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Journal ArticleDOI

Discriminants, resultants and multidimensional determinants , by I. M. Gelfand, M. M. Kapranov and A. Zelevinsky. Pp. 523. £60. 1994. ISBN 3-7643-3660-9 (Birkhäuser)

01 Jul 1995-The Mathematical Gazette (Cambridge University Press (CUP))-Vol. 79, Iss: 485, pp 439-440
About: This article is published in The Mathematical Gazette.The article was published on 1995-07-01. It has received 8 citations till now.
Citations
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Journal ArticleDOI
Zur Izhakian1
TL;DR: The development of the tropical version for the fundamental theorem of algebra leads to the reduced polynomial semiring, a structure that provides a basis for developing a tropical analogue to the classical theory of commutative algebra.
Abstract: This paper introduces the foundations of the polynomial algebra and basic structures for algebraic geometry over the extended tropical semiring. Our development, which includes the tropical version for the fundamental theorem of algebra, leads to the reduced polynomial semiring — a structure that provides a basis for developing a tropical analogue to the classical theory of commutative algebra. The use of the new notion of tropical algebraic com-sets, built upon the complements of tropical algebraic sets, eventually yields the tropical algebraic Nullstellensatz.

30 citations

Posted Content
TL;DR: In this paper, it was shown that the moduli space of rank $2n$ instanton bundles on the 4-dimensional sphere is affine, defined from the well known monad condition.
Abstract: In this note we prove that the moduli space of rank $2n$ symplectic instanton bundles on ${\PP^{2n+1}}$, defined from the well known monad condition, is affine. This result was not known even in the case $n=1$, where the real instanton bundles correspond to self dual Yang Mills $Sp(1)$-connections over the 4-dimensional sphere. The result is proved as a consequence of the existence of an invariant of the multidimensional matrices representing the instanton bundles.

30 citations

Posted Content
22 May 2005
TL;DR: In this article, the authors introduced a new structure of commutative semiring, generalizing the tropical semiring and having an arithmetic that modifies the standard tropical operations, i.e., summation and maximum, and showed that a tropical matrix is invertible if and only if it is regular.
Abstract: This paper introduces a new structure of commutative semiring, generalizing the tropical semiring, and having an arithmetic that modifies the standard tropical operations, i.e. summation and maximum. Although our framework is combinatorial, notions of regularity and invertibility arise naturally for matrices over this semiring; we show that a tropical matrix is invertible if and only if it is regular.

16 citations

Posted Content
TL;DR: In this paper, a selection of tools from modern algebraic geometry, representation theory, the classical invariant theory of binary forms, together with explicit calculations with hypergeometric series and Feynman diagrams, are combined to obtain the following interrelated results.
Abstract: Combining a selection of tools from modern algebraic geometry, representation theory, the classical invariant theory of binary forms, together with explicit calculations with hypergeometric series and Feynman diagrams, we obtain the following interrelated results. A Castelnuovo-Mumford regularity bound and a projective normality result for the locus of hypersufaces that are equally supported on two hyperplanes. The surjectivity of an equivariant map between two plethystic compositions of symmetric powers; a statement which is reminiscent of the Foulkes-Howe conjecture. The nonvanishing of even transvectants of exact powers of generic binary forms. The nonvanishing of a collection of symmetric functions defined by sums over magic squares and transportation matrices with nonnegative integer entries. An explicit set of generators, in degree three, for the ideal of the coincident root locus of binary forms with only two roots of equal multiplicity.

16 citations

DissertationDOI
06 May 2003
TL;DR: In this paper, the authors cite Manuel Abellanas [1] and Günter M. Ziegler [2] as the main sources of inspiration for their work.
Abstract: Acknowledgements It is only fitting to continue the tradition and cite Manuel Abellanas [1] in first place. Without him and his constant encouragement, none of this would have happened. all combis in Berlin, for all the support and the excellent working environment christoph eyrich, for my´sliwska and teaching me all the λ α τ ε χ i know (and then some) Volker Kaibel, for the strongly non-polynomial patience in going through endless details in my manuscripts, and the expertise in shortening or lengthening my proofs Paco Santos, not least for leaving Iñigo a day longer than necessary to come to my exam Ewgenij Gawrilow and Michael Joswig, for polymake the staff and outfitters of the Molotov-cocktail depot, for volatile cakes & coffee Jörg Rambau, for coauthoring Chapter 6 Alexander Schwartz, for reading the manuscript and fixing all the software just in time Bettina Felsner, for making it all work smoothly Lourdes, que tú sabes muy bien todo lo que te tengo que agradecer The place of honor, of course, goes to my PhD advisor, Günter M. Ziegler. He knows better than anyone else just how many wonderful things I learned in these three exciting years in Berlin. Thank you so much for everything, Günter.

13 citations

References
More filters
Journal ArticleDOI
Zur Izhakian1
TL;DR: The development of the tropical version for the fundamental theorem of algebra leads to the reduced polynomial semiring, a structure that provides a basis for developing a tropical analogue to the classical theory of commutative algebra.
Abstract: This paper introduces the foundations of the polynomial algebra and basic structures for algebraic geometry over the extended tropical semiring. Our development, which includes the tropical version for the fundamental theorem of algebra, leads to the reduced polynomial semiring — a structure that provides a basis for developing a tropical analogue to the classical theory of commutative algebra. The use of the new notion of tropical algebraic com-sets, built upon the complements of tropical algebraic sets, eventually yields the tropical algebraic Nullstellensatz.

30 citations

Posted Content
TL;DR: In this paper, it was shown that the moduli space of rank $2n$ instanton bundles on the 4-dimensional sphere is affine, defined from the well known monad condition.
Abstract: In this note we prove that the moduli space of rank $2n$ symplectic instanton bundles on ${\PP^{2n+1}}$, defined from the well known monad condition, is affine. This result was not known even in the case $n=1$, where the real instanton bundles correspond to self dual Yang Mills $Sp(1)$-connections over the 4-dimensional sphere. The result is proved as a consequence of the existence of an invariant of the multidimensional matrices representing the instanton bundles.

30 citations

Journal ArticleDOI
TL;DR: In this paper, an upper bound on the Castelnuovo regularity of the ideal of X ( n, d ) is given, and it is shown that this ideal is r-normal for r ⩾ 2.

28 citations

Posted Content
22 May 2005
TL;DR: In this article, the authors introduced a new structure of commutative semiring, generalizing the tropical semiring and having an arithmetic that modifies the standard tropical operations, i.e., summation and maximum, and showed that a tropical matrix is invertible if and only if it is regular.
Abstract: This paper introduces a new structure of commutative semiring, generalizing the tropical semiring, and having an arithmetic that modifies the standard tropical operations, i.e. summation and maximum. Although our framework is combinatorial, notions of regularity and invertibility arise naturally for matrices over this semiring; we show that a tropical matrix is invertible if and only if it is regular.

16 citations

DissertationDOI
06 May 2003
TL;DR: In this paper, the authors cite Manuel Abellanas [1] and Günter M. Ziegler [2] as the main sources of inspiration for their work.
Abstract: Acknowledgements It is only fitting to continue the tradition and cite Manuel Abellanas [1] in first place. Without him and his constant encouragement, none of this would have happened. all combis in Berlin, for all the support and the excellent working environment christoph eyrich, for my´sliwska and teaching me all the λ α τ ε χ i know (and then some) Volker Kaibel, for the strongly non-polynomial patience in going through endless details in my manuscripts, and the expertise in shortening or lengthening my proofs Paco Santos, not least for leaving Iñigo a day longer than necessary to come to my exam Ewgenij Gawrilow and Michael Joswig, for polymake the staff and outfitters of the Molotov-cocktail depot, for volatile cakes & coffee Jörg Rambau, for coauthoring Chapter 6 Alexander Schwartz, for reading the manuscript and fixing all the software just in time Bettina Felsner, for making it all work smoothly Lourdes, que tú sabes muy bien todo lo que te tengo que agradecer The place of honor, of course, goes to my PhD advisor, Günter M. Ziegler. He knows better than anyone else just how many wonderful things I learned in these three exciting years in Berlin. Thank you so much for everything, Günter.

13 citations