arXiv subjects

M. J. Uray

Publications and source records attributed to M. J. Uray.

3 recordsLinked to original sources

arXiv2020

Algebraic number fields and the LLL algorithm

In this paper we analyze the computational costs of various operations and algorithms in algebraic number fields using exact arithmetic. Let $K$ be an algebraic number field. In the first half of the paper, we calculate the running time and the size of the output of many operations in $K$ in terms of the size of the input and the parameters of $K$. We include some earlier results about these, but we go further than them, e.g. we also analyze some $\mathbb{R}$-specific operations in $K$ like less-than comparison. In the second half of the paper, we analyze two algorithms: the Bareiss algorithm, which is an integer-preserving version of the Gaussian elimination, and the LLL algorithm, which is for lattice basis reduction. In both cases, we extend the algorithm from $\mathbb{Z}^n$ to $K^n$, and give a polynomial upper bound on the running time when the computations in $K$ are performed exactly (as opposed to floating-point approximations).

cs.SC↗

arXiv2020

On the expansivity gap of integer polynomials

Expansive polynomials (whose roots are greater than 1 in modulus) often arise in dynamical systems and other computational problems. This paper examines the expansivity gap (the gap between 1 and the smallest modulus of the roots) of these polynomials, assuming that the coefficients are integers. We give lower bounds on the expansivity gap, using the degree and the coefficient size as parameters. We also construct a family of polynomials which indicate the sharpness of these bounds. As a side-result, we present an explicit condition for deciding expansivity of polynomials, which we find superior to the existing recursive methods for our purpose.

math.NT↗

arXiv2020

Characterization of expansive polynomials by special determinants

A polynomial is expansive if all of its roots lie outside the unit circle. We define some special determinants involving the coefficients of a real polynomial and formulate necessary and sufficient conditions for expansivity using these determinants. We show how these conditions can be turned into an algorithm, which, for integer polynomials, avoids exponential coefficient growth. We also examine the question how close the roots of an expansive polynomial can be to the unit circle if the coefficients are integers. We give several lower bounds on this distance in terms of different measures of the polynomial (e.g. its height). The simplest one is derived by Liouville's inequality, but then we improve this result and give different bounds using our special determinants.

math.NT↗