Relating pp-adic eigenvalues and the local Smith normal form

Conditions are established under which the $p$-adic valuations of the invariant factors (diagonal entries of the Smith form) of an integer matrix are equal to the $p$-adic valuations of the eigenvalues. It is then shown that this correspondence is the typical case for "most" matrices; precise density bounds are given for when the property holds, as well as easy transformations to this typical case.Comment: To appear in Linear Algebra and Its Application

Computing GCRDs of Approximate Differential Polynomials

Differential (Ore) type polynomials with approximate polynomial coefficients are introduced. These provide a useful representation of approximate differential operators with a strong algebraic structure, which has been used successfully in the exact, symbolic, setting. We then present an algorithm for the approximate Greatest Common Right Divisor (GCRD) of two approximate differential polynomials, which intuitively is the differential operator whose solutions are those common to the two inputs operators. More formally, given approximate differential polynomials $f$ and $g$, we show how to find "nearby" polynomials $\widetilde f$ and $\widetilde g$ which have a non-trivial GCRD. Here "nearby" is under a suitably defined norm. The algorithm is a generalization of the SVD-based method of Corless et al. (1995) for the approximate GCD of regular polynomials. We work on an appropriately "linearized" differential Sylvester matrix, to which we apply a block SVD. The algorithm has been implemented in Maple and a demonstration of its robustness is presented.Comment: To appear, Workshop on Symbolic-Numeric Computing (SNC'14) July 201

On computing the Hermite form of a matrix of differential polynomials

Given an n x n matrix over the ring of differential polynomials F(t)[\D;\delta], we show how to compute the Hermite form H of A, and a unimodular matrix U such that UA=H. The algorithm requires a polynomial number of operations in terms of n, deg_D(A), and deg_t(A). When F is the field of rational numbers, it also requires time polynomial in the bit-length of the coefficients

Computing sparse multiples of polynomials

We consider the problem of finding a sparse multiple of a polynomial. Given f in F[x] of degree d over a field F, and a desired sparsity t, our goal is to determine if there exists a multiple h in F[x] of f such that h has at most t non-zero terms, and if so, to find such an h. When F=Q and t is constant, we give a polynomial-time algorithm in d and the size of coefficients in h. When F is a finite field, we show that the problem is at least as hard as determining the multiplicative order of elements in an extension field of F (a problem thought to have complexity similar to that of factoring integers), and this lower bound is tight when t=2.Comment: Extended abstract appears in Proc. ISAAC 2010, pp. 266-278, LNCS 650