Related papers: On Collatz Conjecture for binary polynomials

We prove a binomial formula for Macdonald polynomials and consider applications of it.

In this paper, we will introduce an extension to the Collatz's conjecture. This conjecture may be seen as a general conjecture that unifies the Collatz one together with many other similar conjectures. For instance, we propose our new…

Recently, an analogue over $\mathbb{F}_q[T]$ of Landau's theorem on sums of two squares was considered by Bary-Soroker, Smilansky and Wolf. They counted the number of monic polynomials in $\mathbb{F}_q[T]$ of degree $n$ of the form…

For a long time, Collatz Conjecture has been assumed to be true, although a formal proof has eluded all efforts to date. In this article, evidence is presented that suggests such an assumption is incorrect. By analysing the stopping times…

We prove the Milnor conjecture for Lie groups and the Friedlander conjecture for complex algebraic Lie groups.

The Collatz conjecture is a famous math problem that was introduced by Lothar Collatz in 1937, and nobody has yet succeeded in proving or disproving it. In this article, I will analyze this problem with a new approach and I will discuss my…

Permutation polynomials are an interesting subject of mathematics and have applications in other areas of mathematics and engineering. In this paper, we develop general theorems on permutation polynomials over finite fields. As a…

A class of bilinear permutation polynomials over a finite field of characteristic 2 was constructed in a recursive manner recently which involved some other constructions as special cases. We determine the compositional inverses of them…

In this research, an optimal algorithm for the Collatz conjecture is presented. Properties such as the convergence of the algorithm and an equation that relates the algorithm to the classical Collatz conjecture are obtained. It is validated…

We give an improved polynomial bound on the complexity of the equation solvability problem, or more generally, of finding the value sets of polynomials over finite nilpotent rings. Our proof depends on a result in additive combinatorics,…

This paper investigates whether or not polynomials that are irreducible over $\mathbb{Q}$ and $\mathbb{Z}$ can remain irreducible under substitution by all quadratic polynomials. It answers this question in the negative in the degree 2 and…

We give a new proof of Koll\'ar's conjecture on the pushforward of the dualizing sheaf twisted by a variation of Hodge structure. This conjecture was settled by M. Saito via mixed Hodge modules and has applications in the investigation of…

We provide sufficient conditions for systems of polynomial equations over general (real or complex) algebras to have a solution. This generalizes known results on quaternions, octonions and matrix algebras. We also generalize the…

In this note we prove that the factorization theorem for dominated polynomials previously proved by the authors is equivalent to an alternative factorization scheme that uses classical linear techniques and a linearization process. However,…

This paper contains the proof of difference counterparts of the conjectures due to Keven Kadell on symmetric and anti-symmetric Macdonald polynomials.

In this paper we prove that assuming Schanuel's conjecture, an exponential polynomial in one variable over the algebraic numbers has only finitely many algebraic solutions. This implies a positive answer to Shapiro's conjecture for…

We present combinatorial rules (one theorem and two conjectures) concerning three bases of Z[x1,x2,....]. First, we prove a "splitting" rule for the basis of key polynomials [Demazure '74], thereby establishing a new positivity theorem…

We prove a recent conjecture by Ulas on reducible polynomial substitutions.

We consider the issue of when the L-polynomial of one curve over $\F_q$ divides the L-polynomial of another curve. We prove a theorem which shows that divisibility follows from a hypothesis that two curves have the same number of points…

We study solutions of exponential polynomials over the complex field. Assuming Schanuel's conjecture we prove that certain polynomials have generic solutions in the complex field.