Trigonometric polynomials with simple roots

Abstract

Trogonometric polynimials frequently occur applications in physics, numerical analysis and engineering, since each periodic function can be approximated by a trigonometric polynomial. Additionally, there are many analogies between trigonometric and standard algebraic polynomials. Algorithms in computer algebra depend on methods for the square-free decomposition of polynomials. These methods use polynomial division and cannot be applied directly to trigonometric polynomials. Let P denote the set of odd multiples of π. A trigonometric polynomial T ∗ is a reduced representation of a trigonometric polynomial T if the set of zeros of T in C ⧹ P is the same as the set of zeros of T ∗ in C ⧹ P, and if all zero s of T ∗ are simple zeros. It is shown that a reduced representation of a trigonometric polynomial with rational or algebraic coefficients can be found in polynomial time.