You are previewing General Theory of Algebraic Equations.
O'Reilly logo
General Theory of Algebraic Equations

Book Description

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.

Table of Contents

  1. Cover
  2. Title
  3. Contents
  4. Translator’s Foreword
  5. Dedication from the 1779 edition
  6. Preface to the 1779 edition
  7. Introduction Theory of differences and sums of quantities
  8. Definitions and preliminary notions
  9. About the way to determine the differences of quantities
  10. A general and fundamental remark
  11. Reductions that may apply to the general rule to differentiate quantities when several differentiations must be made.
  12. Remarks about the differences of decreasing quantities
  13. About certain quantities that must be differentiated through a simpler process than that resulting from the general rule
  14. About sums of quantities
  15. About sums of quantities whose factors grow arithmetically
  16. Remarks
  17. About sums of rational quantities with no variable divider
  18. Book One
  19. Section I
  20. About complete polynomials and complete equations
  21. About the number of terms in complete polynomials
  22. Problem I: Compute the value of N(u . . . n)T
  23. 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
  24. Problem II
  25. Problem III
  26. Remark
  27. Initial considerations about computing the degree of the final equation resulting from an arbitrary number of complete equations with the same number of unknowns
  28. Determination of the degree of the final equation resulting from an arbitrary number of complete equations containing
  29. Remarks
  30. Section II
  31. About incomplete polynomials and first-order incomplete equations
  32. 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
  33. Problem IV
  34. Problem V
  35. Problem VI
  36. Problem VII:
  37. Remark
  38. About the sum of some quantities necessary to determine the number of terms of various types of incomplete polynomials
  39. Problem VIII
  40. Problem IX
  41. Problem X
  42. Problem XI
  43. 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
  44. Problem XII
  45. Problem XIII
  46. Problem XIV
  47. Problem XV
  48. Problem XVI
  49. 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
  50. Problem XVII
  51. Problem XVIII
  52. Summary and table of the different values of the number of terms sought in the preceding polynomial and in related quantities
  53. Problem XIX
  54. Problem XX
  55. Problem XXII
  56. 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
  57. Determination of the symptoms indicating which value of the degree of the final equation must be chosen or rejected, among the different available expressions
  58. 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
  59. Application of the preceding theory to equations in three unknowns
  60. General considerations about the degree of the final equation, when considering the other incomplete equations similar to those considered up until now
  61. Problem XXIII
  62. General method to determine the degree of the final equation for all cases of equations of the form (ua . . . n)t = 0
  63. General considerations about the number of terms of other polynomials that are similar to those we have examined
  64. Conclusion about first-order incomplete equations
  65. Section III
  66. About incomplete polynomials and second-, third-, fourth-, etc. order incomplete equations
  67. About the number of terms in incomplete polynomials of arbitrary order
  68. Problem XXIV
  69. 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
  70. Useful notions for the reduction of differentials that enter in the expression of the number of terms of a polynomial with arbitrary order
  71. Problem XXV
  72. Table of all possible values of the degree of the final equations for all possible cases of incomplete, second-order equations in two unknowns
  73. Conclusion about incomplete equations of arbitrary order
  74. Book Two
  75. 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
  76. General observations
  77. A new elimination method for first-order equations with an arbitrary number of unknowns
  78. General rule to compute the values of the unknowns, altogether or separately, in first-order equations, whether these equations are symbolic or numerical
  79. A method to find functions of an arbitrary number of unknowns which are identically zero
  80. About the form of the polynomial multiplier, or the polynomial multipliers, leading to the final equation
  81. About the requirement not to use all coefficients of the polynomial multipliers toward elimination
  82. About the number of coefficients in each polynomial multiplier which are useful for the purpose of elimination
  83. About the terms that may or must be excluded in each polynomial multiplier
  84. About the best use that can be made of the coefficients of the terms that may be cancelled in each polynomial multiplier
  85. Useful considerations to considerably shorten the computation of the coefficients useful for elimination.
  86. 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
  87. General remarks about the symptoms indicating the possibility of lowering the degree of the final equation, and about the way to determine these symptoms
  88. About means to considerably reduce the number of coefficients used for elimination. Resulting simplifications in the polynomial multipliers
  89. More applications, etc.
  90. About the care to be exercised when using simpler polynomial multipliers than their general form (231 and following), when dealing with incomplete equations
  91. More applications, etc.
  92. 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
  93. About polynomial multipliers that are appropriate for elimination using this second method
  94. Details of the method
  95. First general example
  96. Second general example
  97. Third general example
  98. Fourth general example
  99. Observation
  100. Considerations about the factor in the final equation obtained by using the second method
  101. About the means to recognize which coefficients in the proposed equations can appear in the factor of the apparent final equation
  102. Determining the factor of the final equation: How to interpret its meaning
  103. About the factor that arises when going from the general final equation to final equations of lower degrees
  104. Determination of the factor mentioned above
  105. About equations where the number of unknowns is less than the number of equations by two units
  106. Form of the simplest polynomial multipliers used to reach the two condition equations resulting from n equations in n – 2 unknowns
  107. About a much broader use of the arbitrary coefficients and their usefulness to reach the condition equations with lowest literal dimension
  108. About systems of n equations in p unknowns, where p < n
  109. When not all proposed equations are necessary to obtain the condition equation with lowest literal dimension
  110. About the way to find, given a set of equations, whether some of them necessarily follow from the others
  111. About equations that only partially follow from the others
  112. Reflexions on the successive elimination method
  113. About equations whose form is arbitrary, regular or irregular. Determination of the degree of the final equation in all cases
  114. Remark
  115. Follow-up on the same subject
  116. About equations whose number is smaller than the number of unknowns they contain. New observations about the factors of the final equation