Colorful combinatorics and Macdonald polynomials

The non-negative integer cocharge statistic on words was introduced in the 1970s by Lascoux and Schützenberger to combinatorially characterize the Hall–Littlewood polynomials. Cocharge has since been used to explain phenomena ranging from the graded decomposition of Garsia–Procesimodules to the cohomology structure of the Grassmann variety. Although its application to contemporary variations of these problems had been deemed intractable, we prove that the two-parameter, symmetric Macdonald polynomials are generating functions of a distinguished family of colored words. Cocharge adorns one parameter and the second measure its deviation from cocharge on words without color. We use the same framework to expand the plactic monoid, apply Kashiwara’s crystal theory to various Garsia–Haiman modules, and to address problems in $K$-theoretic Schubert calculus.     

Very hyperbolic and stably hyperbolic polynomials

A real polynomial in one variable is hyperbolic if it has only real roots. A hyperbolic polynomial is very hyperbolic if it has hyperbolic primitives of all orders. A polynomial P is stably hyperbolic if $x^{k}P+Q$ is hyperbolic for suitable $k\in \mathbf{N}$ and Q (polynomial of degree $?k?1$). We present some geometric properties of the domains of very hyperbolic and of stably hyperbolic polynomials in the family $x^n+a_1{x}^{n?1}+?+a_n$. To cite this article: V.P. Kostov, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

On the combinatorics of the Boros-Moll polynomials

The Boros-Moll polynomials arise in the evaluation of a quartic integral. The original double summation formula does not imply the fact that the coefficients of these polynomials are positive. Boros and Moll proved the positivity by using Ramanujan’s Master Theorem to reduce the double sum to a single sum. Based on the structure of reluctant functions introduced by Mullin and Rota along with an extension of Foata’s bijection between Meixner endofunctions and bi-colored permutations, we find a combinatorial proof of the positivity. In fact, from our combinatorial argument one sees that it is essentially the binomial theorem that makes it possible to reduce the double sum to a single sum.     

Topology and combinatorics of partitions of masses by hyperplanes

An old problem in combinatorial geometry is to determine when one or more measurable sets in Rd admit an equipartition by a collection of k hyperplanes [B. Grünbaum, Partitions of mass-distributions and convex bodies by hyperplanes, Pacific J. Math. 10 (1960) 1257-1261]. A related topological problem is the question of (non)existence of a map , equivariant with respect to the Weyl group Wk=Bk:=(Z/2)^⊕k?Sk, where U is a representation of Wk and S(U)⊂U the corresponding unit sphere. We develop general methods for computing topological obstructions for the existence of such equivariant maps. Among the new results is the well-known open case of 5 measures and 2 hyperplanes in R⁸ [E.A. Ramos, Equipartitions of mass distributions by hyperplanes, Discrete Comput. Geom. 15 (1996) 147-167]. The obstruction in this case is identified as the element 2Xab∈H₁(D₈;Z)≅Z/4, where Xab is a generator, which explains why this result cannot be obtained by the parity count formulas of Ramos [loc. cit.] or the methods based on either Stiefel-Whitney classes or ideal valued cohomological index theory [E. Fadell, S. Husseini, An ideal-valued cohomological index theory with applications to Borsuk-Ulam and Bourgin-Yang theorems, Ergodic Theory Dynam. Systems ^8* (1988) 73-85].     

Bipartitional polynomials and their applications in combinatorics and statistics

Bipartitional polynomials are multivariable polynomials Y_mn=Y_mn(cy₀₁,cy₁₀,cy₁₁,…,cy_mn), c^k≡c_k, defined by a sum over all partitions of the bipartite number (mn). Recurrence relations, generating functions and some basic properties of these polynomials are given. Applications in Combinatorics and Statistics are briefly indicated.     

On weighted hyperbolic polynomials

The results by Görding, Larsson, Cattabriga, Rodino, Calvo on correctness of Cauchy problem for N-hyperbolic equations are generalized. We prove that in the general case where the vector N = (N ₁, …,N _n ) is different from the vector (1, 0, …, 0), for the correctness of the Cauchy problem more stronger condition is required, which we call weighted hyperbolicity condition. We also discuss the properties of polynomials possessing weighted hyperbolicity property.     

Peripheral polynomials of hyperbolic knots

If K is a hyperbolic knot in S³, an algebraic component of its character variety containing one holonomy of the complete hyperbolic structure of finite volume of S³K is an algebraic curve . The traces of the peripheral elements of K define polynomial functions in , which are related in pairs by polynomials (peripheral polynomials). These are determined by just two adjacent peripheral polynomials. The curves defined by the peripheral polynomials are all birationally equivalent to , with only one possible exception. The canonical peripheral polynomial relating the trace of the meridian with the trace of the canonical longitude of K, is a factor of the A-polynomial.     

Gromov hyperbolic spaces

A mini monograph on Gromov hyperbolic spaces, which need not be geodesic or proper.     

Detecting hyperbolic and definite matrix polynomials

Hyperbolic or more generally definite matrix polynomials are important classes of Hermitian matrix polynomials. They allow for a definite linearization and can therefore be solved by a standard algorithm for Hermitian matrices. They have only real eigenvalues which can be characterized as minmax and maxmin values of Rayleigh functionals, but there is no easy way to test if a given polynomial is hyperbolic or definite or not. Taking advantage of the safeguarded iteration which converges globally and monotonically to extreme eigenvalues we obtain an efficient algorithm that identifies hyperbolic or definite polynomials and enables the transformation to an equivalent definite linear pencil. Numerical examples demonstrate the efficiency of the approach.     

Singular spaces of matrices and their application in combinatorics

We study linear spaces of n×n matrices in which every matrix is singular. Examples are given to illustrate that a characterization of such subspaces would solve various open problems in combinatorics and in computational algebra. Several important special cases of the problem are solved, although often in disguise. Research supported by Hungarian National Research fund No. 1812.     

Interpolation and integral norms of hyperbolic polynomials

The integral norm on the subspace of multivariate trigonometric polynomials with harmonics from the “hyperbolic cross” is equivalent to the interpolation norm taken on a finite set of points whose cardinality increases only slightly faster than the dimension of the subspace. Translated fromMatematicheskie Zametki, Vol. 66, No. 1, pp. 20–29, July, 1999.     

Self-self-dual spaces of polynomials

A space of polynomials V of dimension 7 is called self-dual if the divided Wronskian of any 6-subspace is in V. A self-dual space V has a natural inner product. The divided Wronskian of any isotropic 3-subspace of V is a square of a polynomial. We call V self-self-dual if the square root of the divided Wronskian of any isotropic 3-subspace is again in V. We show that the self-self-dual spaces have a natural non-degenerate skew-symmetric 3-form defined in terms of Wronskians.We show that the self-self-dual spaces correspond to G₂-populations related to the Bethe Ansatz of the Gaudin model of type G₂ and prove that a G₂-population is isomorphic to the G₂ flag variety.     

On a class of hyperbolic polynomials

In the present paper, we investigate whether the roots of a biquadratic equation determined by a pair of real symmetric positive definite matrices of order 3 and a three-dimensional vector of parameters are real. We obtain the explicit representation of the discriminant of such a polynomial as the sum of at most two squares.     

Isometries in hyperbolic spaces

Suppose that f:Hn → Hn (n≥2) maps any r-dimensional hyperplane (1≤rn) into an r-dimensional hyperplane. In this paper, we prove that f is an isometry if and only if f is a surjective map. This result gives an affirmative answer to a recent conjecture due to Li and Yao.     

Difference equations in combinatorics,number theory,and orthogonal polynomials

In this paper we give an overview of some recent developments in the asymptotics of difference equations. Applications to combinatorics and orthogonal polynomials are briefly indicated. Some conjectures and open problems are presented     

The combinatorics of weight systems and characteristic polynomials of isolated quasihomogeneous singularities

Journal of Algebraic Combinatorics - A paper of the first author and Zilke proposed seven combinatorial problems around formulas for the characteristic polynomial and the exponents of an isolated...     

On k-symmetries of hyperbolic spaces

We determine all the possible pointwise k-symmetric spaces of negative constant curvature. In general, such spaces are not k-symmetric.In fact we show that, for all $n?3$, $k\ne 2$, $H^n$ is not k-symmetric, i.e., for any set of selected k-symmetries, one for each point of $H^n$, the regularity condition does not hold.     

Realizing Alexander polynomials by hyperbolic links

We realize a given (monic) Alexander polynomial by a (fibered) hyperbolic arborescent knot and link having any number of components, and by infinitely many such links having at least 4 components. As a consequence, a Mahler measure minimizing polynomial, if it exists, is realized as the Alexander polynomial of a fibered hyperbolic link of at least 2 components. For a given polynomial, we also give an upper bound for the minimal hyperbolic volume of knots/links realizing the polynomial and, in the opposite direction, construct knots of arbitrarily large volume, which are arborescent, or have given free genus at least 2.