Related papers: Rational integrability of trigonometric polynomial…

Let L denote the variety of lattices. In 1982, the second author proved that L is strongly tolerance factorable, that is, the members of L have quotients in L modulo tolerances, although L has proper tolerances. We did not know any other…

We prove that, for fixed n there exist only finitely many embeddings of Q-factorial toric varieties X into P^n that are induced by a complete linear system. The proof is based on a combinatorial result that for fixed nonnegative integers d…

In this paper we explore the general conditions in order that a 2-dimensional natural Hamiltonian system possess a second invariant which is a polynomial in the momenta and is therefore Liouville integrable. We examine the possibility that…

We generalize the Toda lattice (or Toda chain) equation to the square lattice; i.e., we construct an integrable nonlinear equation, for a scalar field taking values on the square lattice and depending on a continuous (time) variable,…

We give an explicit and calculable algebraic model for the block of rational G-spectra on full subgroups when G has identity component a 2-torus T, and component group of order 2 acting non-trivially on H_1(T). The example of particular…

Let $S$ be a rational fraction and let $f$ be a polynomial over a finite field. Consider the transform $T(f)=\operatorname{numerator}(f(S))$. In certain cases, the polynomials $f$, $T(f)$, $T(T(f))\dots$ are all irreducible. For instance,…

In this article we study the fragility of Lagrangian periodic tori for symplectic twist maps of the $2d$-dimensional annulus and prove a rigidity result for completely integrable ones. More specifically, we consider $1$-parameter families…

We show that the one dimensional unitary matrix model with potential of the form $a U + b U^2 + h.c.$ is integrable. By reduction to the dynamics of the eigenvalues, we establish the integrability of a system of particles in one space…

We show that any smooth lattice polytope P with codegree greater or equal than (dim(P)+3)/2 (or equivalently, with degree smaller than dim(P)/2), defines a dual defective projective toric manifold. This implies that P is Q-normal (in the…

Let $I_A \subset K[x_1,\ldots,x_n]$ be a toric ideal. In this paper, we provide a necessary and sufficient condition for the toric variety $V(I_A)$, over an algebraically closed field, to be expressed as the set-theoretic intersection of…

In this note, we prove that the following function space with absolutely convergent Fourier series \[ F_d:=\left\{ f\in L^2([0,1)^d)\:\middle| \: \|f\|:=\sum_{\boldsymbol{k}\in \mathbb{Z}^d}|\hat{f}(\boldsymbol{k})| \max\left(1,\min_{j\in…

We show that a weight variety, which is a quotient of a flag variety by the maximal torus, admits a flat degeneration to a toric variety. In particular, we show that the moduli spaces of spatial polygons degenerate to polarized toric…

Let $L^0$ be the vector space of all (equivalence classes of) real-valued random variables built over a probability space $(\Omega, \mathcal{F}, P)$, equipped with a metric topology compatible with convergence in probability. In this work,…

It is proved that each of compact linear groups of one special type admits a polynomial factorization map onto a real vector space. More exactly, the group is supposed to be non-commutative one-dimensional and to have two connected…

This article provides the basic algebraic background on infinitesimal deformations and presents the proof of the well-known fact that the non-trivial infinitesimal deformations of a $K$-algebra $R$ are parameterized by the elements of…

Let $Q(X)$ be any integral primitive positive definite quadratic form with discriminant $D$ and in $k$ variables where $k\geq4$. We give an upper bound on the number of integral solutions of $Q(X)=n$ for any integer $n$ in terms of $n$, $k$…

A list $\Lambda =\{\lambda _{1},\ldots ,\lambda _{n}\}$ of complex numbers (repeats allowed) is said to be \textit{realizable} if it is the spectrum of an entrywise nonnegative matrix $A$. $\Lambda $ is \textit{diagonalizably realizable} if…

We calculate the Fourier transform of a spherically symmetric exponential function. Our evaluation is much simpler than the known one. We use the polar coordinates and reduce the Fourier transform to the integral of a rational function of…

Let $R$ be a Dedekind ring, $K$ its quotient field, and $L=K(\alpha)$ a finite field extension of $K$ defined by a monic irreducible polynomial $f(x)\in R[x]$. We give an easy version of Dedekind's criterion which computationally improves…

As an extension of the intertwining operator idea, an algebraic method which provides a link between supersymmetric quantum mechanics and quantum (super)integrability is introduced. By realization of the method in two dimensions, two…