Around the 'Fundamental Theorem of Algebra' (extended version)

Abstract: The Fundamental Theorem of Algebra (FTA) asserts that every complex polynomial has as many complex roots, counted with multiplicities, as its degree. A probabilistic analogue of this theorem for real roots of real polynomials, sometimes referred to as the Kac theorem, was found between 1938 and 1943 by J. Littlewood, A. Offord, and M. Kac. In this paper, we present several more versions of FTA: Kac type FTA for Laurent polynomials in one and many variables, Kac type FTA for polynomials on complex reductive groups arising in the context of compact group representations (similar to Laurent polynomials arising in torus representation theory), and FTA for exponential sums in one and many variables. In the case of Laurent polynomials, the result, even in the one-dimensional case, is unexpected: most of the zeros of a real Laurent polynomial are real. This text is a supplemented and more detailed version of \cite{arx}.