General Theory of Algebraic Equations

General Theory of Algebraic Equations

Etienne Bézout
Translated by Eric Feron
Copyright Date: 2006
Published by: Princeton University Press
Pages: 362
https://www.jstor.org/stable/j.ctt7rzdn
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  • Book Info
    General Theory of Algebraic Equations
    Book Description:

    This book provides the first English translation of Bezout's masterpiece, theGeneral 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.

    eISBN: 978-1-4008-2696-4
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-iv)
  2. Table of Contents
    (pp. v-x)
  3. Translator’s Foreword
    (pp. xi-xii)

    This document is a literal translation of Bézout’s seminal work on the theory of algebraic equations in several unknowns. The notation of this translation strictly follows that of the original manuscript. Bézout’s purpose was to provide an in-depth analysis of systems of algebraic equations. His main push was devoted to determining the degree of the final equation in one unknown, resulting from the original set of polynomial equations.

    Translating Bézout’s research centerpiece became necessary to me after attending an illuminating presentation made by Pablo Parrilo at MIT sometime around 2002. His presentation was devoted to polynomially constrained polynomial optimization via...

  4. Dedication from the 1779 edition
    (pp. xiii-xiv)
  5. Preface to the 1779 edition
    (pp. xv-xxiv)
  6. Introduction Theory of differences and sums of quantities
    (pp. 1-14)

    (1.) A function of a given variable is defined as any arithmetic expression involving this variable, irrespective of how it appears in it.

    Thusx,a+bx, (c- 3dx3+fx4)5, (a+fxp+gxq)retc. are functions ofx.

    ConsiderXan arbitrary function ofx, and defineX´ as what becomes ofXwhenxis replaced byx+k; thenX´-Xrepresents the variation ofXwhenxincreases byk.X´-Xis calledthe difference of X. Thus, although strictly speaking, one may not talk about the difference of one quantity, we will adopt this commonly used expression; it means the difference between this quantity, considered in...

  7. Book One
    • SECTION I
      (pp. 15-25)

      (29.) Any polynomial that contains only one unknownxcan be represented in general byaxT+bxT−1+cxT−2· · · +s, whereTis the highest degree ofxanda,b,c, etc. are arbitrary coefficients.

      Likewise, any equation in one unknown can generally be represented byaxT+bxT−1+cxT−2+ · · · +s= 0.

      However, the large number of terms that can enter polynomials and equations, as their degree and the number of unknowns increases, requires us to represent these as compactly as possible. We first present the various notations that we propose to use.

      (30.) We represent any polynomial in one unknown by the abbreviated expression (x)T, which means...

    • SECTION II
      (pp. 26-114)

      (49.) We will not insist on the broad impact of the general theorem (47) about complete equations. We will simply remark that it not gives only a precise expression for the degree of the final equation resulting from an arbitrary number of complete equations with all possible terms and coefficients being present, but also an upper bound on the final degree of any equation, complete or incomplete, which may or may not be reducible due to either the absence of some terms or existing relations among their coefficients.

      (50.) As useful as this upper bound may already be, it is...

    • SECTION III
      (pp. 115-136)

      (165.) No matter how general the polynomials we have dealt with in the previous section are, they do not yet cover all possible polynomials and all possible equations. Their form is not as general as necessary for us to claim that we can determine the lowest degree of the final equation for any algebraic equation.

      It is not enough to specify the highest dimensions reached by the unknowns, taken either alone or in combinations of 2, 3, etc., to capture all the variations that may influence the degree of the final equation. While these have a very significant influence on...

  8. Book Two
    (pp. 137-337)

    (194.) The method expressed in the first book for determining the degree of the final equation strongly indicates that the art of eliminating all variables but one reduces altogether to the elimination method in first-order equations, with an arbitrary number of unknowns. In appearance, it would seem that little remains to be said about this matter, since we know of methods that lead us quickly to the value of all unknowns in first-order equations. But even if these methods had all the perfection we propose to introduce, we would, by leaving our investigations here, leave aside more than one important...