INTRODUCTION TO MODEL THEORY AND TO THE METAMATHEMATICS OF ALGEBRA

By G. DAKAT

CONTENTS INTRODUCTION AND OVERVIEW A DIFFERENTIAL FIELD WITH NO ESCAPE STRATEGY AND MAIN RESULTS ORGANIZATION THE NEXT VOLUME FUTURE CHALLENGES A HISTORICAL NOTE ON TRANSSERIES SOME COMMUTATIVE ALGEBRA THE ZARISKI TOPOLOGY AND NOETHERIANITY RINGS AND MODULES OF FINITE LENGTH INTEGRAL EXTENSIONS AND INTEGRALLY CLOSED DOMAINS LOCAL RINGS KRULL’S PRINCIPAL IDEAL THEOREM REGULAR LOCAL RINGS MODULES AND DERIVATIONS DIFFERENTIALS DERIVATIONS ON FIELD EXTENSIONS VALUED ABELIAN GROUPS ORDERED SETS VALUED VECTOR SPACES ORDERED ABELIAN GROUPS CONTENTS VALUED FIELDS VALUATIONS ON FIELDS PSEUDOCONVERGENCE IN VALUED FIELDS HENSELIAN VALUED FIELDS DECOMPOSING VALUATIONS VALUED ORDERED FIELDS SOME MODEL THEORY OF VALUED FIELDS THE NEWTON TREE OF A POLYNOMIAL OVER A VALUED FIELD DIFFERENTIAL POLYNOMIALS DIFFERENTIAL FIELDS AND DIFFERENTIAL POLYNOMIALS DECOMPOSITIONS OF DIFFERENTIAL POLYNOMIALS OPERATIONS ON DIFFERENTIAL POLYNOMIALS VALUED DIFFERENTIAL FIELDS AND CONTINUITY THE GAUSSIAN VALUATION DIFFERENTIAL RINGS DIFFERENTIALLY CLOSED FIELDS LINEAR DIFFERENTIAL POLYNOMIALS LINEAR DIFFERENTIAL OPERATORS SECOND-ORDER LINEAR DIFFERENTIAL OPERATORS DIAGONALIZATION OF MATRICES SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS DIFFERENTIAL MODULES LINEAR DIFFERENTIAL OPERATORS IN THE PRESENCE OF A VALUATION COMPOSITIONAL CONJUGATION THE RICCATI TRANSFORM JOHNSON’S THEOREM VALUED DIFFERENTIAL FIELDS ASYMPTOTIC BEHAVIOR OF VP ALGEBRAIC EXTENSIONS RESIDUE EXTENSIONS THE VALUATION INDUCED ON THE VALUE GROUP ASYMPTOTIC COUPLES DOMINANT PART THE EQUALIZER THEOREM EVALUATION AT PSEUDOCAUCHY SEQUENCES CONSTRUCTING CANONICAL IMMEDIATE EXTENSIONS DIFFERENTIAL-HENSELIAN FIELDS PRELIMINARIES ON DIFFERENTIAL-HENSELIANITY MAXIMALITY AND DIFFERENTIAL-HENSELIANITY DIFFERENTIAL-HENSEL CON?GURATIONS MAXIMAL IMMEDIATE EXTENSIONS IN THE MONOTONE CASE THE CASE OF FEW CONSTANTS DIFFERENTIAL-HENSELIANITY IN SEVERAL VARIABLES DIFFERENTIAL-HENSELIAN FIELDS WITH MANY CONSTANTS ANGULAR COMPONENTS EQUIVALENCE OVER SUBSTRUCTURES RELATIVE QUANTI?ER ELIMINATION A MODEL COMPANION ASYMPTOTIC FIELDS AND ASYMPTOTIC COUPLES ASYMPTOTIC FIELDS AND THEIR ASYMPTOTIC COUPLES H-ASYMPTOTIC COUPLES APPLICATION TO DIFFERENTIAL POLYNOMIALS BASIC FACTS ABOUT ASYMPTOTIC FIELDS ALGEBRAIC EXTENSIONS OF ASYMPTOTIC FIELDS IMMEDIATE EXTENSIONS OF ASYMPTOTIC FIELDS DIFFERENTIAL POLYNOMIALS OF ORDER ONE EXTENDING H-ASYMPTOTIC COUPLES CLOSED H-ASYMPTOTIC COUPLES 10 H-FIELDS 101 PRE-DIFFERENTIAL-VALUED FIELDS 102 ADJOINING INTEGRALS 103 THE DIFFERENTIAL-VALUED HULL 104 ADJOINING EXPONENTIAL INTEGRALS 105 H-FIELDS AND PRE-H-FIELDS 106 LIOUVILLE CLOSED H-FIELDS 107 MISCELLANEOUS FACTS ABOUT ASYMPTOTIC FIELDS 11 EVENTUAL QUANTITIES, IMMEDIATE EXTENSIONS, AND SPECIAL CUTS 111 EVENTUAL BEHAVIOR 112 NEWTON DEGREE AND NEWTON MULTIPLICITY 113 USING NEWTON MULTIPLICITY AND NEWTON WEIGHT 114 CONSTRUCTING IMMEDIATE EXTENSIONS 115 SPECIAL CUTS IN H-ASYMPTOTIC FIELDS 116 THE PROPERTY OF ?-FREENESS 117 BEHAVIOR OF THE FUNCTION ? 118 SOME SPECIAL DE?NABLE SETS 12 TRIANGULAR AUTOMORPHISMS 121 FILTERED MODULES AND ALGEBRAS 122 TRIANGULAR LINEAR MAPS 123 THE LIE ALGEBRA OF AN ALGEBRAIC UNITRIANGULAR GROUP 124 DERIVATIONS ON THE RING OF COLUMN-FINITE MATRICES 125 ITERATION MATRICES CONTENTS 126 RIORDAN MATRICES 127 DERIVATIONS ON POLYNOMIAL RINGS 128 APPLICATION TO DIFFERENTIAL POLYNOMIALS 13 THE NEWTON POLYNOMIAL 131 REVISITING THE DOMINANT PART 132 ELEMENTARY PROPERTIES OF THE NEWTON POLYNOMIAL 133 THE SHAPE OF THE NEWTON POLYNOMIAL 134 REALIZING CUTS IN THE VALUE GROUP 135 EVENTUAL EQUALIZERS 136 FURTHER CONSEQUENCES OF ?-FREENESS 137 FURTHER CONSEQUENCES OF ?-FREENESS 138 ASYMPTOTIC EQUATIONS 139 SOME SPECIAL H-FIELDS 14 NEWTONIAN DIFFERENTIAL FIELDS 141 RELATION TO DIFFERENTIAL-HENSELIANITY 142 CASES OF LOW COMPLEXITY 143 SOLVING QUASILINEAR EQUATIONS 144 UNRAVELERS 145 NEWTONIZATION 15 NEWTONIANITY OF DIRECTED UNIONS 151 FINITELY MANY EXCEPTIONAL VALUES 152 INTEGRATION AND THE EXTENSION K(X) 153 APPROXIMATING ZEROS OF DIFFERENTIAL POLYNOMIALS 154 PROOF OF NEWTONIANITY 16 QUANTI?ER ELIMINATION 161 EXTENSIONS CONTROLLED BY ASYMPTOTIC COUPLES 162 MODEL COMPLETENESS 163 ??-CUTS AND ??-FIELDS 164 EMBEDDING PRE-??-FIELDS INTO ?-FREE ??-FIELDS 165 THE LANGUAGE OF ??-FIELDS 166 ELIMINATION OF QUANTI?ERS WITH APPLICATIONS INTRODUCTION AND OVERVIEW A DIFFERENTIAL FIELD WITH NO ESCAPE OUR PRINCIPAL OBJECT OF INTEREST IS THE DIFFERENTIAL ?ELD T OF TRANSSERIES TRANSSERIES ARE FORMAL SERIES IN AN INDETERMINATE X > R, SUCH AS ?(X) = ?3 EEX + E EX LOG X + EX LOG2 X + EX LOG3 X + · · · ?X11 + (1) + ? X + X LOG X + X LOG2 X + X LOG3 X + · · · + X2 + X3 + X4 + X5 +