It is shown that for positive real numbers 0 < λ 1 < · · · < λn, 1 β(λ i ,λ j ) , where β(·, ·) denotes the beta function, is infinitely divisible and totally positive. For 1 β(i,j) , the Cholesky decomposition and successive elementary bidiagonal decomposition are computed. Let w(n) be the nth Bell number. It is proved that [w(i + j)] is a totally positive matrix but is infinitely divisible only upto order 4. It is also shown that the symmetrized Stirling matrices are totally positive.
In this article, we compile the work done by various mathematicians on the topic of the fixed divisor of a polynomial. This article explains most of the results concisely and is intended to be an exhaustive survey. We present the results on fixed divisors in various algebraic settings as well as the applications of fixed divisors to various algebraic and number theoretic problems. The work is presented in an orderly fashion so as to start from the simplest case of Z, progressively leading up to the case of Dedekind domains. We also ask a few open questions according to their context, which may give impetus to the reader to work further in this direction. We describe various bounds for fixed divisors as well as the connection of fixed divisors with different notions in the ring of integer-valued polynomials. Finally, we suggest how the generalization of the ring of integer-valued polynomials in the case of the ring of n × n matrices over Z (or Dedekind domain) could lead to the generalization of fixed divisors in that setting.keywords Fixed divisors, Generalized factorials, Generalized factorials in several variables, Common factor of indices, Factoring of prime ideals, Integer valued polynomials NotationsWe fix the notations for the whole paper. R = Integral Domain K = Field of fractions of R N (I) = Cardinality of R/I (Norm of an ideal I ⊆ R) W = {0, 1, 2, 3, . . .} A[x] = Ring of polynomials in n variables (= A[x 1 , . . . , x n ]) with coefficients in the ring A S = Arbitrary (or given) subset of R n such that no non-zero polynomial in K[x] maps it to zero S = S in case when n = 1 Int(S, R) = Polynomials in K[x] mapping S back to R ν k (S) = Bhargava's (generalized) factorial of index k k! S = k th generalized factorial in several variables M m (S) = Set of all m × m matrices with entries in S p = positive prime number Z p = p-adic integers ord p (n) = p-adic ordinal (valuation) of n ∈ Z.
In this article we study the irreducibility of polynomials of the form x n + ǫ 1 x m + p k ǫ 2 , p being a prime number. We will show that they are irreducible for m = 1. We have also provided the cyclotomic factors and reducibility criterion for trinomials of the formThis corrects few of the existing results of W. Ljuggren's on x n + ǫ 1 x m + ǫ 2 .
Let K be a finite extension of a characteristic zero field F. We say that a pair of n × n matrices (A,In this paper, we identify the smallest order circulant matrix representation for any subfield of a cyclotomic field. Furthermore, if p is a prime and K is a subfield of the p-th cyclotomic field, then we obtain a zero-one circulant matrix A of size p × p such that (A, J) represents K, where J is the matrix with all entries 1. In case, the integer n has at most two distinct prime factors, we find the smallest order 0, 1-companion matrix that represents the n-th cyclotomic field. We also find bounds on the size of such companion matrices when n has more than two prime factors.
Abstract. For a fixed positive integer n, let Wn be the permutation matrix corresponding to the permutation 1 2 · · · n − 1 n 2 3 · · · n 1 . In this article, it is shown that a symmetric matrix with rational entries is circulant if, and only if, it lies in the subalgebra ofn . On the basis of this, the singularity of graphs on n-vertices is characterized algebraically. This characterization is then extended to characterize the singularity of directed circulant graphs. The kth power matrix W k n +W −k n defines a circulant graph C k n . The results above are then applied to characterize its singularity, and that of its complement graph. The digraph Cr,s,t is defined as that whose adjacency matrix is circulant circ(a), where a is a vector with the first r-components equal to 1, and the next s, t and n − (r + s + t) components equal to zero, one, and zero respectively. The singularity of this digraph (graph), under certain conditions, is also shown to depend algebraically upon these parameters. A slight generalization of these graphs are also studied.
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