Related papers: Critical points of master functions and integrable…

We consider the population of critical points generated from the critical point of the master function with no variables, which is associated with the trivial representation of the affine Lie algebra $A^2_2$. We describe how the critical…

We consider the population of critical points generated from the critical point of the master function with no variables, which is associated with the trivial representation of the twisted affine Lie algebra $A^{(2)}_{2n}$. The population…

We consider the population of critical points, generated from the critical point of the master function with no variables, which is associated with the trivial representation of the twisted affine Lie algebra $C_n^{(1)}$. The population is…

We consider the master functions associated with one irreducible integrable highest weight representation of a Kac-Moody algebra. We study the generation procedure of new critical points from a given critical point of one of these master…

We consider critical points of master functions associated with integral dominant weights of Kac-Moody algebras and introduce a generating procedure constructing new critical points starting from a given one. The set of all critical points…

Critical points of a master function associated to a simple Lie algebra \g come in families called the populations [MV1]. We prove that a population is isomorphic to the flag variety of the Langlands dual Lie algebra \g^t. The proof is…

In this paper we prove a multiplicity result concerning the critical points of a class of functionals involving local and nonlocal nonlinearities. We apply our result to the nonlinear Schrodinger-Maxwell system and to the nonlinear elliptic…

The hypergeometric solutions to the KZ equation contain a certain symmetric ``master function'', [SV]. Asymptotics of the solutions correspond to critical points of the master function and give Bethe vectors of the inhomogeneous Gaudin…

We consider critical points of a class of functionals on compact four-dimensional manifolds arising from Regularized Determinants for conformally covariant operators, whose explicit form was derived in [10], extending Polyakov's formula.…

We show that computing even very coarse approximations of critical points is intractable for simple classes of nonconvex functions. More concretely, we prove that if there exists a polynomial-time algorithm that takes as input a polynomial…

We consider the problem of computing critical points of the restriction of a polynomial map to an algebraic variety. This is of first importance since the global minimum of such a map is reached at a critical point. Thus, these points…

A positive structure on the varieties of critical points of master functions for KZ equations is introduced. It comes as a combination of the ideas from classical works by G.Lusztig and a previous work by E.Mukhin and the second named…

We prove a Chern-Lashof type formula computing the expected number of critical points of smooth function on a smooth manifold $M$ randomly chosen from a finite dimensional subspace $V\subset C^\infty(M)$ equipped with a Gaussian probability…

We prove an abstract critical point theorem based on a cohomological index theory that produces pairs of nontrivial critical points with nontrivial higher critical groups. This theorem yields pairs of nontrivial solutions that are neither…

We establish an extension of Liouville's classical representation theorem for solutions of the partial differential equation $\Delta u=4 e^{2u}$ and combine this result with methods from nonlinear elliptic PDE to construct holomorphic maps…

Let V $\subset$ C n be an equidimensional algebraic set and g be an n-variate polynomial with rational coefficients. Computing the critical points of the map that evaluates g at the points of V is a cornerstone of several algorithms in real…

Given any n-tuple of complex numbers, one can canonically define a polynomial of degree n+1 that has the entries of this n-tuple as its critical points. In 2002, Beardon, Carne, and Ng studied a map $\theta\colon \mathbb{C}^n\to…

Given a smoothly bounded non-contractible domain $\Omega\subset \mathbb{R}^2$, we prove the existence of positive critical points of the Trudinger-Moser embedding for arbitrary Dirichlet energies. This is done via degree theory, sharp…

A rational function is the ratio of two complex polynomials in one variable without common roots. Its degree is the maximum of the degrees of the numerator and the denominator. Rational functions belong to the same class if one turns into…

Let $p_n$ be a random, degree $n$ polynomial whose roots are chosen independently according to the probability measure $\mu$ on the complex plane. For a deterministic point $\xi$ lying outside the support of $\mu$, we show that almost…