Related papers: From Schanuel's Conjecture to Shapiro's Conjecture
In this paper we prove two conjectures stated by Chao-Ping Chen in [Int. Trans. Spec. Funct. 23:12 (2012), 865--873], using a method for proving inequalities of mixed trigonometric polynomial functions.
We reveal a complex analogue to a result about polynomial solutions to the Dirichlet Problem on ellipsoids in $\mathbb{R}^n$ by showing that the Bergman projection on any ellipsoid in $\mathbb{C}^n$ is such that the projection of any…
Borwein and Choi conjectured that a polynomial $P(x)$ with coefficients $\pm1$ of degree $N-1$ is cyclotomic iff $$P(x)=\pm \Phi_{p_1}(\pm x)\Phi_{p_2}(\pm x^{p_1})\cdots \Phi_{p_r}(\pm x^{p_1p_2\cdots p_{r-1}})$$ where $N=p_1p_2\cdots…
We prove Szpiro's conjecture for elliptic curves over the rationals having $j$-invariant with denominator of logarithmic size with respect to its numerator.
In [1], J. Ax proved a transcendency theorem for certain differential fields of characteristic zero: the differential counterpart of the still open Schanuel's conjecture about the exponential function over the field of complex numbers [11,…
It is conjectured that all separable polynomials with integers coefficients, satisfying some local conditions, take infinitely many square free values on integer arguments. But not a single polynomial of degree greater than $3$ is proven to…
The Casas-Alvero conjecture is about interpolation polynomials. There are some partial proofs of it, but there is not any proof in the general case.In this paper we propose three.
We extend the authors' previous work on Wiener-Wintner double recurrence theorem to the case of polynomials.
Recently, Z.-W. Sun introduced two kinds of polynomials related to the Delannoy numbers, and proved some supercongruences on sums involving those polynomials. We deduce new summation formulas for squares of those polynomials and use them to…
The paper [GLZ] "L-functions of Carlitz modules, resultantal varieties and rooted binary trees" is devoted to a description of some resultantal varieties related to L-functions of Carlitz modules. It contains a conjecture that some of these…
In this paper we state some conjectures about q-Fibonacci polynomials which for q=1 reduce to well-known results about Fibonacci numbers and Fibonacci polynomials.
Schanuel Conjecture contains all ``reasonable" statements that can be made on the values of the exponential function. In particular it implies the Lindemann-Weierstrass Theorem. In my Ph.D. I showed that Schanuel Conjecture has a…
In this paper I prove a conjecture which gives a lower bound for the largest absolute value of the coefficients of the n-th cyclotomic polynomial for some n. Moreover this estimate is essentially sharp.
We give an upper bound for the exponential sum over squarefree integers. This establishes a conjecture by Br\"udern and Perelli.
In 2007, Dmytrenko, Lazebnik and Williford posed two related conjectures about polynomials over finite fields. Conjecture~1 is a claim about the uniqueness of certain monomial graphs. Conjecture~2, which implies Conjecture~1, deals with…
Assuming Schanuel's conjecture, we prove that the complete theory $T_{\exp}$ of the real exponential field is axiomatized by the axioms of definably complete exponential fields satisfying $\exp' = \exp$. This implies the result of Macintyre…
Let R denote the reals, and let h: R^n --> R be a continuous, piecewise-polynomial function. The Pierce-Birkhoff conjecture (1956) is that any such h is representable in the form sup_i inf_j f_{ij}, for some finite collection of polynomials…
We present a proof of the Casas-Alvero conjecture, stating that if a complex polynomial has a root in common with each of its derivatives it must be a multiple of the power of some monomial.
Given a subfield $F$ of $\mathbb{C}$, we study the linear disjointess of the field $E$ generated by iterated exponentials of elements of $\overline{F}$, and the field $L$ generated by iterated logarithms, in the presence of Schanuel's…
In this article, we prove a weighted version of Saitoh's conjecture. As an application, we prove a weighted version of Saitoh's conjecture for higher derivatives.