Why the characteristic polynomial factors
نویسندگان
چکیده
منابع مشابه
Why the Characteristic Polynomial Factors
We survey three methods for proving that the characteristic polynomial of a finite ranked lattice factors over the nonnegative integers and indicate how they have evolved recently. The first technique uses geometric ideas and is based on Zaslavsky’s theory of signed graphs. The second approach is algebraic and employs results of Saito and Terao about free hyperplane arrangements. Finally we con...
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A [ An−1 + p1A n−2 + · · ·+ pn−1 In ] = −pn In . Since A is nonsingular, pn = (−1)n det(A) 6= 0; thus the result follows. Newton’s Identity. Let λ1, λ2, . . . , λn be the roots of the polynomial K(λ) = λ + p1λ n−1 + p2λ n−2 + · · · · · ·+ pn−1λ+ pn. If sk = λ k 1 + λ k 2 + · · ·+ λn, then pk = − 1 k (sk + sk−1 p1 + sk−2 p2 + · · ·+ s2 pk−2p1 + s1 pk−1) . Proof. From K(λ) = (λ − λ1)(λ − λ2) . . ...
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We introduce a new method for showing that the roots of the characteristic polynomial of a finite lattice are all nonnegative integers. Our main theorem gives two simple conditions under which the characteristic polynomial factors in this way. We will see that Stanley’s Supersolvability Theorem is a corollary of this result. We can also use this method to demonstrate the factorization of a poly...
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ABSTRACT. Suppose G is a graph, A(G) its adjacency matrix and f(G, x)=x^n+a_(n-1)x^(n-1)+... is the characteristic polynomial of G. The matching polynomial of G is defined as M(G, x) = x^n-m(G,1)x^(n-2) + ... where m(G,k) is the number of k-matchings in G. In this paper, we determine the relationship between 2k-th coefficient of characteristic polynomial, a_(2k), and k-th coefficient of matchin...
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ژورنال
عنوان ژورنال: Bulletin of the American Mathematical Society
سال: 1999
ISSN: 0273-0979
DOI: 10.1090/s0273-0979-99-00775-2