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Dieses Buch will dem Leser eine Einfuhrung in wichtige Techniken
und Methoden der heutigen reellen Algebra und Geometrie vermitteln.
An Voraussetzungen werden dabei nur Grundkenntnisse der Algebra
erwartet, so dass das Buch fur Studenten mittlerer Semester
geeignet ist.Das erste Kapitel enthalt zunachst grundlegende Fakten
uber angeordnete Koerper und ihre reellen Abschlusse und behandelt
dann verschiedene Methoden zur Bestimmung der Anzahl reeller
Nullstellen von Polynomen. Das zweite Kapitel befasst sich mit
reellen Stellen und gipfelt in Artins Loesung des 17. Hilbertschen
Problems. Kapitel III schliesslich ist dem noch jungen Begriff des
reellen Spektrums und seinen Anwendungen gewidmet."Neben dem 1987
erschienenen "Geometrie algebrique reelle" von J. Bochnak-M. Coste-
M. Roy stellt die vorliegende Monographie das erste Lehrbuch auf
diesem Gebiet dar... Damit liegt eine sehr empfehlenswerte
Einfuhrung...vor..." (H. Mitsch, Monatshefte fur Mathematik 3/111,
1991)
A Mathematician Said Who Can Quote Me a Theorem that's True? For
the ones that I Know Are Simply not So, When the Characteristic is
Two! This pretty limerick ?rst came to my ears in May 1998 during a
talk by T.Y. Lam 1 on ?eld invariants from the theory of quadratic
forms. It is-poetic exaggeration allowed-a suitable motto for this
monograph. What is it about? At the beginning of the seventies I
drew up a specialization theoryofquadraticandsymmetricbilinear
formsover ?elds[32].Let? : K? L?? be a place. Then one can assign a
form? (?)toaform? over K in a meaningful way ? if? has "good
reduction" with respect to? (see1.1). The basic idea is to simply
apply the place? to the coe?cients of?, which must therefore be in
the valuation ring of?. The specialization theory of that time was
satisfactory as long as the ?eld L, and therefore also K, had
characteristic 2. It served me in the ?rst place as the foundation
for a theory of generic splitting of quadratic forms [33], [34].
After a very modest beginning, this theory is now in full bloom. It
became important for the understanding of quadratic forms over
?elds, as can be seen from the book [26]of
Izhboldin-Kahn-Karpenko-Vishik for instance. One should note that
there exists a
theoryof(partial)genericsplittingofcentralsimplealgebrasandreductivealgebraic
groups, parallel to the theory of generic splitting of quadratic
forms (see [29] and the literature cited there).
The present book is devoted to a study of relative Prüfer rings and Manis valuations, with an eye to application in real and p-adic geometry. If one wants to expand on the usual algebraic geometry over a non-algebraically closed base field, e.g. a real closed field or p-adically closed field, one typically meets lots of valuation domains. Usually they are not discrete and hence not noetherian. Thus, for a further develomemt of real algebraic and real analytic geometry in particular, and certainly also rigid analytic and p-adic geometry, new chapters of commutative algebra are needed, often of a non-noetherian nature. The present volume presents one such chapter.
The book is the second part of an intended three-volume treatise on
semialgebraic topology over an arbitrary real closed field R. In
the first volume (LNM 1173) the category LSA(R) or regular
paracompact locally semialgebraic spaces over R was studied. The
category WSA(R) of weakly semialgebraic spaces over R - the focus
of this new volume - contains LSA(R) as a full subcategory. The
book provides ample evidence that WSA(R) is "the" right cadre to
understand homotopy and homology of semialgebraic sets, while
LSA(R) seems to be more natural and beautiful from a geometric
angle. The semialgebraic sets appear in LSA(R) and WSA(R) as the
full subcategory SA(R) of affine semialgebraic spaces. The theory
is new although it borrows from algebraic topology. A highlight is
the proof that every generalized topological (co)homology theory
has a counterpart in WSA(R) with in some sense "the same," or even
better, properties as the topological theory. Thus we may speak of
ordinary (=singular) homology groups, orthogonal, unitary or
symplectic K-groups, and various sorts of cobordism groups of a
semialgebraic set over R. If R is not archimedean then it seems
difficult to develop a satisfactory theory of these groups within
the category of semialgebraic sets over R: with weakly
semialgebraic spaces this becomes easy. It remains for us to
interpret the elements of these groups in geometric terms: this is
done here for ordinary (co)homology.
Dieses Buch will dem Leser eine Einfuhrung in wichtige Techniken
und Methoden der heutigen reellen Algebra und Geometrie vermitteln.
An Voraussetzungen werden dabei nur Grundkenntnisse der Algebra
erwartet, so dass das Buch fur Studenten mittlerer Semester
geeignet ist.Das erste Kapitel enthalt zunachst grundlegende Fakten
uber angeordnete Koerper und ihre reellen Abschlusse und behandelt
dann verschiedene Methoden zur Bestimmung der Anzahl reeller
Nullstellen von Polynomen. Das zweite Kapitel befasst sich mit
reellen Stellen und gipfelt in Artins Loesung des 17. Hilbertschen
Problems. Kapitel III schliesslich ist dem noch jungen Begriff des
reellen Spektrums und seinen Anwendungen gewidmet."Neben dem 1987
erschienenen "Geometrie algebrique reelle" von J. Bochnak-M. Coste-
M. Roy stellt die vorliegende Monographie das erste Lehrbuch auf
diesem Gebiet dar... Damit liegt eine sehr empfehlenswerte
Einfuhrung...vor..." (H. Mitsch, Monatshefte fur Mathematik 3/111,
1991)
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