General theory of algebraic equations:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English French |
| Veröffentlicht: |
Princeton, NJ
Princeton Univ. Press
2006
|
| Schlagworte: | |
| Online-Zugang: | Contributor biographical information Publisher description Table of contents Inhaltsverzeichnis |
| Beschreibung: | XXIV, 337 S. |
| ISBN: | 0691114323 9780691114323 |
Internformat
MARC
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| 240 | 1 | 0 | |a Théorie générale des équations algébriques |
| 245 | 1 | 0 | |a General theory of algebraic equations |c Etienne Bézout ; translated by Eric Feron |
| 264 | 1 | |a Princeton, NJ |b Princeton Univ. Press |c 2006 | |
| 300 | |a XXIV, 337 S. | ||
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| 650 | 4 | |a Équations, Théorie des | |
| 650 | 4 | |a Equations, Theory of | |
| 650 | 0 | 7 | |a Algebraische Gleichung |0 (DE-588)4001162-8 |2 gnd |9 rswk-swf |
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| 856 | 4 | |u http://www.loc.gov/catdir/enhancements/fy0654/2005054518-d.html |3 Publisher description | |
| 856 | 4 | |u http://www.loc.gov/catdir/enhancements/fy0654/2005054518-t.html |3 Table of contents | |
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Datensatz im Suchindex
| _version_ | 1804135516897017856 |
|---|---|
| adam_text | GENERAL THEORY OF ALGEBRAIC EQUATIONS ETIENNE BEZOUT 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 JV(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 DETERMINATION OF THE DEGREE OF THE FINAL
EQUATION RESULTING FROM AN ARBITRARY NUMBER OF COMPLETE EQUATIONS
CONTAINING THE SAME NUMBER OF UNKNOWNS REMARKS SECTION II ABOUT
INCOMPLETE POLYNOMIALS AND FIRST-ORDER INCOMPLETE EQUATIONS 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 PROBLEM IV PROBLEM V
PROBLEM VI PROBLEM VII: WE ASK FOR THE DEGREE OF THE FINAL EQUATION
RESULTING FROM AN ARBITRARY NUMBER N OF EQUATIONS OF THE FORM (U ...
N)* = 0 IN THE SAME NUMBER OF UNKNOWNS REMARK ABOUT THE SUM OF SOME
QUANTITIES NECESSARY TO DETERMINE THE NUMBER OF TERMS OF VARIOUS TYPES
OF INCOMPLETE POLYNOMIALS PROBLEM VIII PROBLEM IX PROBLEM X PROBLEM XI
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 PROBLEM XII PROBLEM XIII PROBLEM
XIV PROBLEM XV 21 22 24 26 28 28 29 32 32 34 35 35 36 36 37 38 39 40 41
42 CONTENTS VII PROBLEM XVI 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 PROBLEM XVII PROBLEM XVIII SUMMARY
AND TABLE OF THE DIFFERENT VALUES OF THE NUMBER OF TERMS SOUGHT IN THE
PRECEDING POLYNOMIAL AND IN RELATED QUANTITIES PROBLEM XIX PROBLEM XX
PROBLEM XXI PROBLEM XXII 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 DETERMINATION OF THE SYMPTOMS INDICATING
WHICH VALUE OF THE DEGREE OF THE FINAL EQUATION MUST BE CHOSEN OR
REJECTED, AMONG THE DIFFERENT AVAILABLE EXPRESSIONS 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 APPLICATION OF THE PRECEDING THEORY
TO EQUATIONS IN THREE UNKNOWNS GENERAL CONSIDERATIONS ABOUT THE DEGREE
OF THE FINAL EQUATION, WHEN CONSIDERING THE OTHER INCOMPLETE EQUATIONS
SIMILAR TO THOSE CONSIDERED UP UNTIL NOW PROBLEM XXIII GENERAL METHOD TO
DETERMINE THE DEGREE OF THE FINAL EQUATION FOR ALL CASES OF EQUATIONS OF
THE FORM (U A ... N)* = 0 GENERAL CONSIDERATIONS ABOUT THE NUMBER OF
TERMS OF OTHER POLYNOMIALS THAT ARE SIMILAR TO THOSE WE HAVE EXAMINED
CONCLUSION ABOUT FIRST-ORDER INCOMPLETE EQUATIONS 42 45 46 47 56 61 62
63 63 65 69 70 71 85 86 94 101 112 VIII CONTENTS SECTION HI 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 CONTENTS 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 REFLEXIONS 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
|
| adam_txt |
GENERAL THEORY OF ALGEBRAIC EQUATIONS ETIENNE BEZOUT 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 JV(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 DETERMINATION OF THE DEGREE OF THE FINAL
EQUATION RESULTING FROM AN ARBITRARY NUMBER OF COMPLETE EQUATIONS
CONTAINING THE SAME NUMBER OF UNKNOWNS REMARKS SECTION II ABOUT
INCOMPLETE POLYNOMIALS AND FIRST-ORDER INCOMPLETE EQUATIONS 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 PROBLEM IV PROBLEM V
PROBLEM VI PROBLEM VII: WE ASK FOR THE DEGREE OF THE FINAL EQUATION
RESULTING FROM AN ARBITRARY NUMBER N OF EQUATIONS OF THE FORM (U .
N)* = 0 IN THE SAME NUMBER OF UNKNOWNS REMARK ABOUT THE SUM OF SOME
QUANTITIES NECESSARY TO DETERMINE THE NUMBER OF TERMS OF VARIOUS TYPES
OF INCOMPLETE POLYNOMIALS PROBLEM VIII PROBLEM IX PROBLEM X PROBLEM XI
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 PROBLEM XII PROBLEM XIII PROBLEM
XIV PROBLEM XV 21 22 24 26 28 28 29 32 32 34 35 35 36 36 37 38 39 40 41
42 CONTENTS VII PROBLEM XVI 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 PROBLEM XVII PROBLEM XVIII SUMMARY
AND TABLE OF THE DIFFERENT VALUES OF THE NUMBER OF TERMS SOUGHT IN THE
PRECEDING POLYNOMIAL AND IN RELATED QUANTITIES PROBLEM XIX PROBLEM XX
PROBLEM XXI PROBLEM XXII 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 DETERMINATION OF THE SYMPTOMS INDICATING
WHICH VALUE OF THE DEGREE OF THE FINAL EQUATION MUST BE CHOSEN OR
REJECTED, AMONG THE DIFFERENT AVAILABLE EXPRESSIONS 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 APPLICATION OF THE PRECEDING THEORY
TO EQUATIONS IN THREE UNKNOWNS GENERAL CONSIDERATIONS ABOUT THE DEGREE
OF THE FINAL EQUATION, WHEN CONSIDERING THE OTHER INCOMPLETE EQUATIONS
SIMILAR TO THOSE CONSIDERED UP UNTIL NOW PROBLEM XXIII GENERAL METHOD TO
DETERMINE THE DEGREE OF THE FINAL EQUATION FOR ALL CASES OF EQUATIONS OF
THE FORM (U A . N)* = 0 GENERAL CONSIDERATIONS ABOUT THE NUMBER OF
TERMS OF OTHER POLYNOMIALS THAT ARE SIMILAR TO THOSE WE HAVE EXAMINED
CONCLUSION ABOUT FIRST-ORDER INCOMPLETE EQUATIONS 42 45 46 47 56 61 62
63 63 65 69 70 71 85 86 94 101 112 VIII CONTENTS SECTION HI 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 CONTENTS 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 REFLEXIONS 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 |
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| author | Bézout, Étienne 1730-1783 |
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| author_sort | Bézout, Étienne 1730-1783 |
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| building | Verbundindex |
| bvnumber | BV021690803 |
| callnumber-first | Q - Science |
| callnumber-label | QA211 |
| callnumber-raw | QA211 |
| callnumber-search | QA211 |
| callnumber-sort | QA 3211 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 230 |
| classification_tum | MAT 120f |
| ctrlnum | (OCoLC)61881203 (DE-599)BVBBV021690803 |
| dewey-full | 512.9/4 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.9/4 |
| dewey-search | 512.9/4 |
| dewey-sort | 3512.9 14 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV021690803 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T15:14:21Z |
| indexdate | 2024-07-09T20:41:45Z |
| institution | BVB |
| isbn | 0691114323 9780691114323 |
| language | English French |
| lccn | 2005054518 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014904872 |
| oclc_num | 61881203 |
| open_access_boolean | |
| owner | DE-703 DE-634 DE-11 DE-91G DE-BY-TUM |
| owner_facet | DE-703 DE-634 DE-11 DE-91G DE-BY-TUM |
| physical | XXIV, 337 S. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Princeton Univ. Press |
| record_format | marc |
| spelling | Bézout, Étienne 1730-1783 Verfasser (DE-588)117590193 aut Théorie générale des équations algébriques General theory of algebraic equations Etienne Bézout ; translated by Eric Feron Princeton, NJ Princeton Univ. Press 2006 XXIV, 337 S. txt rdacontent n rdamedia nc rdacarrier Équations, Théorie des Equations, Theory of Algebraische Gleichung (DE-588)4001162-8 gnd rswk-swf Algebraische Gleichung (DE-588)4001162-8 s DE-604 http://www.loc.gov/catdir/enhancements/fy0654/2005054518-b.html Contributor biographical information http://www.loc.gov/catdir/enhancements/fy0654/2005054518-d.html Publisher description http://www.loc.gov/catdir/enhancements/fy0654/2005054518-t.html Table of contents GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014904872&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bézout, Étienne 1730-1783 General theory of algebraic equations Équations, Théorie des Equations, Theory of Algebraische Gleichung (DE-588)4001162-8 gnd |
| subject_GND | (DE-588)4001162-8 |
| title | General theory of algebraic equations |
| title_alt | Théorie générale des équations algébriques |
| title_auth | General theory of algebraic equations |
| title_exact_search | General theory of algebraic equations |
| title_exact_search_txtP | General theory of algebraic equations |
| title_full | General theory of algebraic equations Etienne Bézout ; translated by Eric Feron |
| title_fullStr | General theory of algebraic equations Etienne Bézout ; translated by Eric Feron |
| title_full_unstemmed | General theory of algebraic equations Etienne Bézout ; translated by Eric Feron |
| title_short | General theory of algebraic equations |
| title_sort | general theory of algebraic equations |
| topic | Équations, Théorie des Equations, Theory of Algebraische Gleichung (DE-588)4001162-8 gnd |
| topic_facet | Équations, Théorie des Equations, Theory of Algebraische Gleichung |
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