General Theory of Algebraic Equations

Etienne Bézout

362 pages, parution le 01/05/2006

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Résumé

This book provides the first English translation of Bezout's masterpiece, the General Theory of Algebraic Equations. It follows, by almost two hundred years, the English translation of his famous mathematics textbooks. Here, Bézout presents his approach to solving systems of polynomial equations in several variables and in great detail. He introduces the revolutionary notion of the "polynomial multiplier," which greatly simplifies the problem of variable elimination by reducing it to a system of linear equations. The major result presented in this work, now known as "Bézout's theorem," is stated as follows: "The degree of the final equation resulting from an arbitrary number of complete equations containing the same number of unknowns and with arbitrary degrees is equal to the product of the exponents of the degrees of these equations."

The book offers large numbers of results and insights about conditions for polynomials to share a common factor, or to share a common root. It also provides a state-of-the-art analysis of the theories of integration and differentiation of functions in the late eighteenth century, as well as one of the first uses of determinants to solve systems of linear equations. Polynomial multiplier methods have become, today, one of the most promising approaches to solving complex systems of polynomial equations or inequalities, and this translation offers a valuable historic perspective on this active research field.

Sommaire

Introduction: Theory of differences and sums of quantities
Book One
Section I
- About complete polynomials and complete equations
- About the number of terms in complete polynomials
- Problem I: Compute the value of N(u . . . n)T About the number of terms of a complete polynomial that can be divided by certain monomials composed of one or more of the unknowns present in this polynomial
- Problem II
- ...
Section II
- About incomplete polynomials and first-order incomplete equations
- About incomplete polynomials and incomplete equations in which each unknown does not exceed a given degree for each unknown. And where the unknowns, combined two-by-two, three-by-three, four-by-four etc., all reach the total dimension of the polynomial or the equation
- Problem IV
- Problem V
- ...
Section III
- About incomplete polynomials and second-, third-, fourth-, etc. order incomplete equations
- About the number of terms in incomplete polynomials of arbitrary order
- Problem XXIV
- About the form of the polynomial multiplier and of the polynomials whose number of terms impact the degree of the final equation resulting from a given number of incomplete equations with arbitrary order
- Useful notions for the reduction of differentials that enter in the expression of the number of terms of a polynomial with arbitrary order
- ...
Book Two
- In which we give a process for reaching the final equation resulting from an arbitrary number of equations in the same number of unknowns, and in which we present many general properties of algebraic quantities and equations
- General observations
- A new elimination method for first-order equations with an arbitrary number of unknowns
- General rule to compute the values of the unknowns, altogether or separately, in first-order equations, whether these equations are symbolic or numerical
- A method to find functions of an arbitrary number of unknowns which are identically zero
- About the form of the polynomial multiplier, or the polynomial multipliers, leading to the final equation
- ...

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Caractéristiques techniques

	PAPIER
Éditeur(s)	Princeton University Press
Auteur(s)	Etienne Bézout
Parution	01/05/2006
Nb. de pages	362
Format	16 x 24
Couverture	Relié
Poids	650g
Intérieur	Noir et Blanc
EAN13	9780691114323
ISBN13	978-0-691-11432-3