Etienne Bezout : Analyse algebrique au si`ecle des Lumi`eres

The topic of this paper is, on the one hand to introduce algebraic analysis results of 'Etienne B'ezout (1730- 1783) not as we know them today but as he found them in his time, and on the other hand to emphasize his innovating viewpoints. We will be concerned with Bezout special way of reducing elimination for any degree systems to finding conditions for linear systems solutions, with his typical use of indeterminate coefficients that he doesn’t compute but looks only for existence and number, with his idea to work on set of polynomials products sums, and with a very personal method to found two equations resultant.

💡 Research Summary

This paper revisits the algebraic analysis of Étienne Bézout (1730‑1783) by presenting his results in the form he originally discovered them, and by highlighting the innovative aspects of his methodology. Bézout’s central contribution was to reduce the elimination problem for systems of arbitrary degree to the study of linear systems, thereby turning the existence of solutions into a problem of linear algebra. He employed indeterminate coefficients not to compute explicit values but solely to ascertain the existence and the number of possible coefficients, a technique that anticipates modern determinant and resultant theory. By considering the whole set of sums of products of polynomials, Bézout was able to derive conditions under which a general polynomial of degree n could be transformed into a binomial form (xⁿ + h yⁿ = 0). This led to his own method for constructing the resultant of two equations, a precursor of what later became known as the Bézout matrix and Sylvester’s resultant matrix. The paper traces Bézout’s intellectual trajectory through his 1762, 1764, and 1765 memoirs, his 1766 textbook Algèbre, and his 1779 treatise Théorie générale des équations algébriques. It shows how he critically engaged with the work of Euler and Lagrange, extending Euler’s conjectures while imposing stricter coefficient conditions, and how his approach differed from the prevailing methods of Descartes and d’Alembert. Although Bézout could not achieve a universal solution for all degrees, he succeeded in solving large families of equations, especially those lacking the second‑highest term, and he provided explicit inductive constructions for degrees up to eight. The analysis emphasizes that Bézout’s use of indeterminate coefficients evolved from a computational tool to a purely existential argument, reducing computational load and foreshadowing modern algebraic geometry’s focus on parameter spaces and moduli. Finally, the paper situates Bézout’s work within the broader context of 18th‑century French mathematics, noting his dual role as a military educator and a researcher, and argues that his methodological innovations laid foundational ideas for later developments in algebraic elimination, resultants, and computational algebra systems.