Kac-Type FTA for Laurent Polynomials

• The paper’s main contribution is establishing a rigorous probabilistic analogue to the Fundamental Theorem of Algebra for Laurent polynomials, determining the expected number and stabilization of real roots.
• It employs integral-geometric methods, notably a Crofton-type formula, to derive exact formulas for real root counts and probabilities across different spectral setups.
• The findings extend to multidimensional settings, linking mixed volume formulas and Newton ellipsoids with applications in toric, symplectic, and random trigonometric polynomial theories.

A Kac-type Fundamental Theorem of Algebra (FTA) for Laurent polynomials provides a rigorous probabilistic analogue to classical algebraic results concerning the root distribution of polynomials. While the classical FTA determines the total (complex) roots of a polynomial, and the Kac theorem estimates the expected real roots for random polynomials with non-negative spectrum, the Kac-type FTA for Laurent polynomials introduces a geometric and probabilistic framework for the expected number and distribution of real roots of random real Laurent polynomials, including generalizations to multidimensional settings. In contrast to the classical case, where the probability that a root is real vanishes asymptotically, the Laurent polynomial case exhibits stabilization: a positive fraction of zeros remain real, even as the maximal exponent grows.

1. Classical Kac Theorem and Real Polynomial Root Statistics

The Kac theorem, originally formulated in the 1938–43 works of J. Littlewood, A. Offord, and M. Kac, addresses the expected real root count of degree-$m$ algebraic polynomials with Gaussian random coefficients: $$f_m(x) = a_0 + a_1 x + \cdots + a_m x^m, \quad a_k \sim N(0,1)\ \text{i.i.d.}$$ The expected number of real zeros $E[N_m]$ obeys the asymptotic relation: $$E[N_m] \sim \frac{2}{\pi} \ln m, \quad m \to \infty$$ The probability that a randomly selected root (among the $m$ complex roots) is real decays to zero: $$P(m) = \frac{E[N_m]}{m} \sim \frac{2}{\pi} \frac{\ln m}{m}, \quad P(m) \to 0$$ The exact expected count is given by the Kac integral formula involving the kernel $K_m(x)$: $$E[N_m] = \int_{-\infty}^{\infty} \rho_m(x)\,dx,\quad \rho_m(x) = \frac{1}{\pi}\frac{ \sqrt{K_m'(x)^2 - K_m(x)K_m''(x)} }{K_m(x)},\quad K_m(x) = \sum_{k=0}^m x^{2k}$$ These results establish that, in the classical case, most zeros of a large-degree random polynomial are non-real.

2. Real Laurent Polynomials: Definitions and Random Model

A real Laurent polynomial of degree $m$ is

$P(z) = \sum_{k=-m}^{m} a_k z^k$

and is "real" if $P(e^{i\theta}) \in \mathbb{R}$ for all $\theta$. The spectrum $\Lambda = \{k : a_k \neq 0\}$ is centrally symmetric; equivalently, $a_k = \overline{a_{-k}}$. Real roots are the zeros on the unit circle $S^1$.

Under the Gaussian model, real Laurent polynomials are identified with trigonometric polynomials: $$T_m(\theta) = a_0 + \sum_{k=1}^m [\alpha_k \cos(k\theta) + \beta_k \sin(k\theta)]$$ where $\alpha_k, \beta_k \sim N(0,1)$ i.i.d., and polynomials are endowed with the $L^2(S^1)$ metric. This identification enables a probabilistic analysis analogous to the Kac setting.

3. Kac-Type FTA for Laurent Polynomials: Precise Theorems

Kazarnovskii established exact formulas for the expected real-root count and the root-by-root probability for random real Laurent polynomials with centrally symmetric spectrum $\Lambda \subset \mathbb{Z}$: $$n = |\Lambda|,\quad \deg f_{\Lambda} = \max_{\lambda \in \Lambda} |\lambda|$$

Main Theorem

$$E[\#(\text{real roots})] = 2 \sqrt{ \frac{1}{n} \sum_{\lambda \in \Lambda} \lambda^2 }$$

$$P(\Lambda) = \frac{1}{\deg f_{\Lambda}} \sqrt{ \frac{1}{n} \sum_{\lambda \in \Lambda} \lambda^2 }$$

A notable case is the full spectrum $\Lambda_m = \{-m, \ldots, m\}$, where

$$E[\#(\text{real roots})] = 2\sqrt{\frac{m(m+1)}{3}}$$

$P(\Lambda_m) = \sqrt{ \frac{m+1}{3m} }$

and as $m \to \infty$, $P(\Lambda_m) \to 1/\sqrt{3} > 1/2$, manifesting an "anti-Kac" phenomenon: a majority of zeros are real even for large $m$ (Kazarnovskii, 11 Oct 2025, Kazarnovskii, 4 Jan 2026).

Spectrum Rescaling

For any dilation $\Lambda \mapsto k\Lambda$, the probability $P(\Lambda)$ remains invariant due to the $S^1$-action.

4. Integral-Geometric Proof via Crofton-Type Formula

The proof utilizes a Crofton-type integral geometry principle. The zero set on $S^1$ is encoded as intersections of a fixed curve in the unit sphere of the coefficient space with random hyperplanes:

• $V = \mathrm{Trig}(\Lambda)$, $n$-dimensional real vector space.
• For $\theta \in [0,2\pi)$, define the evaluation vector $F_{\theta}$, which traces a closed curve $\kappa(S^1)$ on $S^{n-1} \subset \mathbb{R}^n$.
• A random trigonometric polynomial $f$ corresponds to a Gaussian random point $\xi \in \mathbb{R}^n$, with real roots at intersections of $\kappa(S^1)$ and the hyperplane $\{x : \xi \cdot x = 0\}$.

Crofton-type proposition: $$E[\#(K \cap Z(\xi))] = \frac{1}{\pi} \cdot \text{length}(K)$$ where $K = \kappa(S^1)$. Calculating length using the explicit basis yields $$2\pi\sqrt{ \frac{1}{n} \sum_{\lambda \in \Lambda} \lambda^2 }$$, reproducing the main theorem's formulas.

5. Probabilistic Corollaries and Asymptotics

From the closed-form expressions:

• The expected number of real zeros for $\Lambda = \{-m, \ldots, m\}$ is $2\sqrt{m(m+1)/3}$.
• The probability $$P(\Lambda) = \frac{1}{\deg} \sqrt{ (1/n) \sum \lambda^2 }$$.
• For full spectra, as $m \to \infty$, $P(\Lambda_m) \to 1/\sqrt{3}$, indicating that Laurent polynomials retain a positive fraction of real zeros even at high degree.

This behavior contrasts sharply with classical random polynomial models, where the fraction of real roots vanishes in the limit.

6. Multidimensional Generalization and Mixed Volume Formulas

The generalization extends to systems of $n$ real Laurent polynomials in $n$ variables, each with their respective centrally symmetric spectrum $\Lambda_i \subset \mathbb{Z}^n$: $$f_i(z) = \sum_{\lambda \in \Lambda_i} a_{\lambda}^{(i)} z^{\lambda},\quad z \in (\mathbb{C}^{\times})^n$$ Expected real root count on the real torus $T^n = \{ |z_j| = 1 \}$ is given by the mixed volume of Newton ellipsoids (Kazarnovskii, 11 Oct 2025, Kazarnovskii, 4 Jan 2026): $$E[\#(\text{real roots in } T^n)] = n! \cdot (\mathrm{Ell}(\Lambda_1), \ldots, \mathrm{Ell}(\Lambda_n))$$ where $\mathrm{Ell}(\Lambda)$ is the Newton ellipsoid with support function

$$h_{\Lambda}(x) = \sqrt{ \frac{1}{|\Lambda|} \sum_{\lambda \in \Lambda} \langle \lambda, x \rangle^2 },\quad x \in \mathbb{R}^n$$

The probability that a complex root lies on $T^n$ is

$$\mathcal{P}(\Lambda_1,\ldots,\Lambda_n) = \frac{ (\mathrm{Ell}(\Lambda_1),\ldots,\mathrm{Ell}(\Lambda_n)) }{ (\mathrm{conv}(\Lambda_1),\ldots,\mathrm{conv}(\Lambda_n)) }$$

Uniform dilation of the spectra retains a strictly positive limit for the proportion of real roots.

7. Significance, Applications, and Generalizations

• The stabilization of real-root density under the Laurent polynomial model reflects deep geometric invariants (mixed volumes of convex sets derived from spectra).
• The Crofton-type methodology and Newton ellipsoid construction form bridges to broader areas: integral geometry, Finsler geometry, symplectic geometry, toric and reductive-group representation theory.
• Generalizations include exponential sums and polynomial functions on complex reductive groups, where related Crofton and mixed-volume formulas govern density of real solutions.
• Applications arise in random trigonometric polynomial theory, zeros of random sections in complex geometry, and probabilistic versions of classic algebraic theorems.

A plausible implication is that negative powers in the spectrum "stabilize" real root density, and that geometric properties of the Newton ellipsoid can be exploited in related fields where root distributions of random functions are of interest.

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References:

Kazarnovskii, Around the "Fundamental Theorem of Algebra" (Kazarnovskii, 11 Oct 2025); Extended Version (Kazarnovskii, 4 Jan 2026)