Around the "Fundamental Theorem of Algebra"
Boris Kazarnovskii
TL;DR
The paper surveys the classical Kac analogue for real roots of real polynomials and presents two additional Fundamental Theorem of Algebra analogues: one for real Laurent polynomials and another for exponential sums, including their multidimensional forms. It develops probabilistic formulas for the mean number of real zeros and the real-root probability in the Laurent setting, via mappings to real trigonometric polynomials and Crofton-type geometry, and expresses these quantities in terms of the spectrum $\Lambda$ and the Newton ellipsoid $\mathrm{Ell}(\Lambda)$. For exponential sums, it derives asymptotics N$(f,r)$ driven by the Newton polygon $\Delta$ and extends to systems in several variables using the pseudovolume $\mathrm{pvol}$ of convex bodies, connecting to Koushnirenko-type results and mixed volumes. Overall, the work highlights how classical algebraic results extend to probabilistic and geometric contexts, with explicit formulas that link zero counts to convex-geometric invariants like Newton ellipsoids and pseudovolumes.
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, commonly referred to as the Kac theorem, was introduced in 1938 by J. Littlewood and A. Offord. In this paper, we present the Kac theorem and prove two more theorems that can be interpreted as analogues of the FTA: a version of FTA for real Laurent polynomials, and another version for exponential sums. In these two cases, we also provide formulations of multidimensional analogues of corresponding FTA. While these results are not new, they may appear unexpected and are therefore worth highlighting.