\'Etienne B\'ezout : Analyse alg\'ebrique au si\`ecle des Lumi\`eres

TL;DR

This paper explores Étienne Bézout's original algebraic analysis methods, highlighting his innovative approaches to elimination theory, indeterminate coefficients, and resultants, as they were understood in his time.

Contribution

It presents a historical and mathematical analysis of Bézout's original techniques, emphasizing his novel viewpoints and methods in algebraic elimination.

Findings

Bézout's reduction of elimination to linear systems

Use of indeterminate coefficients for existence and count

Personal method for deriving two equations resultant

Abstract

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.