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  • 1
    Electronic Resource
    Electronic Resource
    On the existence of certain smooth toric varieties (1990)
    Springer
    Discrete & computational geometry 5 (1990), S. 255-262 
    Details
    ISSN: 1432-0444
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We prove that the combinatorial types of those cone systems which correspond to complete smooth toric varieties are more restricted than for complete toric varieties: the toric varieties corresponding to essentially all types of cyclic polytopes possess singularities. This yields a negative answer to a problem stated by G. Ewald. Some consequences and problems concerning mathematical programming and the rational cohomology of smooth toric varieties are discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02187789
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  • 2
    Electronic Resource
    Electronic Resource
    Nonrealizability proofs in computational geometry (1990)
    Springer
    Discrete & computational geometry 5 (1990), S. 333-350 
    Details
    ISSN: 1432-0444
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract This paper deals with a class of computational problems in real algebraic geometry. We introduce the concept of final polynomials as a systematic approach to prove nonrealizability for oriented matroids and combinatorial geometries. Hilbert's Nullstellensatz and its real analogue imply that an abstract geometric object is either realizable or it admits a final polynomial. This duality has first been applied by Bokowski in the study of convex polytopes [7] and [11], but in these papers the resulting final polynomials were given without their derivations. It is the objective of the present paper to fill that gap and to describe an algorithm for constructing final polynomials for a large class of nonrealizable chirotopes. We resolve a problem posed in [10] by proving that not every realizable simplicial chirotope admits a solvability sequence. This result shows that there is no easy combinatorial method for proving nonrealizability and thus justifies our final polynomial approach.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02187794
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  • 3
    Electronic Resource
    Electronic Resource
    On the coordinatization of oriented matroids (1986)
    Springer
    Discrete & computational geometry 1 (1986), S. 293-306 
    Details
    ISSN: 1432-0444
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Several important and hard realizability problems of combinatorial geometry can be reduced to the realizability problem of oriented matroids. In this paper we describe a method to find a coordinatization for a large class of realizable cases. This algorithm has been used successfully to decide several geometric realizability problems. It is shown that all realizations found by our algorithm fulfill the isotopy property.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02187702
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  • 4
    Electronic Resource
    Electronic Resource
    Uniform oriented matroids without the isotopy property (1989)
    Springer
    Discrete & computational geometry 4 (1989), S. 97-100 
    Details
    ISSN: 1432-0444
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We give an easy general construction for uniform oriented matroids with disconnected realization space. This disproves the longstanding isotopy conjecture for simple line arrangements or order types in the plane.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02187717
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  • 5
    Electronic Resource
    Electronic Resource
    How to shell a monoid (1998)
    Springer
    Mathematische Annalen 310 (1998), S. 379-393 
    Details
    ISSN: 1432-1807
    Keywords: Mathematics Subject Classification (1991): 14K25, 14K05, 14K10, 14N05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002080050152
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  • 6
    Electronic Resource
    Electronic Resource
    Bounds on degrees of projective schemes (1995)
    Springer
    Mathematische Annalen 302 (1995), S. 417-432 
    Details
    ISSN: 1432-1807
    Keywords: 14A15 ; 14C17 ; 14Q20 ; 13C05 ; 13P10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01444501
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  • 7
    Electronic Resource
    Electronic Resource
    An infinite family of minor-minimal nonrealizable 3-chirotopes (1989)
    Springer
    Mathematische Zeitschrift 200 (1989), S. 583-589 
    Details
    ISSN: 1432-1823
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01160956
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  • 8
    Electronic Resource
    Electronic Resource
    The Cyclohedron Test for Finding Periodic Genes in Time Course Expression Studies (2007)
    Berkeley, Calif. : Berkeley Electronic Press (now: De Gruyter)
    Statistical applications in genetics and molecular biology 6.2007, 1, art21 
    Details
    ISSN: 1544-6115
    Source: Berkeley Electronic Press Academic Journals
    Topics: Biology
    Notes: The problem of finding periodically expressed genes from time course microarray experiments is at the center of numerous efforts to identify the molecular components of biological clocks. We present a new approach to this problem based on the cyclohedron test, which is a rank test inspired by recent advances in algebraic combinatorics. The test has the advantage of being robust to measurement errors, and can be used to ascertain the significance of top-ranked genes. We apply the test to recently published measurements of gene expression during mouse somitogenesis and find 32 genes that collectively are significant. Among these are previously identified periodic genes involved in the Notch/FGF and Wnt signaling pathways, as well as novel candidate genes that may play a role in regulating the segmentation clock. These results confirm that there are an abundance of exceptionally periodic genes expressed during somitogenesis. The emphasis of this paper is on the statistics and combinatorics that underlie the cyclohedron test and its implementation within a multiple testing framework.
    Type of Medium: Electronic Resource
    URL: http://www.bepress.com/sagmb/vol6/iss1/art21
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  • 9
    Electronic Resource
    Electronic Resource
    Computing combinatorial decompositions of rings (1991)
    Springer
    Combinatorica 11 (1991), S. 275-293 
    Details
    ISSN: 1439-6912
    Keywords: 68 C 20 ; 13-04 ; 20 C 05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Using Buchberger's Gröbner basis theory, we obtain explicit algorithms for computing Stanley decompositions, Rees decompositions and Hironaka decompositions of commutative Noetherian rings. These decompositions are of considerable importance in combinatorics, in particular in the theory of Cohen-Macaulay complexes. We discuss several applications of our methods, including a new algorithm for finding primary and secondary invariants of finite group actions on polynomial rings.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01205079
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  • 10
    Electronic Resource
    Electronic Resource
    Gröbner bases and triangulations of the second hypersimplex (1995)
    Springer
    Combinatorica 15 (1995), S. 409-424 
    Details
    ISSN: 1439-6912
    Keywords: 52 B 12 ; 13 P 10 ; 05 C 50
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The algebraic technique of Gröbner bases is applied to study triangulations of the second hypersimplex Δ(2,n). We present a quadratic Gröbner basis for the associated toric idealK(K n ). The simplices in the resulting triangulation of Δ(2,n) have unit volume, and they are indexed by subgraphs which are linear thrackles [28] with respect to a circular embedding ofK n . Forn≥6 the number of distinct initial ideals ofI(K n ) exceeds the number of regular triangulations of Δ(2,n); more precisely, the secondary polytope of Δ(2,n) equals the state polytope ofI(K n ) forn≤5 but not forn≥6. We also construct a non-regular triangulation of Δ(2,n) forn≥9. We determine an explicit universal Gröbner basis ofI(K n ) forn≤8. Potential applications in combinatorial optimization and random generation of graphs are indicated.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01299745
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