Searcharxiv⌕ Search

arXiv subjects

Boris Kazarnovskii

Publications and source records attributed to Boris Kazarnovskii.

16 recordsLinked to original sources
arXiv2026

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

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}.

math.AG↗
arXiv2025

Around the "Fundamental Theorem of Algebra"

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.

math.AG↗
arXiv2025

On real roots of polynomial systems of equations in the context of group theory

The probability that a zero of a random real polynomial of increasing degree is real tends to zero. However, passing from polynomials to Laurent polynomials yields a surprising result: the probability that a root is real tends not to zero, but to $1/\sqrt{3}$. A similar phenomenon has also been observed for systems of Laurent polynomials in several variables. By considering Laurent polynomials as functions associated with torus representations, we describe an analogous phenomenon for representations of any reductive linear group. In the case of a simple group, we provide a formula for the aforementioned limiting probability.

math.AG↗
arXiv2024

The zeros of random sections of real vector bundles

We define integral geometric analogues of the Chern classes for real vector bundle on a smooth real variety. More precisely, we define the Chern densities of a real bundle. These densities are analogues of the Chern forms of a complex vector bundle and inherit some of their properties. \noindent (The text is a summary of a report on the conference PCA'2024 in Euler International Mathematical Institute, St. Petersburg)

math.AG↗
arXiv2021

How many roots of a system of random trigonometric polynomials are real?

The expected number of zeros of a random real polynomial of degree $N$ asymptotically equals $\frac{2}π\log N$. On the other hand, the average fraction of real zeros of a random trigonometric polynomial of increasing degree $N$ converges to not $0$ but to $1/\sqrt 3$. An average number of roots of a system of random trigonometric polynomials in several variables is equal to the mixed volume of some ellipsoids depending on the degrees of polynomials. Comparing this formula with Theorem BKK we prove that the phenomenon of nonzero fraction of real roots remains valid.

math.AG↗
arXiv2020

Crofton formulae for products

It is shown how new integral-geometric formulae can be obtained from the existing formulae of Crofton type. In particular, for classical Crofton formulae in which the answer depends on the Riemannian volume, we obtain generalizations in terms of the mixed Riemannian volume defined in the paper. The method is based on the calculations in the ring of normal densities constructed in the previous work of the authors.

math.DG↗
arXiv2020

Ring of conditions for affine space

The exponential sum (ES) is a linear combination of characters of an additive group $\mathbb C^n$. The exponential analytic set (EAS) is a set of common zeroes of a finite tuple of ESs. We consider ES and EAS as an analogs of Laurent polynomial and of algebraic variety in complex torus $(\mathbb{C}\setminus0)^n$. Respectively we construct the ring of conditions for $\mathbb C^n$ as an analog of the ring of conditions for $(\mathbb{C}\setminus0)^n$. The construction of this ring is based on the definition of associated to EAS algebraic subvariety of some multidimensional torus and on the applying tropical algebraic geometry to this subvariety. Just as in the case of a torus, the ring of conditions is generated by hypersurfaces. This preprint is an extended summary of the article proposed to "Izvestiya: Mathematics".

math.AG↗
arXiv2019

Average number of solutions

Let $X$ be an $n$-dimensional manifold and $V_1,\ldots,V_n\subset C^\infty(X,\mathbb R)$ finite-dimensional vector spaces. For systems of equations $\{f_i = a_i\colon\: f_i\in V_i,\:a_i \in\mathbb R,\:i=1,\ldots,n\}$ we discover a relationship between the average number of their solutions and mixed volumes of convex bodies. To do this, we choose Banach metrics in the spaces $V_i$. Using these metrics, we construct 1) the measure in the space of systems, and 2) Banach convex bodies in $X$, i.e., collections of centrally symmetric convex bodies in the fibers of the cotangent bundle of $X$. It turns out that the average number of solutions is equal to the mixed symplectic volume of Banach convex bodies. Earlier this result was obtained for Euclidean metrics in spaces $V_i$. In Euclidean case, the Banach convex bodies are the collections of ellipsoids.

math.SG↗
arXiv2018

Mixed Hermitian volume and number of common zeros of holomorphic functions

Let $V_i$ be a finite dimensional Hermitian vector space of holomorphic sections of a line bundle $L_i$ on a complex $n$-dimensional manifold $X$. We associate to $V_i$ the non-negative Hermitian quadratic form $g_i$ on $X,$ define a Hermitian mixed volume of $X$ for a "mixing tuple" of $n$ non-negative Hermitian forms, and prove that the average number of common zeroes of $f_1\in V_1,\ldots, f_n\in V_n$ equals to the mixed volume of $X$ for the "mixing tuple" $g_1,\ldots,g_n$. This note is related to arXiv:1802.02741, where the average number of common zeros for real equations are treated in a similar way.

math.DG↗
arXiv2018

Average number of zeros and mixed symplectic volume of Finsler sets

Let $X$ be an $n$-dimensional manifold and $V_1, \ldots, V_n \subset C^\infty(X, \mathbb R)$ finite-dimensional vector spaces with Euclidean metric. We assign to each $V_i$ a Finsler ellipsoid, i.e., a family of ellipsoids in the fibers of the cotangent bundle of $X$. We prove that the average number of isolated common zeros of $f_1 \in V_1, \ldots, f_n \in V_n$ is equal to the mixed symplectic volume of these Finsler ellipsoids. If $X$ is a homogeneous space of a compact Lie group and all vector spaces $V_i$ and their Euclidean metrics are invariant, then the average numbers of zeros satisfy the inequalities, similar to Hodge inequalities for intersection numbers of divisors on a projective variety. This is applied to the eigenspaces of Laplace operator of an invariant Riemannian metric. The proofs are based on a construction of the ring of normal densities on $X$, an analogue of the ring of differential forms. In particular, this construction is used for a generalization of Crofton formula to the product of spheres.

math.DG↗
arXiv2017

An estimate for the average number of common zeros of Laplacian eigenfunctions

On a compact Riemannian manifold $M$ of dimension $n$, we consider $n$ eigenfunctions of the Laplace operator $Δ$ with eigenvalue $λ$. If $M$ is homogeneous under a compact Lie group preserving the metric then we prove that the average number of common zeros of $n$ eigenfunctions does not exceed $c(n)λ^{n/2}{\rm vol}\,M$, the expression known from the celebrated Weyl's law. Moreover, if the isotropy representation is irreducible, then the estimate turns into equality. The constant $c(n)$ is explicitly given. The method of proof is based on the application of Crofton's formula for the sphere.

math.DG↗
arXiv2017

Newton polyhedra, tropical geometry and the ring of condition for $(C^*)^n$

The ring of conditions defined by C. De Concini and C. Procesi is an intersection theory for algebraic cycles in a spherical homogeneous space. In the paper we consider the ring of conditions for the group $(C^*)^n$. Up to a big extend this ring can be reduced to the cohomology rings of smooth projective toric varieties. This ring also can be described using tropical geometry. We recall these known results and provide a new description of this ring in terms of convex integral polyhedra.

math.AG↗
arXiv2016

On common zeros of eigenfunctions of the Laplace operator

We consider the eigenfunctions of the Laplace operator $Δ$ on a compact Riemannian manifold of dimension $n$. For $M$ homogeneous with irreducible isotropy representation and for a fixed eigenvalue of $Δ$ we find the average number of common zeros of $n$ eigenfunctions. For this we compute the volume of the image of $M$ under an equivariant immersion into a sphere.

math.DG↗
arXiv2013

Exponential tropical varieties and complex Monge-Ampere operator

Sometimes it is possible to extend some using Newton polyhedra computations in algebraic geometry from polynomials to exponential sums. For this purpose it is useful to consider analogues of tropical varieties in complex space. These analogues are called exponential tropical varieties (ETV). We construct the ring of ETV. Algebraic tropical varieties form the subring of the ring of ETV. In this paper we connect ETV with the complex Monge-Ampere operator action on the space of piecewise linear functions in complex vector space. We show that all ETV arise as results of such operator action. We give some applications of this connection. One of the applications is a criterion for zero value of a mixed Monge-Ampere operator. This criterion is the modification of the criterion for zero value of a mixed volume of convex bodies. The proof is the modification of A. Khovanskii's unpublished proof of the corresponding theorem on mixed volumes. In the part 1 we give the definition of ETV and detail statements of theorems (without proofs). In the part 2 we prove the theorems on the action of the Monge-Ampere operator.

math.AG↗
arXiv2012

On piecewise pluriharmonic functions

We extend some results on piecewise linear functions on $\C^n$ to piecewise pluriharmonic functions on any complex manifold. We construct a ring generated by currents $h$ and $dd^ch$, where $\{h\}$ is a finite set of piecewise pluriharmonic functions. We prove that, with some restrictions on the set $\{h\}$, the map $\{h\mapsto dd^ch,\ dd^ch\mapsto0\}$ can be continued to the derivation on the ring. As a corollary, the current $dd^cg_1\wedge...\wedge dd^cg_k$ depends on the product of piecewise pluriharmonic functions $g_1,...,g_k$ only.

math.CV↗