of multivariable algebra, similar to Seki's.

To answer criticism, in 1685, Takebe Katahiro (建部 賢弘), one of Seki's pupils, published Hatsubi Sanpō Genkai (発微算法諺解), notes on Hatsubi Sanpō, in which he showed in detail the process of elimination using algebraic symbols.

In 1683, Seki pushed ahead with elimination theory, based on resultants, in the Kaifukudai no Hō (解伏題之法).

In 1690, Izeki Tomotoki (井関 知辰), a mathematician active in Osaka but not in Hashimoto's school, published Sanpō Hakki (算法発揮), in which he gave resultant and Laplace's formula of determinant for the n×n case.

Seki developed his mathematics in competition with mathematicians in Osaka and Kyoto, at the cultural center of Japan.

In comparison with European mathematics, Seki's first manuscript was as early as Leibniz's first commentary on the subject, which treated matrices only up to the 3x3 case.

With elimination theory in hand, a large part of the problems treated in Seki's time became solvable in principle, given the Chinese tradition of geometry almost reduced to algebra.

He also suggested an improvement to Horner's method: to omit higher order terms after some iterations.

In a statistical overview derived from writings by and about Seki Takakazu, OCLC/WorldCat encompasses roughly 50+ works in 50+ publications in three languages and 100+ library holdings.