相关论文: Counting integral matrices with a given characteri…
We obtain an effective analytic formula, with explicit constants, for the number of distinct irreducible factors of a polynomial $f \in \mathbb{Z}[x]$. We use an explicit version of Mertens' theorem for number fields to estimate a related…
We discuss the problem of counting {\em incidence matrices}, i.e. zero-one matrices with no zero rows or columns. Using different approaches we give three different proofs for the leading asymptotics for the number of matrices with $n$ ones…
In this paper we obtained the formula for the number of irreducible polynomials with degree $n$ over finite fields of characteristic two with given trace and subtrace. This formula is a generalization of the result of Cattell et al.(2003)…
We obtain large n asymptotics for products of powers of the absolute values of the characteristic polynomials in the Gaussian Unitary Ensemble of n\times n matrices. Our results can also be interpreted as asymptotics of the determinant of a…
We prove an asymptotic formula for the number of fixed rank matrices with integer coefficients over a number field K/Q and bounded norm. As an application, we derive an approximate Rogers integral formula for discrete sets of module…
We consider the set $\mathcal{M}_n(\mathbb{Z}; H)$ of $n\times n$-matrices with integer elements of size at most $H$ and obtain upper and lower bounds on the number of distinct irreducible characteristic polynomials which correspond to…
This paper proves that the characteristic polynomial is a complete unitary invariant for pairs of projection matrices. Some special cases involving three or more projections are also considered.
We count with a smooth weight the number of $2 \times 2$ integer matrices with a fixed characteristic polynomial with a main term and an error term using bounds for sums of Weyl sums for quadratic roots.
Starting from Montgomery's conjecture, there has been a substantial interest on the connections of random matrix theory and the theory of L-functions. In particular, moments of characteristic polynomials of random matrices have been…
We use spectral analysis to give an asymptotic formula for the number of matrices in SL(n, Z) of height at most T with strong error terms, far beyond the previous known, both for small and large rank.
In this paper we study the asymptotic behavior of the maximum magnitude of a complex random polynomial with i.i.d. uniformly distributed random roots on the unit circle. More specifically, let $\{n_k\}_{k=1}^{\infty}$ be an infinite…
We study characteristic polynomials of symmetric matrices with entries ${i+j\choose i}$ the binomial coefficients, over finite fields.
We study the characteristic polynomial of random permutation matrices following some measures which are invariant by conjugation, including Ewens' measures which are one-parameter deformations of the uniform distribution on the permutation…
We present the asymptotically fastest known algorithms for some basic problems on univariate polynomial matrices: rank, nullspace, determinant, generic inverse, reduced form. We show that they essentially can be reduced to two computer…
This work is in a stream initiated by a paper of Killip and Simon [Ann. of Math. (2003)]. Using methods of Functional Analysis and the classical Szeg\"o Theorem we prove sum rule identities in a very general form. Then, we apply the result…
We present a concentration inequality for linear functionals of noncommutative polynomials in random matrices. Our hypotheses cover most standard ensembles, including Gaussian matrices, matrices with independent uniformly bounded entries…
We identify a relationship between a random walk on a certain Euclidean lattice and incidence matrices of balanced incomplete block designs. We then compute the return probability of the random walk and use it to obtain the asymptotic…
In this text we study the regularity of matrices with special polynomial entries. Barring some mild conditions we show that these matrices are regular if a natural limit size is not exceeded. The proof draws connections to generalized…
We calculate joint moments of the characteristic polynomial of a random unitary matrix from the circular unitary ensemble and its derivative in the case that the power in the moments is an odd positive integer. The calculations are carried…
We calculate the autocorrelation functions (or shifted moments) of the characteristic polynomials of matrices drawn uniformly with respect to Haar measure from the groups U(N), O(2N) and USp(2N). In each case the result can be expressed in…