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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: | 9781400826964 |
|---|---|
| Publisher: | Princeton University Press |
| Publication date: | 01/10/2009 |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 368 |
| File size: | 25 MB |
| Note: | This product may take a few minutes to download. |
About the Author
Read an Excerpt
General Theory of Algebraic Equations
By Etienne Bézout
Princeton University Press
Copyright © 2006 Princeton University PressAll right reserved.
ISBN: 978-0-691-11432-3
Introduction
Theory of differences and sums of quantitiesDefinitions and preliminary notions
(1.) A function of a given variable is defined as any arithmetic expression involving this variable, irrespective of how it appears in it.
Thus x, a + bx, [(c - 3d[x.sup.3] + f[x.sup.4]).sup.5], [(a + fxp + gxq).sup.r] etc. are functions of x.
Consider X an arbitrary function of x, and define X' as what becomes of X when x is replaced by x + k; then X' - X represents the variation of X when x increases by k. X' - X is called the 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 an arbitrary state, and the same quantity, considered in another arbitrary state.
We use the letter d to represent the difference of an arbitrary quantity or function. It will not be used for any other purpose to avoid any confusion. Thus, instead of X' - X, we write dX or d(X).
And toexpress, at the same time, the amount by which the quantity x varies, we thus write d(X) ... [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to express the difference of X when x varies by an amount of k.
We consider increasing quantities here; we will see later what happens when considering decreasing quantities.
Assume the function whose variation or difference under consideration is a function of several variables, x, y or z, whose respective variations are k, l, m; denoting this function by P, we write its difference as d(P) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which means the difference of P when x varies by an amount of k, y by an amount of l, and z by an amount of m.
Applying to X' - X the same ideas as above, assume that x is replaced by x + k' in X' - X. Then X' becomes X'", and X becomes X". Then (X'" - X"L) - (X' - X) is known as the second difference of X, because it is the difference between two successive differences of X.
The second difference will be denoted dd(X) ... [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which means the second difference of X, when x varies first by k and then by k'.
(2.) We will very soon give the rules to determine first differences. But we show right now that the second differences are determined by applying to first differences the same rules as those that generate them.
Indeed, the quantity (X'" - X") - (X' - X) can also be written as follows, (X'" - X') - (X" - X"). Since by assumption X" is what becomes of X' when substituting x + k' for x and, likewise, X" is what becomes of X, we therefore obtain X'" - X' = d(X') ... [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; so [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. However, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], therefore [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] or [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
That is, we must first compute [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]: We must first take the difference of x, when x varies by an amount of k; we then take the difference of the resulting expression, when x varies by an amount of k'.
(3.) The order of the variation of x (whether x varies by an amount of k in the first difference and k' in the second, or vice versa) makes no difference.
Indeed, (X'" - X") - (X' - X) contains [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; it also contains [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Therefore (X'" - X") - (X' - X) or [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. But, by definition, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Thus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], but we also just saw that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; thus
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Assume the function under consideration contains several variables x, y, z, etc., whose first variation is k, l, m, etc., respectively; we call the second difference of this function (whose name I assume to be P) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(4.) To have an idea of the third difference, imagine that x is replaced by x + k" in (X'" - X") - (X' - X). Then if [X.sup.VII], [X.sup.VI], [X.sup.V], [X.sup.IV] are what becomes of X'", X", X' and X with this substitution, the quantity (([X.sup.VII] - [X.sup.VI]) - ([X.sup.V] - [X.sup.IV])) - ((X'" - X") - (X' - X)) is what is called the third difference of X, because it is the difference of two second differences. If k, k', k" are the successive variations of x, the third difference is written [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It is easy to see how this extends to the definition of the fourth, fifth and further differences.
About the way to compute the differences of quantities
(5.) Once the algebraic expression of a quantity is given, it is very easy to compute its difference. For example, assume we want to compute the difference of [x.sup.3] when x varies by k; we just have to evaluate [(x + k).sup.3] and subtract [x.sup.3]. This difference is 3k[x.sup.2] + 3[k.sup.2]x + [k.sup.3]. Computing the difference of a quantity is known as differentiating this quantity.
(6.) The differentiation rules are simply the common rules provided by algebra to compute the power of a binomial expression. But to ease and speed up this computation, we give the following rule, already known for other purposes. It is known that the expansion of the binomial (x + k) to the mth power, is [x.sup.m] + m[x.sup.m-1] [k + m [m.sup.-1]/2 [x.sup.m-2] [k.sup.2] + m [m.sup.-1]/2 [m.sup.-2] [k.sup.3] + etc.
Paying attention to the rules by which those terms are derived from one another, we see that their construction can be performed by using the following rule:
Write on the first line [x.sup.m
Under this line, write m
Multiply by this exponent, and, diminishing the exponent of x by one unit, replace the factor x that currently misses by the factor k, and get in the second line m[x.sup.m-1] k
Under this line, write one half of the current exponent of x; that is, [m.sup.-1]/2
Multiply by the latter, and, diminishing the current exponent of x by one unit, replace the new missing x factor by a new k factor, and get in the third line [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Under this line, write the third of the current exponent of x; that is, [m.sup.-2]/3
Multiply by the latter, and, diminishing the x exponent by one unit, replace the x factor that is missing again by a new factor k, and get in the fourth line [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Keep multiplying according to the same process, successively by one fourth, one fifth, etc. of the exponent of x, and keep lowering the exponent of x by one unit. Replace the missing x factor by a k factor. Then the value of [(x + k).sup.m] is the sum of the first, second, third, fourth etc. lines, until the line where the exponent of x becomes 0 which is obvious by comparison with the first formula.
(7.) Therefore it is sufficient to omit the first line in the result from the preceding rule to obtain the difference of [x.sup.m] where x varies by an amount of k, that is, to obtain the value of [(x + k).sup.m] - [x.sup.m].
(8.) Since the polynomial A[x.sup.p] + B[x.sup.q] + C[x.sup.r] only consists of terms of the form [x.sup.m], computing the difference of such a polynomial can be done by simply applying the rule above given for [x.sup.m].
Thus, to obtain the difference of [x.sup.3] - 5[x.sup.2] + 3x - 6, where x varies by an amount of k, I write as follows:
First line [x.sup.3] - 5[x.sup.2] + 3x - 6 Exponent of x 3 2 1 0
Second line 3[x.sup.2]k - 10xk + 3k Half of exponents of x 2/2 1/2 0/2
Third line 3x[k.sup.2] - 5[k.sup.2] Third of exponents of x 1/3 0/3
Fourth line [k.sup.3]
Thus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which is the sum of lines 2, 3 and 4.
(9.) We can use the same rule to differentiate quantities involving several variables. Thus, we can compute [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] using the method below, by writing successively under each variable its exponent, then one half of its exponent, one third of it, etc. of its according to the line number being computed.
First line [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Second line [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Third line [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] or [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Fourth line [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] or [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Fifth line [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] or [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Sixth line [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] or [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Thus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
(10.) The same rule applies to functions of two variables: Simply compare the result of [(x + k).sup.m] × [(y + l).sup.n] found with this rule, with the result of the expansion of this quantity using ordinary rules of algebra. These indeed lead to
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
By applying our rule, we find as follows:
First line. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Second line. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Third line. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] or [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Fourth line [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] or [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], etc.
We therefore see that the sum of the first, second, third and fourth lines gives exactly the same result.
(11.) We can use the same method to show that the same rule can be applied to an arbitrary number of variables.
We have demonstrated in (2) that it is enough to apply the same rules to first differences to obtain second differences, and that this also holds true for third, fourth, etc. differences; thus the method to compute arbitrary differences reduces to the only rule given in (4). Consider for example the computation of second differences: We want to compute the value of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] I write as follows:
First line [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Second line [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Third line [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] or [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Fourth line [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] or [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Therefore
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Second line [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Third line [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] or [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Therefore
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
A general and fundamental remark
(12.) Whatever the number of variables entering in the quantity to be differentiated, and whatever the dimension these variables can reach, either alone or together, we can generally observe that:
1. If T is the highest dimension reached by these variables, either alone or together, then T - 1 is the highest dimension these variables reach in the first difference, since the rule prescribes to reduce the exponent of the variable of interest by one unit.
Consequently, T - 2 is the highest degree of the variables in the second difference; T - 3 is the highest degree of the variables in the third difference; and in general, T - n is the highest dimension of the variables in the difference of order n. Thus, if the order of the difference has the same exponent as that of the highest dimension of the variables, the degree of the variables in the difference is zero; that is, the difference contains no more variables and is only a function of their respective variations.
For example, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; we see that x and y do not enter in the difference, but their respective variations k and l do.
Likewise, the above rule yields
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
where we see that x and y have vanished and only their respective variations k, k' and [l, l' remain.
2. If there are constant quantities in the function to be differentiated, that is, if there are terms where no variables are present, these terms will not be found in the first derivative, and therefore not in the subsequent differentials either; indeed, the rule prescribes to multiply them by the exponent of the variable, which is zero in that case.
3. The terms where the variables do not exceed, either together or separately, the first dimension, are not to be found in the second difference, since they all become constant by the process of the first differentiation; consequently they will disappear in the second differentiation. For example, assume we must differentiate the quantity a[x.sup.2] + bxy + c[y.sup.2] + ex + fy + g twice; the quantity g is not present in the first difference, which is 2axk + byk + bxl + 2cyl + ek + fl + a[k.sup.2] + bkl + [cl.sup.2]. Likewise, the terms ex and fy do not appear in the second difference, which is 2akk' + bkl' + bk'l + 2cll'. Indeed, during the first differentiation, these terms become ck and fl. Since these terms are constant, they cannot be found in the following difference.
Likewise, the terms where the variables do not exceed, either together or separately, the dimension 2 do not appear in the third difference; in general, the terms where the degree of the variables does not exceed, either together or separately, the dimension n - 1, disappear in the difference of order n.
(Continues...)
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Table of Contents
Translator's Foreword xiDedication 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