Prestel A., Delzell C.N. — Positive Polynomials: From Hilbert's 17th Problem to Real Algebra (Springer Monographs in Mathematics) :: Электронная библиотека попечительского совета мехмата МГУ

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Prestel A., Delzell C.N. — Positive Polynomials: From Hilbert's 17th Problem to Real Algebra (Springer Monographs in Mathematics)

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Название: Positive Polynomials: From Hilbert's 17th Problem to Real Algebra (Springer Monographs in Mathematics)

Автор: Prestel A., Delzell C.N.

Аннотация:

Positivity is one of the most basic mathematical concepts, involved in many areas of mathematics (analysis, real algebraic geometry, functional analysis, etc.). The main objective of the book is to give useful characterizations of polynomials. Beyond basic knowledge in algebra, only valuation theory as explained in the appendix is needed.

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Рубрика: Математика/

Серия: Сделано в холле

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 2001

Количество страниц: 250

Добавлена в каталог: 15.05.2011

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Предметный указатель

, $B(c_{h}, n, s, d, N)$ and bound, on roots      19
-types, space of, $\approx Sper \mathbb{R}[X_{1}, ..., X_{n}]$       110
-types, space of, $\approx Sper\mathbb{R}[X_{1}, ..., X_{n}]$       110
-adic numbers      226
-adic valuation      see Valuation
-module      see module
-semiordering      see Semiordering
$\hat{C}$ ernikov, S.N.      137 188
${\L}$ ocal - Global Principle, homogenous      146
${\L}$ os' theorem      39
${\L}$ os' theorem, special case of      40
Absolute value      16n
Abstract, Positivstellensatz      see Positivstellensatz abstract
Abstract, real Nullstellensatz      see Nullstellensatz real abstract
Acquistapace, F.      159
Affine      see variety
Algebra, real      V 1
algebraic      see Valuation extension
Algebraic, geometry, real      3 110
Algebraic, set      32
Algebraically maximal      see Valued field
Andradas, C.      48 49 110 159
Anisotropic      see Quadratic form
Approximation theorem      see Valuation ring
Archimedean ring, or Archimedean ...      see Module ordering ordered preordering or
Artin - Lang Theorem      49
Artin, E.      V VI 2 3 14 21 29 31 35 36 49 51 75 79 81 90 109
Aut      22
Automorphism group      22
Backmeister      111
Baer - Krull correspondence      29
Baer, R.      29
Basic closed, basic open      see Semialgebraic set
Basis, orthogonal      54
Becker, E.      79 137 177 178
Berg, C.      155 159
Berr, R.      178
Bochnak, J.      48 49 110
Borel measure      see Measure
Bound, on degree      179
Bound, on degree, existence of      183-189
Bound, on degree, in addition      see
Bounded      see $$W_{R}(H_{1}</a></span> <span class=subjpages><a href=$ ... H_{S})$"/>
Bourbaki, N.      245
Br $\ddot{o}$ cker - Prestel Local - Global Principle      158 177
Br $\ddot{o}$ cker - Scheiderer Theorem      49 79
Br $\ddot{o}$ cker, L.      48 49 79 110 137 158
Broglia, F.      159
Browder, F.      250
Brumfiel, G.      48
Cancellation theorem      see Witt E.
Canonical      see Ordering preordering topology or
Cassier, G.      137
Cauchy complete      see Complete Cauchy
Cauchy sequence      24 224 228
Chang, C.C.      40 49 110
Characteristic      see prime ideal or
Characterization Theorem II      144 158 173
Chevalley's Theorem      see Valuation ring
Chinese remainder theorem      211
Choi, M.D.      178 201
Christensen, J.P.R.      155 159
Closure under specialization      105
Closure, real      see Real closure
Coarsening      see Valuation ring
Cofinal      26 228
Cofinality      228 243
Cofinite sets, filter of      37
Compactness in addition.      see $$W_{R}(H_{1}</a></span> <span class=subjpages><a href=$ ... H_{S})$"/>
Compactness of $Sper^{max} A$       107
Compactness theorem of model theory      49
Compactness, property of $\mathbb{R}^{*}$       36 40
Complete cut      see Complete Dedekind
Complete valued field      see Valued field
Complete, Cauchy      24
Complete, Dedekind      9 10 24
Complete, relatively complete relatively valued field      see Valued field
Completion      225; see Valued field
composition      see Valuation ring
Cone      see Positive cone or prepositive cone
Conjugation Theorem      see Valuation extension
Constructible set      85
Constructible topology      see Topology
Continuous solution to Hilbert's 17th problem      see Hilbert D.
Convergence      228
Convex hull      12
Convex subgroup      24
Convex subring      12
Cornelsen, S.      111
Coste      48 49 109 110
Coste-Roy, M.-F.      see Roy M.-F.
Cut      97
Cut valuation      99
Cut, complete      see complete Dedekind
Cut, countably represented      97
Cut, critical      194
Cut, Dedekind      9
Cut, group      98
Cut, positive      97
Dai, Z.D.      201
Daykin, D.E.      201
Decomposition field      see Valuation extension
Decomposition group      see Valuation extension
Decomposition theorem      see Witt E.
Dedekind complete      see Complete Dedekind
Dedekind cut      see cut Dedekind
Definite quadratic form      see Quadratic form
Definition, prenex      34 36
Definition, quantifier-free      32
Definition, semialgebraic      32 34 35
Degree      see bound on
Delzell, C.N.      49 51 110 111 201
Dense subfield      8
Dependence class      see Valuation ring
Dependent      see Valuation ring
Diagonal      see Form of degree 2 $m$

and
Difference of a polynomial      176
DIMENSION      see Quadratic form
Discrete      see Valuation
Divisible      27
Dubois, D.W.      4 109 110 137
Efroymson, G.      49
Elimination of quantifiers      see Tarski A.
Elman, R.      79
Embedding into $\eta_{\alpha}$ -fields      81 95 96 110
Embedding into $\mathbb{R}$       9 28
Embedding of $\mathbb{R}^{(n)}$ into Sper $\mathbb{R}[X_{1},..., X_{n}]$       102
Embedding of function fields into $\mathbb{R}^{*}$       81 96
Endler, O.      245
Equivalent      see Quadratic form
Errata      VI
Euler, L.      29
Exponential function      47
Extension      see
Field ordering      see Ordering field
Field, $\eta_{1}-$       48
Field, $\eta_{\alpha}-$       94-96
Field, $\eta_{\alpha}-$ , embedding into      see Embedding
Field, algebraically closed      15 16
Field, embedding of a      17
Field, extension of a, algebraic      16
Field, extension of a, Galois      15
Field, extension of a, odd-degree      13-14
Field, extension of a, quadratic      13
Field, extension of a, transcendental      14
Field, maximal ordered      see Ordered field
Field, of characteristic not      2 53
Field, of fractions      42 74 109
Field, ordered      see Ordered field
Field, ordering, or valuation ring factorization of polynomials      16

Field, Pythagorean      78
Field, real      12 29
Field, real closed      14 29
Field, SAP      see SAP
Field, uniquely orderable      14
Filter      37
Filter, of cofinite sets      37
Finitely ramified      see valued field
Finiteness theorem      44-48 106 249
Form of degree       49
Form of degree 2 ("diagonal")      172
Form of degree 2 ("diagonal"), 2-isotropic      172
Form of degree 2 ("diagonal"), in addition      see Quadratic form
Form of degree 2 ("diagonal"), Pfister      176
Form of degree 2 ("diagonal"), regular      172
Form of degree 2 ("diagonal"), regular part of a      172
Form of degree 2 ("diagonal"), weakly 2-isotropic      172
Formal deduction      185
Formal derivative      17
Formal Laurent series      226
Formal power series      25 28
Formally real      29
Fraction field      see Field of fractions
Fraction ring      see Ring of fractions
Fuchs, L.      25
Function field, real, in variables      80
Function field, real, in variables, embedding into $\mathbb{R}^{*}$       96
Function field, real, in one variable      72
Function field, real, in one variable, SAP      72
Function field, real, in one variable, Witt's Local - Global Principle      73-74
Function, semialgebraic      see Semialgebraic function
Functional analysis      4 136 137
Functional, linear      152 154 155 158 159
Fundamental ideal      see Witt ring
G $\ddot{o}$ del's Completeness Theorem      185
Galois group      15
Gauss extension      see valuation
General polynomial      91
Gonz $\acute{a}$ lez - Vega, L.      110 111
Greenberg      68
Group      24
Group, 2-group      15
Group, abelian      240
Group, Abelian, rational rank of an      240
Group, in addition      see Ordered group; or ordering group
Grundlagen der Geometric      see Hilbert D.
H $\ddot{o}$ lder, O.      28
Hahn, H.      28
Handelman, D.      137
Harrison basic subset      72 78
Harrison topology      78
Harrison, D.K.      79 249
Hausdorff, F.      110
Hensel's lemma      218
Henselian      see valued field
Henselization (or Henselian closure)      see Valued field
Hilbert 17th problem      V VI 2 31 36 49 51 74 75 81
Hilbert 17th problem, $C^{r}$ variation in solutions of      111
Hilbert 17th problemcontinuous solution      31 81 91-94 110 111
Hilbert 17th problemgeneralization      3 53 74-77 90
Hilbert 17th problemgeneralization (abstract version)      88
Hilbert, D.      V 2 28 29n 49-51
Hilbert, Grundlagen der Geometrie      28 50 5n
Hodges, W.      49
Hoffmann, D.      80
Hyperbolic      see Quadratic form
Ideal, real      84
Immediate extension      see Valuation extension
Indefinite quadratic form      see Quadratic form
Independent      see Valuation ring
integral      242
Integral domain, ordered      see Ordered integral domain
International Congress      V
interval      17; see Topology
Irreducible polynomial      13
Isotropic 2-isotropic      see Form of degree 2 $m$

and
Jacobi, Thomas      124 127 137 153 158 159 170 177
Jacobson, N.      68
Kadison - Dubois Representation Theorem      see Representation Theorem
Kadison, R. V.      4 109 247
Keisler, H.J.      40 49 110
Knebusch      29 79
Krivine maximal real spectrum      109
Krivine, J. - L.      3 109 110 137 159
Krivine, Positivstellensatz      109
Krivine, real Nullstellensatz      109
Krull dimension      143
Krull valuation      see Valuation
Krull, W.      29 249
Kuhlmann, F. - V.      245
Kuhlmann, S.      159
Lagrange, J.L.      50
Lam, T.Y.      29 79 80 159 178 201
Landau, E.      29 29n 49n
Lang, S.      49 68 247
Laurent series, formal      226
Leading coefficient      134
Leicht, J.      79
Length of sum of squares      179
Level 2      see Module (pre)ordering or
Linear functional      154
Linear ordering      see Ordering
Linear representation      4 139 152
Local - Global Principle      see Br $\ddot{o}$ cker - Prestel Pfister or
Local ring      see Ring
Logic, first-order      185
Lojasiewicz, S.      49 111
Lombardi, H.      110 111 201
Lorenz, F.      79
Mah $\acute{e}$ , L.      201
Marshall      137 159
Maximal ordered      14
McEnerney, J.      49
Measure, positive Borel      152
Metric      223
Minkowski's theorem      137 157
Minkowski, H.      49 50 188 250
Model theory      110 201
Model, complete      49
Module of level 2      161
Module of level 2, Archimedean      164
Module, - ( a preordering)      113
Module, - ( a preordering), generated by $a_{1}, ..., a_{m}$       113
Module, - ( a preordering), maximal      114
Module, - ( a preordering), support of a      114
Module, - ( of level 2)      161
Module, - ( of level 2), support of a      161
Module, quadratic      4 113 115
Module, quadratic, Archimedean      5 116 129 144
Module, quadratic, of $\mathbb{R}[x_{1}, ..., x_{n}]:=\mathbb{R}[X_{1}, ..., X_{n}]/I$       117
Moment problem      152-157 159
Monoid, multiplicative, of a ring      86
Multi-degree      134
Multiplicative monoid      see Monoid
Multiplicative set      108
Nilradical of a ring      65
Non-critical      194
Nonstandard      118
Normal subgroup      24
Nullstellensatz, abstract      88 109
Nullstellensatz, due to Krivine      109
Nullstellensatz, real      81 90 110 250 254
Number field      29n
Oberwolfach      VI
Order-embedded      9
Order-embedding      17
Order-extension      12
Order-isomorphism      17
Order-preserving      17
Ordered field      7

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