Related papers: A remark on a theorem of M. Haiman
We present a restricted variable generalization of Warning's Second Theorem (a result giving a lower bound on the number of solutions of a low degree polynomial system over a finite field, assuming one solution exists). This is analogous to…
We calculate the E-polynomials of certain twisted GL(n,C)-character varieties M_n of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lie-type GL(n,F_q) and a theorem proved in the…
A general method based on the polynomial deformations of the Lie algebra sl(2,R) is proposed in order to exhibit the quasi-exactly solvability of specific Hamiltonians implied by quantum physical models. This method using the…
In this letter, we prove an inequality involving alternating binomial logarithmic sums by exploiting the variance of the logarithm of the maximum of independent and identically distributed exponential random variables. This inequality was…
Twisted Lie algebroid cohomologies, i.e. with values in representations, are shown to be Lie algebroid homotopy-invariant. Several important classes of examples are discussed. As an application, a generalized version of the Poincar\'e lemma…
Limiting cases are studied of the Koornwinder-Macdonald multivariable generalization of the Askey-Wilson polynomials. We recover recently and not so recently introduced families of hypergeometric orthogonal polynomials in several variables…
We establish sharp estimates that adapt the polynomial method to arbitrary varieties. These include a partitioning theorem, estimates on polynomials vanishing on fixed sets and bounds for the number of connected components of real algebraic…
The paired Hayman's conjecture of different types are considered. More accurately speaking, the zeros of a pair of $f^nL(z,g)-a_1(z)$ and $g^mL(z,f)-a_2(z)$ are characterized using different methods from those previously employed, where $f$…
We introduce a number of new tools for the study of relatively hyperbolic groups. First, given a relatively hyperbolic group G, we construct a nice combinatorial Gromov hyperbolic model space acted on properly by G, which reflects the…
Since the introduction of Stein's method in the early 1970s, much research has been done in extending and strengthening it; however, there does not exist a version of Stein's original method of exchangeable pairs for multivariate normal…
In this note, we give an alternate proof of the multinomial theorem using a probabilistic approach. Although the multinomial theorem is basically a combinatorial result, our proof may be simpler for a student familiar with only basic…
We give an alternate proof of the main theorem of Kawamata's paper: Pluricanonical systems on minimal algebraic varieties. Our proof also works for varieties in class $\mathcal C$. We note that our proof is completely different from…
We give combinatorial proofs of two multivariate Cayley--Hamilton type theorems. The first one is due to Phillips (Amer. J. Math., 1919) involving $2k$ matrices, of which $k$ commute pairwise. The second one regards the mixed discriminant,…
We construct a class of permutation polynomials of $\bF_{2^m}$ that are closely related to Dickson polynomials.
We bound the greatest common divisor of two coprime multivariable polynomials evaluated at algebraic numbers, generalizing work of Levin, and going towards conjectured inequalities of Silverman and Vojta. As an application, we prove results…
The generating polynomial of permutations of size $n$, counted by the number of alternating runs, has a root at $-1$ of multiplicity $\lfloor (n-2)/2 \rfloor$ for all $n \ge 2$. This result can be derived by combining the David--Barton…
We introduce the notions of alternating roots of polynomials and alternating polynomials over a Cayley-Dickson algebra, and prove a connection between the alternating roots of a given polynomial and the roots of the corresponding…
We use the Gromov-Witten invariants and a nonsqueezing theorem by the author to affirm a conjecture by P.Biran on the Lagrangian barriers.
The multivariate Hahn polynomials are constructed explicitly as the common eigenvectors of a family of second order difference operators. They are orthogonal with respect to the hypergeometric multinomial distribution. The main difference…
We construct examples of commuting ordinary scalar differential operators with polynomial coefficients that are related to a spectral curve of an arbitrary genus g and to an arbitrary even rank r = 2k, and also to an arbitrary rank of the…