Related papers: Near NP-Completeness for Detecting p-adic Rational…

Sumsets are central objects in additive combinatorics. In 2007, Granville asked whether one can efficiently recognize whether a given set $S$ is a sumset, i.e. whether there is a set $A$ such that $A+A=S$. Granville suggested an algorithm…

We study the repartition of the roots of a random p-adic polynomial in an algebraic closure of Qp.We prove that the mean number of roots generating a fixed finite extension K of Qp depends mostly on the discriminant of K, an extension…

We investigate algebraic and arithmetic properties of a class of sequences of sparse polynomials that have binomial coefficients both as exponents and as coefficients. In addition to divisibility and irreducibility results we also consider…

Bernstein-Sato polynomial of a hypersurface is an important object with numerous applications. It is known, that it is complicated to obtain it computationally, as a number of open questions and challenges indicate. In this paper we propose…

We investigate, using the weighted linear sieve, the distribution of almost-primes among the residue classes (mod p) that generate the multiplicative group of reduced residue classes. We are concerned with finding an upper bound for the…

We develop a theory of arithmetic Newton polygons of higher order, that provides the factorization of a separable polynomial over a $p$-adic field, together with relevant arithmetic information about the fields generated by the irreducible…

We study monic univariate polynomials whose coefficients are analytic functions of a real variable and whose roots lie in a specified analytic curve. These include characteristic polynomials of unitary and hermitian matrices whose entries…

Suppose $k,p\!\in\!\mathbb{N}$ with $p$ prime and $f\!\in\!\mathbb{Z}[x]$ is a univariate polynomial with degree $d$ and all coefficients having absolute value less than $p^k$. We give a Las Vegas randomized algorithm that computes the…

This paper presents the following results on sets that are complete for NP. 1. If there is a problem in NP that requires exponential time at almost all lengths, then every many-one NP-complete set is complete under length-increasing…

How many operations do we need on the average to compute an approximate root of a random Gaussian polynomial system? Beyond Smale's 17th problem that asked whether a polynomial bound is possible, we prove a quasi-optimal bound $\text{(input…

We consider the problem of evaluating certain exponential sums. These sums take the form $\sum_{x_1,...,x_n \in Z_N} e^{f(x_1,...,x_n) {2 \pi i / N}} $, where each x_i is summed over a ring Z_N, and f(x_1,...,x_n) is a multivariate…

This paper is devoted to finding solutions of polynomial equations in roots of unity. It was conjectured by S. Lang and proved by M. Laurent that all such solutions can be described in terms of a finite number of parametric families called…

Given a polynomial $x \in {\mathbb R}^n \mapsto p(x)$ in $n=2$ variables, a symbolic-numerical algorithm is first described for detecting whether the connected component of the plane sublevel set ${\mathcal P} = \{x : p(x) \geq 0\}$…

We investigate two well known dynamical systems that are designed to find roots of univariate polynomials by iteration: the methods known by Newton and by Ehrlich-Aberth. Both are known to have found all roots of high degree polynomials…

In this note we consider roots of multivariate polynomials over a finite grid. When given information on the leading monomial with respect to a fixed monomial ordering, the footprint bound [8, 5] provides us with an upper bound on the…

We exhibit a probabilistic algorithm which solves a polynomial system over the rationals defined by a reduced regular sequence. Its bit complexity is roughly quadratic in the B\'ezout number of the system and linear in its bit size. Our…

There are several numerical methods for computing approximate zeros of a given univariate polynomial. In this paper, we develop a simple and novel method for determining sharp upper bounds on errors in approximate zeros of a given…

In this paper we partially settle our conjecture from [1] (math.SP/0701143) on roots of eigenpolynomials for degenerate exactly-solvable operators. Namely, for any such operator, we establish a lower bound (which supports our conjecture)…

We consider the problem of minimising the number of states in a multiplicity tree automaton over the field of rational numbers. We give a minimisation algorithm that runs in polynomial time assuming unit-cost arithmetic. We also show that a…

In this paper, we compute the tightest possible bounds on the probability that the optimal value of a combinatorial optimization problem in maximization form with a random objective exceeds a given number, assuming only knowledge of the…