Related papers: Exponential prefixed polynomial equations
Let $P(x) \in \mathbb{Z}[x]$ be a polynomial. We give an easy and new proof of the fact that the set of primes $p$ such that $p \mid P(n)$, for some $n \in \mathbb{Z}$, is infinite. We also get analog of this result for some special…
The notion of Ehrhart tensor polynomials, a natural generalization of the Ehrhart polynomial of a lattice polytope, was recently introduced by Ludwig and Silverstein. We initiate a study of their coefficients. In the vector and matrix…
This is an expository paper aiming to introduce Zilber's Exponential Closedness conjecture to a general audience. Exponential Closedness predicts when (systems of) equations involving addition, multiplication, and exponentiation have…
We consider polynomial equations, or systems of polynomial equations, with integer coefficients, modulo prime numbers $p$. We offer an elementary approach based on a counting method. The outcome is a weak form of the Lang-Weil lower bound…
We formulate an exponential Diophantine equation, which is is some sense one order higher that Fermat's Last Theorem. We also give three examples of solutions to this exponential Diophantine equation and formulate a conjecture.
We investigate which polynomials can possibly occur as factors in the denominators of rational solutions of a given partial linear difference equation (PLDE). Two kinds of polynomials are to be distinguished, we call them /periodic/ and…
A polynomial triangle is an array whose inputs are the coefficients in integral powers of a polynomial. Although polynomial coefficients have appeared in several works, there is no systematic treatise on this topic. In this paper we plan to…
Cubic and quartic non-autonomous differential equations with continuous piecewise linear coefficients are considered. The main concern is to find the maximum possible multiplicity of periodic solutions. For many classes, we show that the…
This paper explores the Prouhet-Tarry-Escott problem (PTE), the Generalized PTE problem (GPTE), and the Fermat form of Generalized PTE problem (FPTE). The GPTE problem extends the PTE problem by allowing different sets of exponents, while…
In this note we determine all power series $F(X)\in 1+X\F_p[[X]]$ such that $(F(X+Y))^{-1} F(X)F(Y)$ has only terms of total degree a multiple of $p$. Up to a scalar factor, they are all the series of the form $F(X)=E_p(cX)\cdot G(X^p)$ for…
We investigate exponential sums over singular binary quartic forms, proving an explicit formula for the finite field Fourier transform of this set. Our formula shares much in common with analogous formulas proved previously for other vector…
A new family of polynomials, called cumulant polynomial sequence, and its extensions to the multivariate case is introduced relied on a purely symbolic combinatorial method. The coefficients of these polynomials are cumulants, but depending…
This work proposes elementary proofs of several related primes counting problems, based on an elementary weighted sieve. The subsets of primes considered here are the followings: the subset of twin primes PT = {p and p + 2 are primes}, the…
We investigate a scalar characteristic exponential polynomial with complex coefficients associated with a first order scalar differential-difference equation. Our analysis provides necessary and sufficient conditions for allocation of the…
We give conditions ensuring that the Fatou set and the complement of the fast escaping set of an exponential polynomial $f$ have finite Lebesgue measure. Essentially, these conditions are designed such that $|f(z)|\ge\exp(|z|^\alpha)$ for…
We study differential equations $F(y,...,y^{(n)})=0$ where $F(Y_0,...,Y_n)$ is a formal series in $Y_0,...,Y_n$ with coefficients in some field of \emph{generalized power series} $\mathds{K}_r$ with finite rank $r\in\mathbb{N}^*$. Our…
We consider the problem of evaluating certain exponential sums. These sums take the form $\sum_{x_1,...,x_n \in Z_N} e^{f(x_1,...,x_n) {2 \pi i / N}} $, where each x_i is summed over a ring Z_N, and f(x_1,...,x_n) is a multivariate…
Under the generalized Lindel\"of Hypothesis in the t- and q-aspects, we bound exponential sums with coefficients of Dirichlet series belonging to a certain class. We use these estimates to establish a conditional result on squares of Hecke…
A $1$-prefix normal word is a binary word with the property that no factor has more $1$s than the prefix of the same length; a $0$-prefix normal word is defined analogously. These words arise in the context of indexed binary jumbled pattern…
The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an…