General theory of algebraic equations

362 Pages·2006·1.378 MB·English

by  Bezout E.

General Theory of
Algebraic Equations

General Theory of
Algebraic Equations
Etienne B´ezout
Translated by Eric Feron
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD

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
vi CONTENTS
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 (ua...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
CONTENTS vii
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 (ua...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
viii CONTENTS
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
CONTENTS ix
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