Matrix polynomials, generalized Jacobians, and graphical zonotopes

A matrix polynomial is a polynomial in a complex variable $\lambda$ with coefficients in $n \times n$ complex matrices. The spectral curve of a matrix polynomial $P(\lambda)$ is the curve $\{ (\lambda, \mu) \in \mathbb{C}^2 \mid \mathrm{det}(P(\lambda) - \mu \cdot \mathrm{Id}) = 0\}$. The set of matrix polynomials with a given spectral curve $C$ is known to be closely related to the Jacobian of $C$, provided that $C$ is smooth. We extend this result to the case when $C$ is an arbitrary nodal, possibly reducible, curve. In the latter case the set of matrix polynomials with spectral curve $C$ turns out to be naturally stratified into smooth pieces, each one being an open subset in a certain generalized Jacobian. We give a description of this stratification in terms of purely combinatorial data and describe the adjacency of strata. We also make a conjecture on the relation between completely reducible matrix polynomials and the canonical compactified Jacobian defined by V.Alexeev.Comment: 19 pages, 7 figure

Curvature of Poisson pencils in dimension three

A Poisson pencil is called flat if all brackets of the pencil can be simultaneously locally brought to a constant form. Given a Poisson pencil on a 3-manifold, we study under which conditions it is flat. Since the works of Gelfand and Zakharevich, it is known that a pencil is flat if and only if the associated Veronese web is trivial. We suggest a simpler obstruction to flatness, which we call the curvature form of a Poisson pencil. This form can be defined in two ways: either via the Blaschke curvature form of the associated web, or via the Ricci tensor of a connection compatible with the pencil. We show that the curvature form of a Poisson pencil can be given by a simple explicit formula. This allows us to study flatness of linear pencils on three-dimensional Lie algebras, in particular those related to the argument translation method. Many of them appear to be non-flat.Comment: 14 pages, 1 figur

Algebraic geometry and stability for integrable systems

In 1970s, a method was developed for integration of nonlinear equations by means of algebraic geometry. Starting from a Lax representation with spectral parameter, the algebro-geometric method allows to solve the system explicitly in terms of theta functions of Riemann surfaces. However, the explicit formulas obtained in this way fail to answer qualitative questions such as whether a given singular solution is stable or not. In the present paper, the problem of stability for equilibrium points is considered, and it is shown that this problem can also be approached by means of algebraic geometry

Stability of relative equilibria of multidimensional rigid body

It is a classical result of Euler that the rotation of a torque-free three-dimensional rigid body about the short or the long axis is stable, whereas the rotation about the middle axis is unstable. This result is generalized to the case of a multidimensional body

Generalized argument shift method and complete commutative subalgebras in polynomial Poisson algebras

The Mischenko-Fomenko argument shift method allows to construct commutative subalgebras in the symmetric algebra $S(\mathfrak g)$ of a finite-dimensional Lie algebra $\mathfrak g$. For a wide class of Lie algebras, these commutative subalgebras appear to be complete, i.e. they have maximal transcendence degree. However, for many algebras, Mischenko-Fomenko subalgebras are incomplete or even empty. In this case, we suggest a natural way how to extend Mischenko-Fomenko subalgebras, and give a completeness criterion for these extended subalgebras

Stability in bi-Hamiltonian systems and multidimensional rigid body

The presence of two compatible Hamiltonian structures is known to be one of the main, and the most natural, mechanisms of integrability. For every pair of Hamiltonian structures, there are associated conservation laws (first integrals). Another approach is to consider the second Hamiltonian structure on its own as a tensor conservation law. The latter is more intrinsic as compared to scalar conservation laws derived from it and, as a rule, it is "simpler". Thus it is natural to ask: can the dynamics of a bi-Hamiltonian system be understood by studying its Hamiltonian pair, without studying the associated first integrals?\par In this paper, the problem of stability of equilibria in bi-Hamiltonian systems is considered and it is shown that the conditions for nonlinear stability can be expressed in algebraic terms of linearization of the underlying Poisson pencil. This is used to study stability of stationary rotations of a free multidimensional rigid body.Comment: Journal of Geometry and Physics, 201

Flat bi-Hamiltonian structures and invariant densities

A bi-Hamiltonian structure is a pair of Poisson structures $\mathcal P$, $\mathcal Q$ which are compatible, meaning that any linear combination $\alpha \mathcal P + \beta \mathcal Q$ is again a Poisson structure. A bi-Hamiltonian structure $(\mathcal P, \mathcal Q)$ is called flat if $\mathcal P$ and $\mathcal Q$ can be simultaneously brought to a constant form in a neighborhood of a generic point. We prove that a generic bi-Hamiltonian structure $(\mathcal P, \mathcal Q)$ on an odd-dimensional manifold is flat if and only if there exists a local density which is preserved by all vector fields Hamiltonian with respect to $\mathcal P$, as well as by all vector fields Hamiltonian with respect to $\mathcal Q$.Comment: 10 pages; Lett Math Phys (2016

Singularities of integrable systems and nodal curves

The relation between integrable systems and algebraic geometry is known since the XIXth century. The modern approach is to represent an integrable system as a Lax equation with spectral parameter. In this approach, the integrals of the system turn out to be the coefficients of the characteristic polynomial $\chi$ of the Lax matrix, and the solutions are expressed in terms of theta functions related to the curve $\chi = 0$. The aim of the present paper is to show that the possibility to write an integrable system in the Lax form, as well as the algebro-geometric technique related to this possibility, may also be applied to study qualitative features of the system, in particular its singularities.Comment: 30 pages, 1 figure, 2 tables. Partially published as Singularities of integrable systems and algebraic curves, International Mathematics Research Notices, doi:10.1093/imrn/rnw168 (2016

A note on relative equilibria of multidimensional rigid body

It is well known that a rotation of a free generic three-dimensional rigid body is stationary if and only if it is a rotation around one of three principal axes of inertia. As it was noted by many authors, the analogous result is true for a multidimensional body: a rotation is stationary if and only if it is a rotation in the principal axes of inertia, provided that the eigenvalues of the angular velocity matrix are pairwise distinct. However, if some eigenvalues of the angular velocity matrix of a stationary rotation coincide, then it is possible that this rotation has a different nature. A description of such rotations is given in the present paper

Characterization of steady solutions to the 2D Euler equation

Steady fluid flows have very special topology. In this paper we describe necessary and sufficient conditions on the vorticity function of a 2D ideal flow on a surface with or without boundary, for which there exists a steady flow among isovorticed fields. For this we introduce the notion of an antiderivative (or circulation function) on a measured graph, the Reeb graph associated to the vorticity function on the surface, while the criterion is related to the total negativity of this antiderivative. It turns out that given topology of the vorticity function, the set of coadjoint orbits of the symplectomorphism group admitting steady flows with this topology forms a convex polytope. As a byproduct of the proposed construction, we also describe a complete list of Casimirs for the 2D Euler hydrodynamics: we define generalized enstrophies which, along with circulations, form a complete set of invariants for coadjoint orbits of area-preserving diffeomorphisms on a surface.Comment: 34 pages, 13 figure