We show that the ring of multisymmetric functions over a commutative ring is isomorphic to the ring generated by the coefficients of the characteristic polynomial of polynomials in commuting generic matrices. As a consequence we give a surjection from the ring of invariants of several matrices to the ring of multisymmetric functions generalizing a classical result of H.Weyl and F.Junker. We als...

We describe the set of characteristic polynomials of abelian varieties of dimension 3 over finite fields.

An analytical function of a matrix with repeated eigenvalues is expressed in terms of constituent matrices. Two approaches to computing the constituent matrices are then presented. For a special case of a companion matrix, the computation can be greatly simplified. 1.FORMULATION For a given nxn constant matrix A, eigenvalues are determined by solving the characteristic polynomial of degree n, d...

We generalize the construction of connected branched polymers and the notion of the volume of the space of connected branched polymers studied by Brydges and Imbrie [BI], and Kenyon and Winkler [KW] to any hyperplane arrangement A. The volume of the resulting configuration space of connected branched polymers associated to the hyperplane arrangement A is expressed through the value of the chara...

We describe an algorithm to compute the zeta function of any Cab curve over any finite field Fpn . The algorithm computes a p-adic approximation of the characteristic polynomial of Frobenius by computing in the Monsky-Washnitzer cohomology of the curve and thus generalizes Kedlaya’s algorithm for hyperelliptic curves. For fixed p the asymptotic running time for a Cab curve of genus g over Fpn i...

The paper concerns the problem of decomposition of a large-scale linear dynamic system into two subsystems. An equivalent problem is to split the characteristic polynomial of the original system into two polynomials of lower degrees. Conditions are found concerning the coefficients of the original polynomial which must be fulfilled for its factorization. It is proved that knowledge of only one ...

This paper clarifies the hierarchical structure of the sharp constants for the discrete Sobolev inequality on a weighted complete graph. To this end, we introduce a generalized-graph Laplacian A = I − B on the graph, and investigate two types of discrete Sobolev inequalities. The sharp constants C0(N; a) and C0(N) were calculated through the Green matrix G(a) = (A + aI)−1(0 < a < ∞) and the pse...

The Lax pair of the Ruijsenaars-Schneider model with interaction potential of trigonometric type based on Dn Lie algebra is presented. We give a general form for the Lax pair and prove partial results for small n. Liouville integrability of the corresponding system follows a series of involutive Hamiltonians generated by the characteristic polynomial of the Lax matrix. The rational case appears...

A matrix D is diagonal if all its off-diagonal entries are zero. If D is diagonal, then its eigenvalues are the diagonal entries, and the characteristic polynomial of D is fD(x) = ∏i=1(x−dii), where dii is the (i, i) diagonal entry of D. A matrix A is diagonalisable if there is an invertible matrix Q such that QAQ−1 is diagonal. Note that A and QAQ−1 always have the same eigenvalues and the sam...

We analyze the precision of the characteristic polynomial χ(A) of an n × n p-adic matrix A using differential precision methods developed previously. When A is integral with precision O(p), we give a criterion (checkable in time O (̃n)) for χ(A) to have precision exactly O(p). We also give a O (̃n) algorithm for determining the optimal precision when the criterion is not satisfied, and give examp...