General Theory of Algebraic Equations

Etienne Bezout (1730-1783) is credited with the invention of the determinant (named Bezoutian by Sylvester) as well as several key innovations to solve simultaneous polynomial equations in many unknowns. By the time of his death, he was a member of the French Academy of Sciences and the Examiner of the Guards of the Navy and of the Corps of Artillery. Eric Feron Dutton/Ducoffe Professor of Aerospace Engineering at Georgia Institute of Technology, and Visiting Professor of Aerospace Engineering at Massachusetts Institute of Technology, where he is affiliated with the Laboratory for Information and Decision Systems and the Operations Research Center. He is also an Adviser to the French Academy of Technologies. His interests span numerical analysis, optimization, systems analysis, and their applications to aerospace engineering.

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