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General Theory of Algebraic Equations - Etienne Bezout

General Theory of Algebraic Equations

By: Etienne Bezout, Eric Feron (Translator)

Hardcover

Published: 2nd April 2006
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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, Bezout 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 "Bezout'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.

"This is not a book to be taken to the office, but to be left at home, and to be read on weekend, as a romance. We already know the plot, but here we meet all the characters, major and minor."--Cicero Fernandes de Carvalho, Mathematical Reviews "Bezout's classic General Theory of Algebraic Equations is ... an immortal evergreen of astonishing actual relevance... [I]ts first English translation is utmost welcome and appropriate, and a great gain for the international mathematical community, both today and in the future."--Werner Kleinert, Zentralblatt MATH

Translator's Forewordp. xi
Dedication from the 1779 editionp. xiii
Preface to the 1779 editionp. xv
Introduction: Theory of differences and sums of quantitiesp. 1
Definitions and preliminary notionsp. 1
About the way to determine the differences of quantitiesp. 3
A general and fundamental remarkp. 7
Reductions that may apply to the general rule to differentiate quantities when several differentiations must be madep. 8
Remarks about the differences of decreasing quantitiesp. 9
About certain quantities that must be differentiated through a simpler process than that resulting from the general rulep. 10
About sums of quantitiesp. 10
About sums of quantities whose factors grow arithmeticallyp. 11
Remarksp. 11
About sums of rational quantities with no variable dividerp. 12
About complete polynomials and complete equationsp. 15
About the number of terms in complete polynomialsp. 16
Compute the value of N(u...n)[superscript T]p. 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 polynomialp. 17
p. 17
p. 19
Remarkp. 20
Initial considerations about computing the degree of the final equation resulting from an arbitrary number of complete equations with the same number of unknownsp. 21
Determination of the degree of the final equation resulting from an arbitrary number of complete equations containing the same number of unknownsp. 22
Remarksp. 24
About incomplete polynomials and first-order incomplete equationsp. 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 equationp. 28
p. 28
p. 29
p. 32
We ask for the degree of the final equation resulting from an arbitrary number n of equations of the form (u[superscript a]...n)[superscript t] = 0 in the same number of unknownsp. 32
Remarkp. 34
About the sum of some quantities necessary to determine the number of terms of various types of incomplete polynomialsp. 35
p. 35
p. 36
p. 36
p. 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 equationp. 38
p. 39
p. 40
p. 41
p. 42
Problem XVIp. 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 polynomialp. 45
p. 46
p. 47
Summary and table of the different values of the number of terms sought in the preceding polynomial and in related quantitiesp. 56
p. 61
p. 62
p. 63
p. 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 termsp. 65
Determination of the symptoms indicating which value of the degree of the final equation must be chosen or rejected, among the different available expressionsp. 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 valuesp. 70
Application of the preceding theory to equations in three unknownsp. 71
General considerations about the degree of the final equation, when considering the other incomplete equations similar to those considered up until nowp. 85
p. 86
General method to determine the degree of the final equation for all cases of equations of the form (u[superscript a]...n)[superscript t] = 0p. 94
General considerations about the number of terms of other polynomials that are similar to those we have examinedp. 101
Conclusion about first-order incomplete equationsp. 112
About incomplete polynomials and second-, third-, fourth-, etc. order incomplete equationsp. 115
About the number of terms in incomplete polynomials of arbitrary orderp. 118
p. 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 orderp. 119
Useful notions for the reduction of differentials that enter in the expression of the number of terms of a polynomial with arbitrary orderp. 121
p. 122
Table of all possible values of the degree of the final equations for all possible cases of incomplete, second-order equations in two unknownsp. 127
Conclusion about incomplete equations of arbitrary orderp. 134
p. 137
General observationsp. 137
A new elimination method for first-order equations with an arbitrary number of unknownsp. 138
General rule to compute the values of the unknowns, altogether or separately, in first-order equations, whether these equations are symbolic or numericalp. 139
A method to find functions of an arbitrary number of unknowns which are identically zerop. 145
About the form of the polynomial multiplier, or the polynomial multipliers, leading to the final equationp. 151
About the requirement not to use all coefficients of the polynomial multipliers toward eliminationp. 153
About the number of coefficients in each polynomial multiplier which are useful for the purpose of eliminationp. 155
About the terms that may or must be excluded in each polynomial multiplierp. 156
About the best use that can be made of the coefficients of the terms that may be cancelled in each polynomial multiplierp. 158
Other applications of the methods presented in this book for the General Theory of Equationsp. 160
Useful considerations to considerably shorten the computation of the coefficients useful for eliminationp. 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 equationp. 174
General remarks about the symptoms indicating the possibility of lowering the degree of the final equation, and about the way to determine these symptomsp. 191
About means to considerably reduce the number of coefficients used for elimination. Resulting simplifications in the polynomial multipliersp. 196
More applications, etc.p. 205
About the care to be exercised when using simpler polynomial multipliers than their general form (231 and following), when dealing with incomplete equationsp. 209
More applications, etc.p. 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 unknownsp. 221
About polynomial multipliers that are appropriate for elimination using this second methodp. 223
Details of the methodp. 225
First general examplep. 226
Second general examplep. 228
Third general examplep. 234
Fourth general examplep. 237
Observationp. 241
Considerations about the factor in the final equation obtained by using the second methodp. 251
About the means to recognize which coefficients in the proposed equations can appear in the factor of the appearent final equationp. 253
Determining the factor of the final equation: How to interpret its meaningp. 269
About the factor that arises when going from the general final equation to final equations of lower degreesp. 270
Determination of the factor mentioned abovep. 274
About equations where the number of unknowns is less than the number of equations by two unitsp. 276
Form of the simplest polynomial multipliers used to reach the two condition equations resulting from n equations in n - 2 unknownsp. 278
About a much broader use of the arbitrary coefficients and their usefulness to reach the condition equations with lowest literal dimensionp. 301
About systems of n equations in p unknowns, where p < np. 307
When not all proposed equations are necessary to obtain the condition equation with lowest literal dimensionp. 314
About the way to find, given a set of equations, whether some of them necessarily follow from the othersp. 316
About equations that only partially follow from the othersp. 318
Reflexions on the successive elimination methodp. 319
About equations whose form is arbitrary, regular or irregular. Determination of the degree of the final equation in all casesp. 320
Remarkp. 327
Follow-up on the same subjectp. 328
About equations whose number is smaller than the number of unknowns they contain. New observations about the factors of the final equationp. 333
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780691114323
ISBN-10: 0691114323
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 368
Published: 2nd April 2006
Publisher: Princeton University Press
Country of Publication: US
Dimensions (cm): 23.5 x 15.2  x 3.81
Weight (kg): 0.62