# General Theory of Algebraic Equations

by Etienne BezoutISBN-10: 0691114323

ISBN-13: 9780691114323

Pub. Date: 03/13/2006

Publisher: Princeton University Press

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

## Overview

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.

## Product Details

- ISBN-13:
- 9780691114323
- Publisher:
- Princeton University Press
- Publication date:
- 03/13/2006
- Pages:
- 362
- Product dimensions:
- 6.30(w) x 9.30(h) x 1.20(d)

## Table of Contents

Translator's Foreword xi

Dedication from the 1779 edition xiii

Preface to the 1779 edition xv

Introduction: Theory of differences and sums of quantities 1

Definitions and preliminary notions 1

About the way to determine the differences of quantities 3

A general and fundamental remark 7

Reductions that may apply to the general rule to differentiate quantities when several differentiations must be made. 8

Remarks about the differences of decreasing quantities 9

About certain quantities that must be differentiated through a simpler process than that resulting from the general rule 10

About sums of quantities 10

About sums of quantities whose factors grow arithmetically 11

Remarks 11

About sums of rational quantities with no variable divider 12

Book One

Section I

About complete polynomials and complete equations 15

About the number of terms in complete polynomials 16

Problem I: Compute the value of N(u . . . n)^{T} 16

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 17

Problem II 17

Problem III 19

Remark 20

Initial considerations about computing the degree of the final equation resulting from an arbitrary number of complete equations with the same number of unknowns 21

Determination of the degree of the final equation resulting from an arbitrary number of complete equations containing the same number of unknowns 22

Remarks 24

Section II

About incomplete polynomials and first-order incomplete equations 26

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 28

Problem IV 28

Problem V 29

Problem VI 32

Problem VII: We ask for the degree of the final equation resulting from an arbitrary number n of equations of the form (u ^{a} . . . n)^{t} = 0 in the same number of unknowns 32

Remark 34

About the sum of some quantities necessary to determine the number of terms of various types of incomplete polynomials 35

Problem VIII 35

Problem IX 36

Problem X 36

Problem XI 37

About incomplete polynomials, and incomplete equations, in which two of the unknowns (the same in each polynomial or equation) share the following characteristics:

(1) The degree of each of these unknowns does not exceed a given number (different or the same for each unknown);

(2) These two unknowns, taken together, do not exceed a given dimension;

(3) The other unknowns do not exceed a given degree (different or the same for each), but, when combined groups of two or three among themselves as well as with the first two, they reach all possible dimensions until that of the polynomial or the equation 38

Problem XII 39

Problem XIII 40

Problem XIV 41

Problem XV 42

Problem XVI 42

About incomplete polynomials and equations, in which three of the unknowns satisfy the following characteristics:

(1) The degree of each unknown does not exceed a given value, different or the same for each;

(2) The combination of two unknowns does not exceed a given dimension, different or the same for each combination of two of these three unknowns;

(3) The combination of the three unknowns does not exceed a given dimension.

We further assume that the degrees of the n - 3 other unknowns do not exceed given values; we also assume that the combination of two, three, four, etc. of these variables among themselves or with the first three reaches all possible dimensions, up to the dimension of the polynomial 45

Problem XVII 46

Problem XVIII 47

Summary and table of the different values of the number of terms sought in the preceding polynomial and in related quantities 56

Problem XIX 61

Problem XX 62

Problem XXI 63

Problem XXII 63

About the largest number of terms that can be cancelled in a given polynomial by using a given number of equations, without introducing new terms 65

Determination of the symptoms indicating which value of the degree of the final equation must be chosen or rejected, among the different available expressions 69

Expansion of the various values of the degree of the final equation, resulting from the general expression found in (104), and expansion of the set of conditions that justify these values 70

Application of the preceding theory to equations in three unknowns 71

General considerations about the degree of the final equation, when considering the other incomplete equations similar to those considered up until now 85

Problem XXIII 86

General method to determine the degree of the final equation for all cases of equations of the form (u ^{a} . . . n)^{t} = 0 94

General considerations about the number of terms of other polynomials that are similar to those we have examined 101

Conclusion about first-order incomplete equations 112

Section III

About incomplete polynomials and second-, third-, fourth-, etc. order incomplete equations 115

About the number of terms in incomplete polynomials of arbitrary order 118

Problem XXIV 118

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 119

Useful notions for the reduction of differentials that enter in the expression of the number of terms of a polynomial with arbitrary order 121

Problem XXV 122

Table of all possible values of the degree of the final equations for all possible cases of incomplete, second-order equations in two unknowns 127

Conclusion about incomplete equations of arbitrary order 134

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 137

General observations 137

A new elimination method for first-order equations with an arbitrary number of unknowns 138

General rule to compute the values of the unknowns, altogether or separately, in first-order equations, whether these equations are symbolic or numerical 139

A method to find functions of an arbitrary number of unknowns which are identically zero 145

About the form of the polynomial multiplier, or the polynomial multipliers, leading to the final equation 151

About the requirement not to use all coefficients of the polynomial multipliers toward elimination 153

About the number of coefficients in each polynomial multiplier which are useful for the purpose of elimination 155

About the terms that may or must be excluded in each polynomial multiplier 156

About the best use that can be made of the coefficients of the terms that may be cancelled in each polynomial multiplier 158

Other applications of the methods presented in this book for the General Theory of Equations 160

Useful considerations to considerably shorten the computation of the coefficients useful for elimination. 163

Applications of previous considerations to different examples; interpretation and usage of various factors that are encountered in the computation of the coefficients in the final equation 174

General remarks about the symptoms indicating the possibility of lowering the degree of the final equation, and about the way to determine these symptoms 191

About means to considerably reduce the number of coefficients used for elimination. Resulting simplifications in the polynomial multipliers 196

More applications, etc. 205

About the care to be exercised when using simpler polynomial multipliers than their general form (231 and following), when dealing with incomplete equations 209

More applications, etc. 213

About equations where the number of unknowns is lower by one unit than the number of these equations. A fast process to find the final equation resulting from an arbitrary number of equations with the same number of unknowns 221

About polynomial multipliers that are appropriate for elimination using this second method 223

Details of the method 225

First general example 226

Second general example 228

Third general example 234

Fourth general example 237

Observation 241

Considerations about the factor in the final equation obtained by using the second method 251

About the means to recognize which coefficients in the proposed equations can appear in the factor of the apparent final equation 253

Determining the factor of the final equation: How to interpret its meaning 269

About the factor that arises when going from the general final equation to final equations of lower degrees 270

Determination of the factor mentioned above 274

About equations where the number of unknowns is less than the number of equations by two units 276

Form of the simplest polynomial multipliers used to reach the two condition equations resulting from n equations in n - 2 unknowns 278

About a much broader use of the arbitrary coefficients and their usefulness to reach the condition equations with lowest literal dimension 301

About systems of n equations in p unknowns, where p < n 307

When not all proposed equations are necessary to obtain the condition equation with lowest literal dimension 314

About the way to find, given a set of equations, whether some of them necessarily follow from the others 316

About equations that only partially follow from the others 318

Re exions on the successive elimination method 319

About equations whose form is arbitrary, regular or irregular. Determination of the degree of the final equation in all cases 320

Remark 327

Follow-up on the same subject 328

About equations whose number is smaller than the number of unknowns they contain. New observations about the factors of the final equation 333

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