Related papers: Efficient Computation of the Characteristic Polyno…
We discuss rather systematically the principle, implicit in earlier works, that for a "random" element in an arithmetic subgroup of a (split, say) reductive algebraic group over a number field, the splitting field of the characteristic…
We consider the problem of finding a dense submatrix of a matrix with i.i.d. Gaussian entries, where density is measured by average value. This problem arose from practical applications in biology and social sciences…
We present an efficient computational representation of central simple algebras using Brauer factor sets. Using this representation and polynomial quantum algorithms for number theoretical tasks such as factoring and $S$-unit group…
In this work we describe an efficient implementation of a hierarchy of algorithms for Gaussian elimination upon dense matrices over the field with two elements. We discuss both well-known and new algorithms as well as our implementations in…
We introduce the characteristic numbers and the chromatic polynomial of a tensor. Our approach generalizes and unifies the chromatic polynomial of a graph and of a matroid, characteristic numbers of quadrics in Schubert calculus, Betti…
Form an $n \times n$ matrix by drawing entries independently from $\{\pm1\}$ (or another fixed nontrivial finitely supported distribution in $\mathbf{Z}$) and let $\phi$ be the characteristic polynomial. Conditionally on the extended…
By using the Poisson formula for resultants and the variants of chip-firing game on graphs, we provide a combinatorial method for computing a class of of resultants, i.e. the characteristic polynomials of the adjacency tensors of starlike…
An improved characteristic set algorithm for solving Boolean polynomial systems is proposed. This algorithm is based on the idea of converting all the polynomials into monic ones by zero decomposition, and using additions to obtain…
We give an efficient algorithm to enumerate all sets of $r\ge 1$ quadratic polynomials over a finite field, which remain irreducible under iterations and compositions.
We present a new, practical algorithm for computing the determinant of a non-singular dense, uniform matrix over Z; the aim is to achieve better practical efficiency, which is always at least as good as currently known methods. The…
In this article we describe a new inductive approach to compute the chromatic polynomial of simple graphs and the characteristic polynomial of central hyperplane arrangements.
In this paper we study the problem of computing a Kolchin characteristic set of a radical differential ideal. The central part of the article is the presentation of algorithms solving this problem in two principal cases: for ordinary…
We give an easy algorithm which calculates the generating function of the cocharacter sequence of the T-ideal of the polynomial identities of the algebra of upper triangular matrices over a field of characteristic zero. Applying this…
This note provides new methods for constructing quadratic nonresidues in finite fields of characteristic p. It will be shown that there is an effective deterministic polynomial time algorithm for constructing quadratic nonresidues in finite…
In this work we relate the deterministic complexity of factoring polynomials (over finite fields) to certain combinatorial objects we call m-schemes. We extend the known conditional deterministic subexponential time polynomial factoring…
We present an efficient numerical method for computing Hamiltonian matrix elements between non-orthogonal Slater determinants, focusing on the most time-consuming component of the calculation that involves a sparse array. In the usual case…
We propose a new definition of characteristic polynomials of tensors based on a partition function of Grassmann variables. This new notion of characteristic polynomial addresses general tensors including totally antisymmetric ones, but not…
The computation of the sparse principal component of a matrix is equivalent to the identification of its principal submatrix with the largest maximum eigenvalue. Finding this optimal submatrix is what renders the problem…
We compute the leading order asymptotic of the maximum of the characteristic polynomial for i.i.d. matrices with real or complex entries. In particular, this result is new even for real Ginibre matrices, which was left as an open problem in…
A random matrix is likely to be well conditioned, and motivated by this well known property we employ random matrix multipliers to advance some fundamental matrix computations. This includes numerical stabilization of Gaussian elimination…