TAILIEUCHUNG - Báo cáo toán học: " Sweeping the cd-Index and the Toric h-Vector"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Sweeping the cd-Index and the Toric h-Vector. | Sweeping the cd-Index and the Toric h-Vector Carl W. Lee Department of Mathematics University of Kentucky Lexington Kentucky USA lee@ Submitted Nov 24 2009 Accepted Mar 15 2011 Published Mar 24 2011 Mathematics Subject Classification 52B05 Abstract We derive formulas for the cd-index and the toric h-vector of a convex polytope P from a sweeping by a hyperplane. These arise from interpreting the corresponding S-shelling of the dual of P. We describe a partition of the faces of the complete truncation of P to reflect explicitly the nonnegativity of its cd-index and what its components are counting. One corollary is a quick way to compute the toric h-vector directly from the cd-index that turns out to be an immediate consequence of formulas of Bayer and Ehrenborg. We also propose an extended toric h-vector that fully captures the information in the flag h-vector. 1 Introduction By sweeping a hyperplane across a simple convex d-polytope P the h-vector h P ho . hd of its dual can be computed the edges in P are oriented in the direction of the sweep and hj equals the number of vertices of outdegree i. Moreover the nonempty faces of P can be partitioned to explicitly reflect the formula for the h-vector. For a general convex polytope in place of the h-vector one often considers the flag f-vector and flag h-vector as well their encoding into the cd-index and also the toric h-vector which does not contain the full information of the flag h-vector . In this paper we derive formulas for the cd-index and for the toric h-vector of a convex polytope P from a sweeping of P Theorems 2 3 4 and 6 . These arise from interpreting the corresponding S-shelling 14 of the dual of P. We describe a partition of the faces of the complete truncation of P to provide an interpretation of what the components of the cd-index are counting Theorem 1 and Corollary 1 . One corollary Theorem 5 is a quick way to compute the toric h-vector directly from the cd-index that turns out to be an .

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