Knebusch, Manfred
Overview
Works:  37 works in 162 publications in 2 languages and 2,836 library holdings 

Roles:  Author, Editor, Creator 
Classifications:  QA3, 512.944 
Publication Timeline
.
Most widely held works by
Manfred Knebusch
Locally semialgebraic spaces
by
Hans Delfs(
Book
)
14 editions published in 1985 in English and German and held by 467 WorldCat member libraries worldwide
14 editions published in 1985 in English and German and held by 467 WorldCat member libraries worldwide
Weakly semialgebraic spaces
by
Manfred Knebusch(
Book
)
13 editions published in 1989 in English and held by 454 WorldCat member libraries worldwide
The book is the second part of an intended threevolume 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 Kgroups, 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
13 editions published in 1989 in English and held by 454 WorldCat member libraries worldwide
The book is the second part of an intended threevolume 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 Kgroups, 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
Specialization of quadratic and symmetric bilinear forms
by
Manfred Knebusch(
)
11 editions published in 2010 in English and held by 408 WorldCat member libraries worldwide
The specialization theory of quadratic and symmetric bilinear forms over fields and the subsequent generic splitting theory of quadratic forms were invented by the author in the mid1970's. They came to fruition in the ensuing decades and have become an integral part of the geometric methods in quadratic form theory. This book comprehensively covers the specialization and generic splitting theories. These theories, originally developed for fields of characteristic different from 2, are explored here without this restriction. In addition to chapters on specialization theory, generic splitting t
11 editions published in 2010 in English and held by 408 WorldCat member libraries worldwide
The specialization theory of quadratic and symmetric bilinear forms over fields and the subsequent generic splitting theory of quadratic forms were invented by the author in the mid1970's. They came to fruition in the ensuing decades and have become an integral part of the geometric methods in quadratic form theory. This book comprehensively covers the specialization and generic splitting theories. These theories, originally developed for fields of characteristic different from 2, are explored here without this restriction. In addition to chapters on specialization theory, generic splitting t
Manis valuations and Prüfer extensions I : a new chapter in commutative algebra
by
Manfred Knebusch(
Book
)
19 editions published between 2002 and 2004 in English and held by 406 WorldCat member libraries worldwide
The present book is devoted to a study of relative Prfer rings and Manis valuations, with an eye to application in real and padic geometry. If one wants to expand on the usual algebraic geometry over a nonalgebraically closed base field, e.g. a real closed field or padically 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 padic geometry, new chapters of commutative algebra are needed, often of a nonnoetherian nature. The present volume presents one such chapter
19 editions published between 2002 and 2004 in English and held by 406 WorldCat member libraries worldwide
The present book is devoted to a study of relative Prfer rings and Manis valuations, with an eye to application in real and padic geometry. If one wants to expand on the usual algebraic geometry over a nonalgebraically closed base field, e.g. a real closed field or padically 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 padic geometry, new chapters of commutative algebra are needed, often of a nonnoetherian nature. The present volume presents one such chapter
Manis valuations and Prüfer extensions II
by
Manfred Knebusch(
)
5 editions published in 2014 in English and held by 272 WorldCat member libraries worldwide
This volume is a sequel to 'Manis Valuation and Prüfer Extensions I,' LNM1791. The Prüfer extensions of a commutative ring A are roughly those commutative ring extensions R / A,where commutative algebra is governed by Manis valuations on R with integral values on A. These valuations then turn out to belong to the particularly amenable subclass of PM (=PrüferManis) valuations. While in Volume I Prüfer extensions in general and individual PM valuations were studied, now the focus is on families of PM valuations. One highlight is the presentation of a very general and deep approximation theorem for PM valuations, going back to Joachim Gräter's work in 1980, a farreaching extension of the classical weak approximation theorem in arithmetic. Another highlight is a theory of so called 'Kronecker extensions,' where PM valuations are put to use in arbitrary commutative ring extensions in a way that ultimately goes back to the work of Leopold Kronecker
5 editions published in 2014 in English and held by 272 WorldCat member libraries worldwide
This volume is a sequel to 'Manis Valuation and Prüfer Extensions I,' LNM1791. The Prüfer extensions of a commutative ring A are roughly those commutative ring extensions R / A,where commutative algebra is governed by Manis valuations on R with integral values on A. These valuations then turn out to belong to the particularly amenable subclass of PM (=PrüferManis) valuations. While in Volume I Prüfer extensions in general and individual PM valuations were studied, now the focus is on families of PM valuations. One highlight is the presentation of a very general and deep approximation theorem for PM valuations, going back to Joachim Gräter's work in 1980, a farreaching extension of the classical weak approximation theorem in arithmetic. Another highlight is a theory of so called 'Kronecker extensions,' where PM valuations are put to use in arbitrary commutative ring extensions in a way that ultimately goes back to the work of Leopold Kronecker
Algebraic theory of quadratic forms : generic methods and Pfister forms
by
Manfred Knebusch(
Book
)
13 editions published in 1980 in English and held by 258 WorldCat member libraries worldwide
13 editions published in 1980 in English and held by 258 WorldCat member libraries worldwide
Wittrings
by
Manfred Knebusch(
Book
)
11 editions published between 1972 and 1982 in 3 languages and held by 209 WorldCat member libraries worldwide
11 editions published between 1972 and 1982 in 3 languages and held by 209 WorldCat member libraries worldwide
Grothendieck und Wittringe von nichtausgearteten symmetrischen Bilinearformen
by
Manfred Knebusch(
Book
)
7 editions published in 1970 in German and English and held by 79 WorldCat member libraries worldwide
7 editions published in 1970 in German and English and held by 79 WorldCat member libraries worldwide
Einführung in die reelle Algebra
by
Manfred Knebusch(
Book
)
8 editions published in 1989 in German and held by 71 WorldCat member libraries worldwide
8 editions published in 1989 in German and held by 71 WorldCat member libraries worldwide
Ordnungen in lokalen Jordanalgebren vom Grade 2 und 3
by
Manfred Knebusch(
Book
)
7 editions published in 1964 in German and held by 31 WorldCat member libraries worldwide
7 editions published in 1964 in German and held by 31 WorldCat member libraries worldwide
Manis valuations and Prüfer extensions
by
Manfred Knebusch(
Book
)
4 editions published in 2014 in English and held by 21 WorldCat member libraries worldwide
4 editions published in 2014 in English and held by 21 WorldCat member libraries worldwide
Dominance and Transmissions in Supertropical Valuation Theory
by
Zur Izhakian(
)
3 editions published in 2011 in English and held by 18 WorldCat member libraries worldwide
3 editions published in 2011 in English and held by 18 WorldCat member libraries worldwide
Supertropical Linear Algebra
by
Zur Izhakian(
)
3 editions published in 2010 in English and German and held by 18 WorldCat member libraries worldwide
3 editions published in 2010 in English and German and held by 18 WorldCat member libraries worldwide
Monoid Valuations and Value Ordered Supervaluations
by
Zur Izhakian(
)
3 editions published in 2011 in English and German and held by 18 WorldCat member libraries worldwide
3 editions published in 2011 in English and German and held by 18 WorldCat member libraries worldwide
Supertropical Semirings and Supervaluations
by Zur Ishakian(
)
1 edition published in 2010 in German and held by 15 WorldCat member libraries worldwide
1 edition published in 2010 in German and held by 15 WorldCat member libraries worldwide
Supertropical Quadratic Forms I
by
Zur Izhakian(
)
3 editions published in 2013 in English and German and held by 14 WorldCat member libraries worldwide
3 editions published in 2013 in English and German and held by 14 WorldCat member libraries worldwide
Manis valuations and prüfer extensions
by
Manfred Knebusch(
Book
)
1 edition published in 2002 in English and held by 13 WorldCat member libraries worldwide
1 edition published in 2002 in English and held by 13 WorldCat member libraries worldwide
Wittringe
by
Manfred Knebusch(
Book
)
5 editions published between 1972 and 1978 in 3 languages and held by 12 WorldCat member libraries worldwide
5 editions published between 1972 and 1978 in 3 languages and held by 12 WorldCat member libraries worldwide
Manis valuations and Prüfer extensions
by
Manfred Knebusch(
Book
)
7 editions published in 2002 in English and Undetermined and held by 10 WorldCat member libraries worldwide
7 editions published in 2002 in English and Undetermined and held by 10 WorldCat member libraries worldwide
Manis valuations and Prüfer extensions
by
Manfred Knebusch(
Book
)
1 edition published in 2002 in English and held by 8 WorldCat member libraries worldwide
1 edition published in 2002 in English and held by 8 WorldCat member libraries worldwide
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Associated Subjects
Algebra Algebraic fields Algebraic spaces Algebraic topology Approximation theory Bilinear forms Categories (Mathematics) Commutative algebra Commutative rings Forms, Pfister Forms, Quadratic Geometry, Algebraic Graph labelings Group theory Homotopy theory Mathematical statistics Mathematics Prüfer rings Rings (Algebra) Witt rings