相关论文: Quantum Search for Zeros of Polynomials

An approach is proposed for bounding the number of zeros that solutions of linear differential systems with polynomial coefficients may have. A bound is obtained in a special case which improves upon currently existing.

A simple algorithm to compute all the zeros of a generic polynomial is proposed.

We consider multivariable polynomials over a fixed number field, linear in some of the variables. For a system of such polynomials satisfying certain technical conditions we prove the existence of search bounds for simultaneous zeros with…

The classical problem of counting the number of real zeros of a real polynomial was solved a long time ago by Sturm. The analogous problem of counting the number of zeros that a polynomial has on the unit circle is, however, still an open…

To cater to the needs of (Zero Knowledge) proofs for (mathematical) proofs, we describe a method to transform formal sentences in 2x2-matrices over multivariate polynomials with integer coefficients, such that usual proof-steps like…

Computing the real solutions to a system of polynomial equations is a challenging problem, particularly verifying that all solutions have been computed. We describe an approach that combines numerical algebraic geometry and sums of squares…

The purpose of this paper is to present three new methods for finding all simple zeros of polynomials simultaneously. First, we give a new method for finding simultaneously all simple zeros of polynomials constructed by applying the…

A novel method with two variations is proposed with which the number of positive and negative zeros of a polynomial with real coefficients and degree $n$ can be restricted with significantly better determinacy than that provided by the…

We calculate the zeros of an exponential polynomial of three variables by a classical algorithm and quantum algorithms which are based on the method of van Dam and Shparlinski, they treated the case of two variables, and compare with the…

Using a variety of matrix techniques, the problem of locating the left eigenvalues of the quaternion companion matrices are investigated in this paper. In a recent paper, Dar et al. [6], proved that the zeros of a quaternionic polynomial…

The usual methods for root finding of polynomials are based on the iteration of a numerical formula for improvement of successive estimations. The unpredictable nature of the iterations prevents to search roots inside a pre-specified region…

Minimal annihilating polynomials are very useful in a wide variety of algorithms in exact linear algebra. A new efficient method is proposed for calculating the minimal annihilating polynomials for all the unit vectors, for a square matrix…

In this paper, we develop a new deflation technique for refining or verifying the isolated singular zeros of polynomial systems. Starting from a polynomial system with an isolated singular zero, by computing the derivatives of the input…

A quantum algorithm is a set of instructions for a quantum computer, however, unlike algorithms in classical computer science their results cannot be guaranteed. A quantum system can undergo two types of operation, measurement and quantum…

We consider quantum interpolation of polynomials. We imagine a quantum computer with black-box access to input/output pairs (x_i, f(x_i)), where f is a degree-d polynomial, and we wish to compute f(0). We give asymptotically tight quantum…

Motivated by Wilmshurst's conjecture, we investigate the zeros of harmonic polynomials. We utilize a certified counting approach which is a combination of two methods from numerical algebraic geometry: numerical polynomial homotopy…

This article introduces quantum computation by analogy with probabilistic computation. A basic description of the quantum search algorithm is given by representing the algorithm as a C program in a novel way.

In this paper we investigate the distribution of zeros of Boubaker polynomials.

We study global distribution of zeros for a wide range of ensembles of random polynomials. Two main directions are related to almost sure limits of the zero counting measures, and to quantitative results on the expected number of zeros in…

We investigate the behavior of fractional derivatives of polynomials. In particular, we consider the locations and the asymptotic behaviour of their zeros and give bounds for their Mahler measure.