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

Boris Kazarnovskii

TL;DR

The paper develops probabilistic refinements of the Fundamental Theorem of Algebra across three frontiers: Laurent polynomials, polynomials on compact groups, and exponential sums. It introduces Newton ellipsoids and Monge–Ampère–type geometry to express expected real-root counts and real-root probabilities via mixed volumes, providing multidimensional Kac-type results that extend Bernstein–Kushnirenko–Khovanskii theory. For group representations, it extends these ideas to real $\pi$-polynomials and derives analogous BKK-type formulas and asymptotics for the proportion of real roots, relying on weight polytopes and Newton bodies. In exponential sums, it derives density formulas for zeros in terms of Newton polygons and polyhedral data, yielding a precise analogue of FTA in multiple variables via Monge–Ampère currents and Crofton-type theorems. Together, the results reveal deep connections between root distributions and convex-geometric invariants across algebraic and analytic settings.

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}.

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

TL;DR

The paper develops probabilistic refinements of the Fundamental Theorem of Algebra across three frontiers: Laurent polynomials, polynomials on compact groups, and exponential sums. It introduces Newton ellipsoids and Monge–Ampère–type geometry to express expected real-root counts and real-root probabilities via mixed volumes, providing multidimensional Kac-type results that extend Bernstein–Kushnirenko–Khovanskii theory. For group representations, it extends these ideas to real -polynomials and derives analogous BKK-type formulas and asymptotics for the proportion of real roots, relying on weight polytopes and Newton bodies. In exponential sums, it derives density formulas for zeros in terms of Newton polygons and polyhedral data, yielding a precise analogue of FTA in multiple variables via Monge–Ampère currents and Crofton-type theorems. Together, the results reveal deep connections between root distributions and convex-geometric invariants across algebraic and analytic settings.

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}.
Paper Structure (21 sections, 20 theorems, 91 equations, 1 table)

This paper contains 21 sections, 20 theorems, 91 equations, 1 table.

Table of Contents

  1. Introduction
  2. Kac type theorem for Laurent polynomials
  3. Laurent polynomials of one variable
  4. The Bernstein--Kushnirenko--Khovanskii theorem
  5. A multidimensional Kac theorem for Laurent polynomials
  6. Proof Theorems \ref{['thmMy1']}, \ref{['thmMy2']}, \ref{['thmMy3']}, \ref{['thmMy4']}
  7. Theorem \ref{['thmMy1']}
  8. Theorem \ref{['thmMy2']}
  9. Theorems \ref{['thmMy3']}, \ref{['thmMy4']}
  10. Kac type theorems in the context of groups
  11. Introduction
  12. Theorem BKK for reductive groups
  13. Newton ellipsoid of group representation
  14. Asymptotics of the proportion of real roots
  15. FTA for exponential sums
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $P$ be a random real polynomial of degree $m$ in one variable whose coefficients are independent Gaussian random variables with zero mean and unit variance. Then, as $m\to\infty$, the expected number of real roots of $P$ is asymptotic to $\frac{2}{\pi}\log m$. Equivalently, where $\mathcal{P}(m)$ denotes the probability that a root of $P$ is real.

Theorems & Definitions (46)

  • Theorem 1.1: Kac
  • Remark 1.1
  • Remark 1.2
  • Theorem 2.1
  • proof
  • Remark 2.1
  • Remark 2.2
  • Proposition 2.1
  • proof
  • proof
  • ...and 36 more