Polynomial-exponential Equations Research Articles

Solutions of equations involving the modular 𝑗 function

Inspired by work done for systems of polynomial exponential equations, we study systems of equations involving the modular j j function. We show general cases in which these systems have solutions, and then we look at certain situations in which the modular Schanuel conjecture implies that these systems have generic solutions. An unconditional result in this direction is proven for certain polynomial equations on j j with algebraic coefficients.

Quasi-periodic invariant 2-tori in a delayed BAM neural network

In this paper, we consider a four-neuron bi-directional associative memory (BAM, for short) neural network with two delays. We choose connection weights and the sum of delays as bifurcation parameters and derive the critical values where a double Hopf bifurcation may occur by analyzing the associated characteristic equation which is a fourth-degree polynomial exponential equation. Meanwhile, we obtain some parameter conditions on the existence of invariant 2-tori of the truncated normal form near the bifurcation point by the center manifold theorem and normal form method. Despite the fact that the higher-degree terms may destroy the invariant 2-tori of the truncated normal form, we prove that the neural network model has quasi-periodic invariant 2-tori for most of the parameter set where the truncated normal form possesses invariant 2-tori in a sufficiently small neighborhood of the bifurcation point. Numerical examples and simulations are given to support the theoretical analysis.

A Bond Work index mill ball charge and closing screen product size distributions for grinding crystalline grains

Ball mill charge size distributions originally specified for Bond grindability tests to determine the Bond Work index (BWi) are not commercially available. Those proposed to date do not match all of Bond's (1961) original specifications of total ball mass, number and surface area. An alternative mill ball charge is proposed that closely approximates Bond's original total ball mass, number of balls and ball surface area.Results of 30 Bond Work index tests of six pure materials (calcite, magnesite, labradorite (feldspar), quartz, andalusite and glass) using closing screen apertures (P1) values of 500, 250, 125, 90 and 63μm are analysed. The samples were selected on the basis of having distinct hardness's (Mohs hardness 3 to 7.5), being relatively free of crystallographic defects and having distinct cleavage properties The 80 percentile size of the fines produced (P80) concur with those of published values.The trend based of P80 values for P1 values of 150 to 44μm recommended by Bond poorly fit with published P1 values >250μm. It is demonstrated that the BWi test P1 values of mono-mineralogical or mono-material samples determines the P80 value obtained. Based on test results of this investigation a polynomial exponential equation relating P80 to P1 values with an R2 correlation factor of 0.9977 is presented. This relation is independent of the tested material hardness and crystallographic defects and planes of weakness.

A polynomial–exponential equation related to the Ramanujan–Nagell equation

For any given odd prime p and a fixed positive integer D prime to p, we study the equation \(x^2+D^m=p^n\) in positive integers x, m and n. We use a classical work of Dem’janenko in 1965 on a certain quadratic Diophantine equation together with some results concerning the existence of primitive divisors of Lucas sequences to examine our equation when D is a product of \(p-1\) and a square.

The orbit intersection problem for linear spaces and semiabelian varieties

Let f1, f2 : CN −→ CN be affine maps fi(x) := Aix + yi (where each Ai is an N -by-N matrix and yi ∈ CN ), and let x1,x2 ∈ AN (C) such that xi is not preperiodic under the action of fi for i = 1, 2. If none of the eigenvalues of the matrices Ai is a root of unity, then we prove that the set {(n1, n2) ∈ N0 : f n1 1 (x1) = f n2 2 (x2)} is a finite union of sets of the form {(m1k + `1,m2k + `2) : k ∈ N0} where m1,m2, `1, `2 ∈ N0. Using this result, we prove that for any two self-maps Φi(x) := Φi,0(x) + yi on a semiabelian variety X defined over C (where Φi,0 ∈ End(X) and yi ∈ X(C)), if none of the eigenvalues of the induced linear action DΦi,0 on the tangent space at 0 ∈ X is a root of unity (for i = 1, 2), then for any two non-preperiodic points x1, x2, the set {(n1, n2) ∈ N0 : Φ n1 1 (x1) = Φ n2 2 (x2)} is a finite union of sets of the form {(m1k + `1,m2k + `2) : k ∈ N0} where m1,m2, `1, `2 ∈ N0. We give examples to show that the above condition on eigenvalues is necessary and introduce certain geometric properties that imply such a condition. Our method involves an analysis of certain systems of polynomial-exponential equations and the padic exponential map for semiabelian varieties.

Certifying solutions to square systems of polynomial-exponential equations

Smale's α-theory certifies that Newton iterations will converge quadratically to a solution of a square system of analytic functions based on the Newton residual and all higher order derivatives at the given point. Shub and Smale presented a bound for the higher order derivatives of a system of polynomial equations based in part on the degrees of the equations. For a given system of polynomial-exponential equations, we consider a related system of polynomial-exponential equations and provide a bound on the higher order derivatives of this related system. This bound yields a complete algorithm for certifying solutions to polynomial-exponential systems, which is implemented in alphaCertified. Examples are presented to demonstrate this certification algorithm.

Polynomial–exponential equations and Zilber's conjecture

Assuming Schanuel's conjecture, we prove that any polynomial exponential equation in one variable must have a solution that is transcendental over a given finitely generated field. With the help of some recent results in Diophantine geometry, we obtain the result by proving (unconditionally) that certain polynomial exponential equations have only finitely many rational solutions. This answers affirmatively a question of David Marker, who asked, and proved in the case of algebraic coefficients, whether at least the one-variable case of Zilber's strong exponential-algebraic closedness conjecture can be reduced to Schanuel's conjecture.

On the Diophantine equation [formula omitted

Recently, mixed polynomial–exponential equations similar to the one in the title have been considered by many authors. In these results a certain non-coprimality condition plays an important role. In this paper we completely solve the title equation for odd positive integers x with x<50. Since we avoid the mentioned non-coprimality condition, this can be considered as a partial completion of the above mentioned results. It seems that the deep effective tools (such as Baker's method) alone are not capable to handle the problem. We combine local arguments and Baker's method to prove our results.

RATIONAL SOLUTIONS OF POLYNOMIAL-EXPONENTIAL EQUATIONS

We give a description of solutions of polynomial-exponential equations where the variables vary over rational numbers. Using this, we also prove a finiteness result.

Polynomial-exponential equations involving multi-recurrences

In this paper we consider polynomial-exponential Diophantine equations of the form\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$G_n^{(0)} y^d + G_n^{(1)} y^{d - 1} + \cdots + G_n^{(d - 1)} y + G_n^{(d)} = 0$$ \end{document}whereGn(i)are multi-recurrences, i.e. polynomial-exponential functions in variablesn= (n1,...,nk). Under suitable (but restrictive) conditions we prove that there are finitely many multi-recurrencesHn(1),...,Hn(s)such that for all solutions (n1,...,nk,y) ∈ ℕk× ℤ we either have\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$H_n^{(i)} = 0 or y = H_n^{(j)}$$ \end{document}for certain 1 ≦i,j≦s, respectively. This generalizes earlier results of this type on such equations. The proof uses a recent result by Corvaja and Zannier.

Polynomial-exponential equations involving several linear recurrences

Polynomial-exponential equations involving several linear recurrences

Ternary Diophantine Equations via Galois Representations and Modular Forms

AbstractIn this paper, we develop techniques for solving ternary Diophantine equations of the shape Axn + Byn = Cz2, based upon the theory of Galois representations and modular forms. We subsequently utilize these methods to completely solve such equations for various choices of the parameters A, B andC. We conclude with an application of our results to certain classical polynomial-exponential equations, such as those of Ramanujan–Nagell type.

Polynomial-exponential equations and linear recurrences

Let K be an algebraic number eld and let (Gn) be a linear recurring sequence dened by Gn = 1 n +P2(n) n + +Pt(n) n , where 1; 1; : : : ; t are non-zero elements of K and where Pi(x) 2 K(x) for i = 2; : : : ; t. Furthermore let f(z; x) 2 K(z; x) monic in x. In this paper we want to study the polynomial{exponential Diophantine equation f(Gn; x) = 0. We want to use a quantitative version of W. M. Schmidt's Subspace Theorem (due to J.-H. Evertse (8)) to calculate an upper bound for the number of solutions (n; x) under some additional assumptions.

The Number of Solutions of Polynomial-Exponential Equations

We will give explicit bounds for the number of solutions of polynomial-exponential equations. In contrast to earlier work, the bounds are independent of the coefficients of the equations, and they are of only single exponential growth in the number of coefficients.

The set of solutions of a polynomial-exponential equation

l∈λ Pl(x)αl = 0 (λ ∈ P). Denote by S(P) the set of solutions of (1.1)P which do not satisfy (1.1)Q for any proper refinement Q of P (notice that every solution of (1.1) lies in S(P) for some partition P). Then let G(P) be the subgroup of Z consisting of x such that αl = α x m whenever l and m lie in the same set λ of P. Let d be the degree of the field K, and for l ∈ Λ let δl be the total degree of the polynomial Pl. Set

Polynomial–Exponential Equations in Two Variables

We consider equations of the formF(x, y, αx, βy)=0 in integer unknownsxandy, whereFis a polynomial with complex coefficients, andαandβare non-zero complex numbers. We prove, outside of some exceptional cases, that if such an equation has infinitely many solutions, thenαandβare algebraic.

On polynomial-exponential equations

Let M(K) be the set of absolute values of K, normalized such that they extend the standard or a p-adic absolute value of Q. Let S c M(K) be the subset consisting of the archimedean absolute values, as well as of the non-archimedean absolute values v e M(K) having I~ol~ 4= 1 for some ~ej. Thus S is a finite set. An S-unit is an element ~ e K* having I~1~--1 for every v$S. Then in particular the ~o, and therefore every a tXe with x E Z ' , is an S-unit. Therefore (1.2) is related to the S-unit equation