Uniform approximation by polynomials with integer coefficients

Approximation by Polynomials with Coefficients

Chebyshev Polynomials with Integer Coefficients

We study the asymptotic structure of polynomials with integer coef cients and smallest uniform norms on an interval of the real line Introducing methods of the weighted potential theory into this problem we improve the bounds for the multiplicities of some factors of the integer Chebyshev polynomials Introduction Let Pn C and Pn Z be the sets of algebraic polynomials of degree at most n respect...

Small Polynomials with Integer Coefficients

The primary goal of this paper is the study of polynomials with integer coefficients that minimize the sup norm on the set E. In particular, we consider the asymptotic behavior of these polynomials and of their zeros. Let Pn(C) and Pn(Z) be the classes of algebraic polynomials of degree at most n, respectively with complex and with integer coefficients. The problem of minimizing the uniform nor...

Uniform approximation by (quantum) polynomials

We show that quantum algorithms can be used to re-prove a classical theorem in approximation theory, Jackson’s Theorem, which gives a nearly-optimal quantitative version of Weierstrass’s Theorem on uniform approximation of continuous functions by polynomials. We provide two proofs, based respectively on quantum counting and on quantum phase estimation.

On D5-polynomials with integer coefficients

We give a family of D5-polynomials with integer coefficients whose splitting fields over Q are unramified cyclic quintic extensions of quadratic fields. Our polynomials are constructed by using Fibonacci, Lucas numbers and units of certain cyclic quartic fields.