By Christos A. Athanasiadis, Victor V. Batyrev, Dimitrios I. Dais, Martin Henk, and Francisco Santos

This quantity includes unique study and survey articles stemming from the Euroconference "Algebraic and Geometric Combinatorics". The papers talk about a variety of difficulties that illustrate interactions of combinatorics with different branches of arithmetic, equivalent to commutative algebra, algebraic geometry, convex and discrete geometry, enumerative geometry, and topology of complexes and in part ordered units. one of the subject matters lined are combinatorics of polytopes, lattice polytopes, triangulations and subdivisions, Cohen-Macaulay telephone complexes, monomial beliefs, geometry of toric surfaces, groupoids in combinatorics, Kazhdan-Lusztig combinatorics, and graph colorations. This ebook is geared toward researchers and graduate scholars drawn to quite a few elements of contemporary combinatorial theories

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Other properties of the Euler characteristic can be found in [95, Chapters 3 and 4]. e. ~PA~P8 = 0). A classical open problem is to give necessary and sufficient conditions for the exactness of this complex. M c C o y ' s theorem solves the question when F " = 0. In [49], using Theorem 35, the problem is solved when R [ T ] is integrally closed and rankk A = rankR F - 1. In [23], for the Noetherian case, and [95, p. A. Hermida-Alonso THEOREM 49. S u p p o s e that (C) 0 ~ 9 F, ~o, > Ft-I ~o,_, ---~ 999 -+ Fi ~o, >F0, where t ~ 1, is a c o m p l e x o f finite f r e e R - m o d u l e s .

See [87] and [95, p. 225]. 2. MacRae's invariant and the characteristic polynomial 0 ~ F I lit > F,-I tit-I > ""--+ FI 111 > Fo it() Let > M--+O be a finite free resolution of M and 99 an endomorphism of M. Then there exist endomorphisms qgi" Fi ~ Fi that make the diagram 0 > El 0 > Ft lit > Ft-t "'> Ft-! A. Hermida-Alonso commutative. Considering the associated characteristic sequences it follows that the diagram of R [ T ] - m o d u l e s 0 0 0 0 >... >F~[TI ~'>fo[r] ~O>M[r] >0 0 ~ F, [T] ~' > .

0) t . Hence 27 is feedback equivalent to a linear system of the form ((,, ,)(d)) d2 * * 0 b2 a2 , 0 . 0 Iterating the process one has the following result, see [29]. THEOREM 102. Let 27 -- (A, b) be a single input n-dimensional linear system that is weakly reachable 9 Then there exist nonzero elements dl . . . dn o f R such that E is feedback equivalent to a system o f the f o r m Z A=(A A b ~a)- all a12 "" al,n-I aln dl de a22 "" a2, n - I a2n 0 9 .. 0 0 0 0 ". -. ,,, a,,,, 0 0 Moreover the ordered sequence o f ideals {(dl), (d2) .