Related papers: From Schanuel's Conjecture to Shapiro's Conjecture

We generalize the completely monotone conjecture ([CG15]) from Shannon entropy to the Tsallis entropy up for orders up to at least four. To this end, we employ the algorithm ([J\"un16, JM06a]) which employs the technique of systematic…

We prove two polynomial identities which are particular cases of a conjecture arising in the theory of L-functions of twisted Carlitz modules. This conjecture is stated in earlier papers of the second author.

In this paper, we obtain new results on the critical points of a polynomial. We discuss the Sendov conjecture for polynomials of degree nine.

Fox's conjecture (1962) states that the sequence of absolute values of the coefficients of the Alexander polynomial of alternating links is trapezoidal. While the conjecture remains open in general, a number of special cases have been…

We present positivity conjectures for the Schur expansion of Jack symmetric functions in two bases given by binomial coefficients. Partial results suggest that there are rich combinatorics to be found in these bases, including Eulerian…

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…

Pseudoexponential fields are exponential fields similar to complex exponentiation satisfying the Schanuel Property, which is the abstract statement of Schanuel's Conjecture, and an adapted form of existential closure. Here we show that if…

If $\mathcal{P}$ is a lattice polytope (i.e., $\mathcal{P}$ is the convex hull of finitely many integer points in $\mathbb{R}^d$), Ehrhart's famous theorem (1962) asserts that the integer-point counting function $|t \mathcal{P} \cap…

We report results on various techniques which allow to compute the expansion into Legendre (or in general Gegenbauer) polynomials in an efficient way. We describe in some detail the algebraic/symbolic approach already presented in Ref.1 and…

We reduce the Nowicki conjecture on the Weitzenb\"ock derivation of polynomial algebras to well-known problem of the classical invariant theory.

We prove in this paper the Ax-Schanuel conjecture for all admissible variations of mixed Hodge structures.

In this paper we establish an exponential covering theorem implying a conjecture formulated by A. Zygmund circa 1935 whose three-dimensional case was obtained by the first named author in 1978.

We prove Samuel's conjecture on certain Graham positivity of the expansion coefficient of two double Schubert polynomials in three sets of variables by establishing a refined version of Graham's positivity theorem. As a corollary, we prove…

We investigate necessary and sufficient conditions for an arbitrary polynomial of degree $n$ to be trivial, i.e. to have the form $a(z-b)^n$. These results are related to an open problem, conjectured in 2001 by E. Casas- Alvero. It says,…

In this paper we study the genralized q-Euler numbers and polynomials. From our results, we derive some interesting congruences related tothe generalized q-Euler numbers.

We prove a conjecture of H.Widom stated in [W] (math/0108008) about the reality of eigenvalues of certain infinite matrices arising in asymptotic analysis of large Toeplitz determinants. As a byproduct we obtain a new proof of A.Okounkov's…

In this paper we prove the Dynamical Mordell-Lang Conjecture for polynomial endomorphisms of the affine plane.

We give a new demonstration of Loewner's characterization of polynomials, solving in the positive a conjecture proposed by Laird and McCann in 1984.

The celebrated Mason's conjecture states that the sequence of independent set numbers of any matroid is log-concave, and even ultra log-concave. The strong form of Mason's conjecture was independently solved by Anari, Liu, Oveis Gharan and…

We provide explicit combinatorial formulas for the Chow polynomial and for the augmented Chow polynomial of uniform matroids, thereby proving a conjecture by Ferroni. These formulas refine existing formulas by Hampe and by Eur, Huh, and…