相关论文: Quantum Search for Zeros of Polynomials

We obtain several Cauchy-like and Pellet-like results for the zeros of a general complex polynomial by considering similarity transformations of the squared companion matrix and the reformulation of the zeros of a scalar polynomial as the…

We present a polynomial-time quantum algorithm for obtaining the energy spectrum of a physical system, i.e. the differences between the eigenvalues of the system's Hamiltonian, provided that the spectrum of interest contains at most a…

We design a homotopy continuation algorithm, that is based on numerically tracking Viro's patchworking method, for finding real zeros of sparse polynomial systems. The algorithm is targeted for polynomial systems with coefficients…

We propose a quantum algorithm for solving combinatorial search problems that uses only a sequence of measurements. The algorithm is similar in spirit to quantum computation by adiabatic evolution, in that the goal is to remain in the…

Detection of symmetry is vital to problem solving. Most of the problems of computer vision and computer graphics and machine intelligence in general, can be reduced to symmetry detection problem. Unstructured search problem can also be…

The literature on quaternionic polynomials and, in particular, on methods for determining and classifying their zero-sets, is fast developing and reveals a growing interest on this subject. In contrast, polynomials defined over the algebra…

We propose a general methodology for testing whether a given polynomial with integer coefficients is identically zero. The methodology evaluates the polynomial at efficiently computable approximations of suitable irrational points. In…

Univariate polynomial root-finding is a classical subject, still important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the…

Quantum modifications to black holes on scales comparable to the horizon size, or even more radical physics, are apparently needed to reconcile the existence of black holes with the principles of quantum mechanics. This piece gives an…

Nuclear magnetic resonance techniques are used to realize a quantum algorithm experimentally. The algorithm allows a simple NMR quantum computer to determine global properties of an unknown function requiring fewer function ``calls'' than…

The main aim of this work is to apply the matrix approach of ortho\-gonal polynomials associated with infinite Hermitian definite positive matrices in relation with an important question regarding the location of zeros of Sobolev orthogonal…

The authors present a unified method for calculating the zeros of the classical orthogonal polynomials based upon the electrostatic interpretation and its connection to the energy minimization problem. Examples are given with error…

We propose polynomial-time algorithms for finding nontrivial zeros of quadratic forms with four variables over rational function fields of characteristic 2. We apply these results to find prescribed quadratic subfields of quaternion…

Poles of a multi-input multi-output (MIMO) linear system can be computed by solving an eigenvalue problem; however, the problem of computing its invariant zeros is equivalent to a generalized eigenvalue problem. This paper revisits the…

We survey results on the distribution of zeros of random polynomials and of random holomorphic sections of line bundles, especially for large classes of probability measures on the spaces of holomorphic sections. We provide furthermore some…

We investigate the zeros of two one-parameter families of harmonic functions and describe how the number of zeros depends on the parameter. Our functions have the property that all zeros lie on certain rays in the complex plane and thus we…

For polynomials of degree two which have no zeros, the method of accompanying variables is developed and zeros of associated vector polynomials are determined. Our flexible method uses a wide variety of possible vector-valued vector…

We derive a useful result about the zeros of the $k$-polar polynomials on the unit circle; in particular we obtain a ring shaped region containing all the zeros of these polynomials. Some examples are presented.

Quantum mechanical search induces polynomial speed up in an unsorted database search process. In case of classical linear search the computational time increases with the dimensionality of the query. However, quantum parallelism, inherent…

A fast and weakly stable method for computing the zeros of a particular class of hypergeometric polynomials is presented. The studied hypergeometric polynomials satisfy a higher order differential equation and generalize Laguerre…