Addition of sets via symmetric polynomials — A polynomial method

AbstractLet A1,…,Ah be finite non-empty subsets of a field K and let sk(x1,…,xh) be the elementary symmetric polynomial of degree k in h indeterminates. Here we present some estimates for the cardinality of the sets of the images of all h-tuples of A1×⋯×Ah by the polynomial sk, with and without the restriction that the elements of the h-tuples are pairwise distincts

Structure of the sets of mutually unbiased bases with cyclic symmetry

Mutually unbiased bases that can be cyclically generated by a single unitary operator are of special interest, since they can be readily implemented in practice. We show that, for a system of qubits, finding such a generator can be cast as the problem of finding a symmetric matrix over the field $\mathbb{F}_2$ equipped with an irreducible characteristic polynomial of a given Fibonacci index. The entanglement structure of the resulting complete sets is determined by two additive matrices of the same size.Comment: 20 page

Moving least squares via orthogonal polynomials

A method for moving least squares interpolation and differentiation is presented in the framework of orthogonal polynomials on discrete points. This yields a robust and efficient method which can avoid singularities and breakdowns in the moving least squares method caused by particular configurations of nodes in the system. The method is tested by applying it to the estimation of first and second derivatives of test functions on random point distributions in two and three dimensions and by examining in detail the evaluation of second derivatives on one selected configuration. The accuracy and convergence of the method are examined with respect to length scale (point separation) and the number of points used. The method is found to be robust, accurate and convergent.Comment: Extensively revised in response to referees' comment