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

Ordered field Characterization Theorem I      142 158 172
Ordered field, Archimedean      8
Ordered field, Archimedean, embedding into $\mathbb{R}$       9 28
Ordered field, characteristic of an      8
Ordered field, history of      28
Ordered field, maximal      14
Ordered group      24
Ordered group, Archimedean      8 28
Ordered group, Archimedean, in addition      see ordering of ... x_{n}]:=\mathbb{R}[X_{1} ... X_{n}]/I$"/>
Ordered group, extension of an      12-16
Ordered group, extension of an, Archimedean      26n
Ordered group, extension of an, odd-degree      13-14
Ordered group, extension of an, quadratic      13
Ordered group, extension of an, transcendental      14
Ordered group, field      3 7
Ordered group, group      24
Ordered group, history of      28
Ordered group, induced      12
Ordered group, lexicographic      134
Ordered group, linear      1 7
Ordered group, of $\bar{\mathcal{O}}^{p}$ , canonical      190
Ordered group, of $\mathbb{Q}(X_{1})$       9
Ordered group, of $\mathbb{R}(X_{1})$       8-9
Ordered group, of $\mathbb{R}[x_{1}, ..., x_{n}]:=\mathbb{R}[X_{1}, ..., X_{n}]/I$ , Archimedean      117
Ordered group, of $\mathcal{O}$ , canonical      190 191
Ordered group, of $\mathcal{O}/m$ , canonical      26
Ordered group, of level 2      177 247
Ordered group, support of an      81
Ordered group, unique      14 25
Ordered integral domain      29
Ordered skew-field      28
Ordered subfield      12 21
Ordering, $\eta_{1}-$       48
Orthogonal basis      54
Orthogonal sum      see Quadratic form
P $\acute{o}$ lya's Theorem      131 136 137
Pfister $2^{n}$ -bound      68 69 80 183
Pfister form      see Form of degree 2 $m$

and
Pfister Local - Global Principle      53 66 74 79
Pfister, A.      69
Positive Borel measure      see Measure
Positive cone      see Ordering
Positive cone of a field      10
Positive cone of a ring      81-85
Positive semidefinite      2 49
Positive semidefinite, on Sper       2
Positive semidefinite, over       36 74
Positivstellensatz      90 109 110 136 201 247 249 250 254
Positivstellensatz, abstract      81 88 109
Positivstellensatz, abstract, generalized      86
Positivstellensatz, due to Krivine      110
Positivstellensatz, weak (for modules of level 2)      164
Positivstellensatz, weak (for quadratic modules)      116
Power series, formal      25 28
Powers, V.      201
Prenex definition      see Definition prenex
Prenex statement      35
Preordering, Archimedean      4 116
Preordering, of $C(\mathcal{X}^{max}_{T}, \mathbb{R})$ , canonical      4
Preordering, of $\mathbb{R}[X_{1}, ..., X_{n}]$ , canonical      1 4
Preordering, of $\mathbb{R}[x_{1}, ..., x_{n}]:=\mathbb{R}[X_{1}, ..., X_{n}]/I$ , Archimedean      117
Preordering, of a field      10
Preordering, of a ring      1 81
Preordering, of if odd      175
Preordering, of level 2      161
Preordering, of level 2, on , extends to an ordering of level      2
Preordering, support of a      81
Prepositive cone of a field      10 29
Prepositive cone of a ring      1 81-82
Prepositive cone, in addition      see Preordering
Preprime, Archimedean      130-133
Prestel, A.      29 49 79 110 111 137 158 158n 159 178 201 230n 245
Prie ${\ss}$ - Crampe      28 29
Prime ideal, characteristic of a      61
Primitive polynomial      218
Principal ultrafilter      see Ultrafilter principal
PRODUCT      see Quadratic form
Projection theorem      33
Projection theorem, over       34
PSD      49
Psd, over       36
Putinar      137 159
Pythagoras number      80 179;
Pythagoras number 2th      178
Pythagorean field      78
Quadratic form      see Form of degree 2 $m$

Quadratic form, , the value set of a      69
Quadratic form, anisotropic      55
Quadratic form, associated symmetric bilinear form      54
Quadratic form, definite w.r.t. a semiordering      139
Quadratic form, definition of a      53
Quadratic form, diagonal      54
Quadratic form, dimension of a      53
Quadratic form, equivalent      54
Quadratic form, hyperbolic      56
Quadratic form, indefinite over a field, totally      73 76n
Quadratic form, indefinite w.r.t. a semiordering      139
Quadratic form, indefinite w.r.t. an ordering      62
Quadratic form, isotropic      55
Quadratic form, matrix of a      53
Quadratic form, orthogonal sum of      56
Quadratic form, Pfister form      69
Quadratic form, product of      59
Quadratic form, regular      56
Quadratic form, regular part of a      141 142
Quadratic form, representing a      55
Quadratic form, round      68
Quadratic form, signature of a      62 63
Quadratic form, similar      59
Quadratic form, similarity class of a      60
Quadratic form, total signature of a      65
Quadratic form, totally indefinite      see Quadratic form indefinite
Quadratic form, weakly      74 79 139 158
Quadratic form, weakly isotropic      see Quadratic form isotropic
Quadratic module      see Module quadratic
Quadratic system of representatives      28
Quantifier-free definition      see Definition quantifier-free
Quasi-compact      85
Quotient field      see Field of
Quotient group      25
R $\acute{e}$ cio      49 249
Radical of an ideal      65
Radical of an ideal, real      88
Ramification index      see Valuation extension
Rank 1      see Valuation
Rational rank      240
REAL      see Algebra algebraic field function ideal Nullstellensatz radical ring or
Real algebraic numbers      42
Real closed      14 29
Real closure, existence      16
Real closure, notation for      21
Real closure, uniqueness      21
Real closure, VI      16
Recursively enumerable      185
refinement      see Valuation ring
Regular      see Form of degree 2 $m$

and
Regular part see form of degree 2 $m$

and
Relatively algebraically closed      23
Relatively complete      see Valued field
Relatively separably closed      228
Representation Theorem for modules of level 2 (Jacobi)      170
Representation Theorem for preorderings      4 109 121
Representation Theorem for preprimes      132
Representation Theorem for quadratic modules (Jacobi)      127
Representation Theorem, history of the      136-137
Representation Theorem, Stone's      137
Representing       see Quadratic form
Residue degree      see Valuation extension
Residue field      see Valuation
Residue map      see Valuation

Ressel, P.      155 159
Reznick, B.      178 201
Ribenboim, P.      245
Riesz representation theorem      153
Ring of fractions      108
Ring, local      167 203 253
Ring, real      82
Ring, semireal      1 82
Risler, J. - J.      110
Roots, bound on      19
Rosenberg, A.      79
round      see quadratic form
Roy, M. - F.      48 49 109 110
Rudin, W.      153
Ruiz, J.      48 49 110
SAP      78 79
SAP of a real function field in one variable      72
SAP, valuation theoretic characterization of      79 158n
Saturation      40 49 94 110
SCALE      50n
Scharlau, W.      159
Scheiderer, C.      29 49 79 80
Schilling, O.F.G.      245
Schm $\ddot{u}$ dgen's Theorem      4 5 123 136 137 139 144 150 159 185 192 196 254
Schm $\ddot{u}$ dgen, K.      137 159
Schmid, J.      178 201
Schreier      0 14 21 29 79 109
Schwartz, N.      137
Schweighofer, M., VI      137 201
Semialgebraic definition      see Definition semialgebraic
Semialgebraic function      94n
Semialgebraic set      31-36
Semialgebraic set, basic closed      32
Semialgebraic set, basic open      32
Semialgebraic set, bounded and basic closed      see $$W_{R}(h_{1}</a></span> <span class=subjpages><a href=$ ... h_{s})$"/>
Semiordering      5 79 113 115 137
Semiordering, - ( a preordering)      114
Semiordering, - ( a preordering), support of a      115
Semiordering, of $\mathbb{R}[x_{1}, ..., x_{n}]:=\mathbb{R}[X_{1}, ..., X_{n}]/I$ , Archimedean      117
Semiordering, of level 2      163
Semiordering, of level 2, Archimedean      164
Semiordering, of level 2, determines a valuation ring      166
Semiordering, of level 2, history of      177
Semiordering, of level 2, support of a      163
Semireal ring      1 82
Separable closure      213n
Separable polynomials      218
Serre, J.P.      29
Seventeenth problem      see Hilbert D.
Siegel, C.L.      29n
Signature      see Quadratic form
Similar      see Quadratic form
Similarity classes      see Quadratic form
Space of orderings      65; see Spectrum real
Space of orderings, weak topology on      66
Specialization      2 3 31 81 89 103
Specialization, closure under      105
Spectral topology      see Topology
Spectrum, real      VI 2 4 84 109;
Spectrum, real, embeds in $\{0,1\}^{A}$       107
Spectrum, real, maximal      3
Spectrum, real, maximal: compactness of      107
Spectrum, real, maximal: found by Krivine      109
Spectrum, real, of $R[X_{1}]$       84
Spectrum, real, of $\mathbb{R}[X_{1}, ..., X_{n}]$       81 101 102 110
Spectrum, real, of $\mathbb{R}[X_{1}, ..., X_{n}]/I$       106-107
Spectrum, real, of $\mathbb{R}[X_{zero},..., X_{n}]:\approx$ space of -types      110
Spectrum, real, of $\mathbb{Z}$       84
Spectrum, real, of a field      84
Spectrum, real, topologies on      see Topology on Sper $A$

Spectrum, Zariski      61
Spectrum, Zariski, of $\mathbb{Z}$       84
Spectrum, Zariski, of a field      84
Standard part      103 189
Statement, prenex      35
Stengle, G.      110 201
Stone - Weierstra ${\ss}$ Theorem      123 134 159
Strong approximation property      see SAP
Sturm sequence      18
Sturm theorem      18
Subfield, dense      see Dense subfield
Subfield, ordered      see Ordered subfield
Subgroup, convex      see Convex subgroup
Subring, convex      see Convex subring
Subsemiring      86n
Sum of 2th powers      161
Sum of squares      11
Sum of squares, length of      179
Support      see Module (pre) preprime or
Support of a subset of a ring      81
Sylow subgroup      15
Tarski elimination of quantifiers      33 49 79
Tarski Transfer Principle      2 3 23 31 35 49 110
Tarski, A.      49
Tarski, generalized      34
Topology on $R^{(n)}$ , interval      32
Topology on $\mathbb{R}^{(n)}$ , discrete      104
Topology on $\mathbb{R}^{(n)}$ , interval      104
Topology on Semi - Sper , constructible      119
Topology on Semi - Sper , spectral      119
Topology on Sper , constructible (or "canonical")      4 85
Topology on Sper , spectral      85
Topology product (preserves Hausdorffness, (quasi-) compactness)      107
Torsion element      66
Torsion element, $2^{n}$       65
Torsion subgroup      66
Total signature      see Quadratic form
Totally indefinite      see Quadratic form
Totally isotropic      see Quadratic form
Totally positive      29
Totally real      50
Transcendence degree      96
Transcendental      see valuation extension
Transfer principle      see Tarski A.
Tressl      110
Tsen - Lang Theorem      68
Ultrafilter      37
Ultrafilter, principal      37
Ultrapower      38
Ultraproduct      36-41
Uniquely orderable      see Ordering unique
Valuation      27 204
Valuation ring      12 26 203;
Valuation ring, Approximation Theorem      234
Valuation ring, canonical      175 194 etc.)
Valuation ring, Chevalley's Theorem      167 205
Valuation ring, coarsening of a      230
Valuation ring, composition      232
Valuation ring, dependence class of a      233
Valuation ring, dependent      233
Valuation ring, determined by a semiordering of level 2      166
Valuation ring, determined by a valuation      204
Valuation ring, independent      234
Valuation ring, refinement of a      231
Valuation ring, trivial      12 203
Valuation, -adic      204
Valuation, discrete      242
Valuation, extension of a      206
Valuation, extension of a, algebraic      207-213
Valuation, extension of a, Conjugation Theorem      212
Valuation, extension of a, decomposition field of a      213
Valuation, extension of a, decomposition group of a      213
Valuation, extension of a, existence of an      207
Valuation, extension of a, Gauss      218
Valuation, extension of a, immediate      217
Valuation, extension of a, ramification index e of an      207
Valuation, extension of a, residue degree of an      207
Valuation, extension of a, transcendental      236-242
Valuation, Krull      29
Valuation, of rank 1      223 242

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