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| In mathematics, the '''Dehn–Sommerville equations''' are a complete set of linear relations between the numbers of faces of different dimension of a [[simplicial polytope]]. For polytopes of dimension 4 and 5, they were found by [[Max Dehn]] in 1905. Their general form was established by [[Duncan MacLaren Young Sommerville|Duncan Sommerville]] in 1927. The Dehn–Sommerville equations can be restated as a symmetry condition for the [[h-vector|''h''-vector'']] of the simplicial polytope and this has become the standard formulation in recent combinatorics literature. By duality, analogous equations hold for [[simple polytope]]s.
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| == Statement ==
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| Let ''P'' be a ''d''-dimensional [[simplicial polytope]]. For ''i'' = 0, 1, ..., ''d''−1, let ''f''<sub>''i''</sub> denote the number of ''i''-dimensional [[face (geometry)|faces]] of ''P''. The sequence
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| : <math> f(P)=(f_0,f_1,\ldots,f_{d-1}) </math>
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| is called the '''''f''-vector''' of the polytope ''P''. Additionally, set
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| : <math> f_{-1}=1, f_d=1. </math>
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| Then for any ''k'' = −1, 0, …, ''d''−2, the following '''Dehn–Sommerville equation''' holds:
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| :<math>\sum_{j=k}^{d-1} (-1)^{j} \binom{j+1}{k+1} f_j = (-1)^{d-1}f_k. </math>
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| When ''k'' = −1, it expresses the fact that [[Euler characteristic]] of a (''d'' − 1)-dimensional [[simplicial sphere]] is equal to 1 + (−1)<sup>''d''−1</sup>.
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| Dehn–Sommerville equations with different ''k'' are not independent. There are several ways to choose a maximal independent subset consisting of <math>\left[\frac{d+1}{2}\right]</math> equations. If ''d'' is even then the equations with ''k'' = 0, 2, 4, …, ''d''−2 are independent. Another independent set consists of the equations with ''k'' = −1, 1, 3, …, ''d''−3. If ''d'' is odd then the equations with ''k'' = −1, 1, 3, …, ''d''−2 form one independent set and the equations with ''k'' = −1, 0, 2, 4, …, ''d''−3 form another.
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| == Equivalent formulations ==
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| {{main|h-vector}}
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| Sommerville found a different way to state these equations:
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| <math> \sum_{i=-1}^{k-1}(-1)^{d+i}\binom{d-i-1}{d-k} f_i =
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| \sum_{i=-1}^{d-k-1}(-1)^{i}\binom{d-i-1}{k} f_i, </math>
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| where 0 ≤ k ≤ ½(d−1). This can be further facilitated introducing the notion of ''h''-vector of ''P''. For ''k'' = 0, 1, …, ''d'', let
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| : <math> h_k = \sum_{i=0}^k (-1)^{k-i}\binom{d-i}{k-i}f_{i-1}. </math> | |
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| The sequence
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| : <math>h(P)=(h_0,h_1,\ldots,h_d)</math>
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| is called the [[h-vector|''h''-vector]] of ''P''. The ''f''-vector and the ''h''-vector uniquely determine each other through the relation
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| : <math> \sum_{i=0}^{d}f_{i-1}(t-1)^{d-i}=\sum_{k=0}^{d}h_{k}t^{d-k}. </math>
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| Then the Dehn–Sommerville equations can be restated simply as
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| : <math> h_k = h_{d-k} \quad\textrm{for}\quad 0\leq k\leq d. </math>
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| The equations with 0 ≤ k ≤ ½(d−1) are independent, and the others are manifestly equivalent to them.
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| [[Richard P. Stanley|Richard Stanley]] gave an interpretation of the components of the ''h''-vector of a simplicial convex polytope ''P'' in terms of the [[projective variety|projective]] [[toric variety]] ''X'' associated with (the dual of) ''P''. Namely, they are the dimensions of the even [[intersection cohomology]] groups of ''X'':
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| : <math> h_k=\operatorname{dim}_{\mathbb{Q}}\operatorname{IH}^{2k}(X,\mathbb{Q}) </math>
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| (the odd [[intersection cohomology]] groups of ''X'' are all zero). In this language, the last form of the Dehn–Sommerville equations, the symmetry of the ''h''-vector, is a manifestation of the [[Poincaré duality]] in the intersection cohomology of ''X''.
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| ==References==
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| * [[Branko Grünbaum]], ''Convex polytopes''. Second edition. Graduate Texts in Mathematics, 221, Springer, 2003 ISBN 0-387-00424-6
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| * [[Richard P. Stanley|Richard Stanley]], ''Combinatorics and commutative algebra''. Second edition. Progress in Mathematics, 41. Birkhäuser Boston, Inc., Boston, MA, 1996. x+164 pp. ISBN 0-8176-3836-9
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| * [[Günter M. Ziegler|G. Ziegler]], ''Lectures on Polytopes'', [[Springer-Verlag|Springer]], 1998. ISBN 0-387-94365-X
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| {{DEFAULTSORT:Dehn-Sommerville equations}}
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| [[Category:Polyhedral combinatorics]]
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Hi there, I am Yoshiko Villareal but I never truly liked that title. Playing croquet is some thing I will by no means give up. Years ago we moved to Arizona but my wife desires us to move. His day occupation is a cashier and his salary has been truly fulfilling.
Also visit my website extended car warranty (mouse click the next web page)