TAILIEUCHUNG - Báo cáo toán học: "Generating functions attached to some infinite matrices"

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: Generating functions attached to some infinite matrices. | Generating functions attached to some infinite matrices Paul Monsky Brandeis University Waltham MA 02454-9110 USA monsky@ Submitted Aug 9 2010 Accepted Dec 13 2010 Published Jan 5 2011 Mathematics Subject Classification 05E40 and 05A15 Abstract Let V be an infinite matrix with rows and columns indexed by the positive integers and entries in a field F. Suppose that Vij only depends on i j and is 0 for i j large. Then Vn is defined for all n and one has a generating function G 52 a1 1 Vn zn. Ira Gessel has shown that G is algebraic over F z . We extend his result allowing Vij for fixed i j to be eventually periodic in i rather than constant. This result and some variants of it that we prove will have applications to Hilbert-Kunz theory. 1 Introduction Throughout A is a ring with identity element 1. Suppose that wi j i and j ranging over the positive integers are in A and that wi j 0 whenever i j lies outside a fixed finite set. Then if W is the infinite matrix wij one may speak of Wn for all n 0 and one gets a generating function G W 22 anzn in A z where an is the 1 1 entry in the matrix Wn. We shall prove Theorem I. Suppose that Wij 0 if i j ị 1 0 1 and that wi 1j 1 wi j unless i j 1. Suppose further that A Ms F F a field so that G W may be viewed as an s by s matrix with entries in F z . Then these matrix entries are algebraic over F z . Corollary. Let F be a field and Vij i and j ranging over the positive integers be in F. Suppose a Vij 0 whenever i j lies outside a fixed finite set. b For fixed r in Z viii r is an eventually periodic function of i. Then if V is the matrix vi j the generating function G V is algebraic over F z . THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P5 1 Proof. To derive the corollary we choose s so that 1 Vij 0 whenever i s and j 2s or j s and i 2s. 2 vi sj s vi j whenever i j s 2. We then write the initial 2s by 2s block in V as DC with A B C D in Ms F . Our choice of s tells us that V is built out of s by s blocks where .

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