Sie haben keine Artikel im Warenkorb.

Specialization of Quadratic and Symmetric Bilinear Forms von Manfred Knebusch

CHF 65.55
Verlag: Springer London
ISBN: 978-1-4471-2586-0
GTIN: 9781447125860
Einband: Kartonierter Einband (Kt)
Verfügbarkeit: Folgt in ca. 5 Arbeitstagen
+ -
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? (see§1.1). 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).

From the reviews:

"In the present book Knebusch develops a theory of specialization covering all the situations occurring by combining the cases when the characteristic of the field L is equal or unequal to 2 with the cases when the forms are either symmetric bilinear or quadratic and have a prescribed type of reduction under the place. ? The book is a welcome addition to the literature on quadratic forms supporting the tendency of presenting results for forms over all fields independently of the field characteristic." (K. Szymiczek, Mathematical Reviews, Issue 2011 h)

"The author developed a specialization theory and a subsequent generic splitting theory of quadratic and symmetric bilinear forms over fields. ? This book comprehensively covers the theory without the restriction of the characteristic. ? offers results on specialization with respect to quadratic places providing a look on further research. A book for the specialist!" (H. Mitsch, Monatshefte für Mathematik, Vol. 164 (3), November, 2011)

"This is an important monograph in which the author has done an excellent job putting together in one place many important results on specialization of quadratic and bilinear forms and generic splitting of quadratic forms that were previously dispersed in several research articles. It belongs on the shelf of any mathematician interested on the algebraic, geometric and arithmetic aspects of quadratic forms. The book under review joins the class of the ever expanding literature on quadratic forms ? ." (Feline Zaldivar, The Mathematical Association of America, January, 2011)

"The present book represents the state of the art in the theory of specialization of quadratic forms, containing a substantial amount of previously unpublished new results. It is a welcome addition to the literature as this is the first monograph in which this original and important approach to the algebraic theory of quadratic forms hasbeen given such a comprehensive treatment." (Detlev Hoffmann, Zentralblatt MATH, Vol. 1203, 2011)


*
*
*
*
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? (see§1.1). 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).

From the reviews:

"In the present book Knebusch develops a theory of specialization covering all the situations occurring by combining the cases when the characteristic of the field L is equal or unequal to 2 with the cases when the forms are either symmetric bilinear or quadratic and have a prescribed type of reduction under the place. ? The book is a welcome addition to the literature on quadratic forms supporting the tendency of presenting results for forms over all fields independently of the field characteristic." (K. Szymiczek, Mathematical Reviews, Issue 2011 h)

"The author developed a specialization theory and a subsequent generic splitting theory of quadratic and symmetric bilinear forms over fields. ? This book comprehensively covers the theory without the restriction of the characteristic. ? offers results on specialization with respect to quadratic places providing a look on further research. A book for the specialist!" (H. Mitsch, Monatshefte für Mathematik, Vol. 164 (3), November, 2011)

"This is an important monograph in which the author has done an excellent job putting together in one place many important results on specialization of quadratic and bilinear forms and generic splitting of quadratic forms that were previously dispersed in several research articles. It belongs on the shelf of any mathematician interested on the algebraic, geometric and arithmetic aspects of quadratic forms. The book under review joins the class of the ever expanding literature on quadratic forms ? ." (Feline Zaldivar, The Mathematical Association of America, January, 2011)

"The present book represents the state of the art in the theory of specialization of quadratic forms, containing a substantial amount of previously unpublished new results. It is a welcome addition to the literature as this is the first monograph in which this original and important approach to the algebraic theory of quadratic forms hasbeen given such a comprehensive treatment." (Detlev Hoffmann, Zentralblatt MATH, Vol. 1203, 2011)


Autor Knebusch, Manfred / Unger, Thomas (Übers.)
Verlag Springer London
Einband Kartonierter Einband (Kt)
Erscheinungsjahr 2012
Seitenangabe 208 S.
Lieferstatus Folgt in ca. 5 Arbeitstagen
Ausgabekennzeichen Englisch
Abbildungen Paperback
Masse H23.5 cm x B15.5 cm x D1.2 cm 324 g
Auflage 2010
Reihe Algebra and Applications

Über den Autor Manfred Knebusch

¿Manfred Knebusch is Professor Emeritus at the University of Regensburg. He has written nine books and more than 80 papers on the algebraic theory of quadratic forms over rings and fields, valuation theory, real algebra and real algebraic geometry. His current research focusses on tropical geometry. Claus Scheiderer is Professor at Konstanz University. His primary research interests are real algebraic geometry and convex algebraic geometry. Thomas Unger is Associate Professor at University College Dublin. His research interests include quadratic and hermitian forms, algebras with involution, and noncommutative real algebra and geometry.

Weitere Titel von Manfred Knebusch

Alle Bände der Reihe "Algebra and Applications"