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A well-known cancellation problem of Zariski asks when, for two given domains (fields) and over a field k, a k-isomorphism of () and () implies a k-isomorphism of and . The main results of this article give affirmative answer to the two low-dimensional cases of this problem:1. Let K be an affine field over an algebraically closed field k of any characteristic. Suppose , then .2. Let M be a 3-dimensional affine algebraic variety over an algebraically closed field k of any characteristic. Let be the coordinate ring of M. Suppose , then , where is the field of fractions of A.In the case of zero characteristic these results were obtained by Kang in [Ming-chang Kang, A note on the birational cancellation problem, J. Pure Appl. Algebra 77 (1992) 141–154; Ming-chang Kang, The cancellation problem, J. Pure Appl. Algebra 47 (1987) 165–171]. However, the case of finite characteristic is first settled in this article, that answered the questions proposed by Kang in [Ming-chang Kang, A note on the birational cancellation problem, J. Pure Appl. Algebra 77 (1992) 141–154; Ming-chang Kang, The cancellation problem, J. Pure Appl. Algebra 47 (1987) 165–171]. 相似文献
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Sarah Crown Rundell 《Journal of Combinatorial Theory, Series A》2012,119(5):1095-1109
In this paper, we study the homology of the coloring complex and the cyclic coloring complex of a complete k-uniform hypergraph. We show that the coloring complex of a complete k-uniform hypergraph is shellable, and we determine the rank of its unique nontrivial homology group in terms of its chromatic polynomial. We also show that the dimension of the homology group of the cyclic coloring complex of a complete k-uniform hypergraph is given by a binomial coefficient. Further, we discuss a complex whose r-faces consist of all ordered set partitions where none of the contain a hyperedge of the complete k-uniform hypergraph H and where . It is shown that the dimensions of the homology groups of this complex are given by binomial coefficients. As a consequence, this result gives the dimensions of the multilinear parts of the cyclic homology groups of . 相似文献
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For a set of distances a set A in the plane is called D-avoiding if no pair of points of A is at distance for some i. We show that the density of A is exponentially small in k provided the ratios are all small enough. We also show that there exists a largest D-avoiding set, and give an algorithm to compute the maximum density of a D-avoiding set for any D. 相似文献
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Jean-Pierre Tignol 《Comptes Rendus Mathematique》2006,342(2):89-92
Let A be a central simple algebra of degree 4 over a field k of characteristic 2 and let be the quadratic form on A given by the second coefficient of the reduced characteristic polynomial. We show that A uniquely determines a 2-fold Pfister form and a 4-fold Pfister form such that in the Witt group of k, where is the form . The form is the norm form of the quaternion algebra Brauer-equivalent to , and is hyperbolic if and only if A is cyclic. To cite this article: J.-P. Tignol, C. R. Acad. Sci. Paris, Ser. I 342 (2006). 相似文献
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A subgroup H of a group G is said to permute with the subgroup K of G if . Subgroups H and K are mutually permutable (totally permutable) in G if every subgroup of H permutes with K and every subgroup of K permutes with H (if every subgroup of H permutes with every subgroup of K). If H and K are mutually permutable and , then H and K are totally permutable. A subgroup H of G is S-permutable in G if H permutes with every Sylow subgroup of G. A group G is called a PST-group if S-permutability is a transitive relation in G. Let be the set of prime divisors of the order of a finite group G with the set of prime divisors of the order of the normal subgroup N of G. A set of Sylow subgroups , , form a strong Sylow system with respect to N if is a mutually permutable product for all and . We show that a finite group G is a solvable PST-group if and only if it has a normal subgroup N such that is nilpotent and G has a strong Sylow system with respect to N. It is also shown that G is a solvable PST-group if and only if G has a normal solvable PST-subgroup N and is a solvable PST-group. 相似文献
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If is a tree then – T denotes the additive hereditary property consisting of all graphs that does not contain T as a subgraph. For an arbitrary vertex v of T we deal with a partition of T into two trees , , so that , , , , and . We call such a partition a of T. We study the following em: Given a graph G belonging to –T. Is it true that for any -partition , of T there exists a partition of the vertices of G such that and ? This problem provides a natural generalization of Δ-partition problem studied by L. Lovász ([L. Lovász, On decomposition of graphs. Studia Sci. Math. Hungar. 1 (1966) 237–238]) and Path Partition Conjecture formulated by P. Mihók ([P. Mihók, Problem 4, in: M. Borowiecki, Z. Skupien (Eds.), Graphs, Hypergraphs and Matroids, Zielona Góra, 1985, p. 86]). We present some partial results and a contribution to the Path Kernel Conjecture that was formulated with connection to Path Partition Conjecture. 相似文献
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Let G be a non-Abelian, connected, nilpotent Lie group. Then there exist and such that , contrary to what happens for the group . Moreover, the set of zero divisors is a total subset of . This result is first proven for the Heisenberg group where it is based on the existence of non-trivial Schwartz functions f satisfying for . To cite this article: J. Ludwig et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006). 相似文献
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Marc Chaperon Santiago López de Medrano José Lino Samaniego 《Comptes Rendus Mathematique》2005,340(11):827-832
Under fairly general hypotheses, we investigate by elementary methods the structure of the p-periodic orbits of a family of transformations near when and has a simple eigenvalue which is a primitive p-th root of unity. To cite this article: M. Chaperon et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005). 相似文献
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