Related papers: Representing an integer and its powers in two unre…

We give solutions of a Diophantine equation containing factorials, which can be written as a cubic form, or as a sum of binomial coefficients. We also give some solutions to higher degree forms and relate some solutions to an unsolvable…

Given any line in the plane, we strengthen the Littlewood conjecture by two logarithms for almost every point on the line, thereby generalising the fibre result of Beresnevich, Haynes, and Velani. To achieve this, we prove an effective…

We improve on Jarn\'{\i}k's inequality between uniform Diophantine exponent $\alpha $ and ordinary Diophantine exponent $\beta$ for a system of $ n\ge 2$ real linear forms in two integer variables. Jarn\'{\i}k (1949, 1954) proved that…

By the theory of elliptic curves, we study the nontrivial rational parametric solutions and rational solutions of the Diophantine equations $z^2=f(x)^2 \pm f(y)^2$ for some simple Laurent polynomials $f$.

We study systems of polynomial equations in several classes of finitely generated rings and algebras. For each ring $R$ (or algebra) in one of these classes we obtain an interpretation by systems of equations of a ring of integers $O$ of a…

Let $1 < k < 33 / 29$. We prove that if $\lambda_1$, $\lambda_2$ and $\lambda_3$ are non-zero real numbers, not all of the same sign and that $\lambda_1 / \lambda_2$ is irrational and $\varpi$ is any real number, then for any $\eps > 0$ the…

We establish a q-generalization of Gordon's theorem that the space of diagonal coinvariants has a quotient identified with a perfect representation of the rational double affine Hecke algebra. It leads to a simple proof of his theorem and…

We refine a recent result of Parsell on the values of the form $\lambda_1p_1 + \lambda_2p_2 + \mu_1 2^{m_1} + ...m + \mu_s 2^{m_s}, $ where $p_1,p_2$ are prime numbers, $m_1,...c, m_s$ are positive integers, $\lambda_1 / \lambda_2$ is…

The original version of this paper did not take into account that there may be solutions (x_0, y_o)in Z X Z of f(x,y) = x^3 + p(y)x + q(y) = 0 even though w_0 = (-3D(y_0))^(1/2) is irrational.

We prove a strong simultaneous Diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes.

A basic question of Diophantine approximation, which is the first issue we discuss, is to investigate the rational approximations to a single real number. Next, we consider the algebraic or polynomial approximations to a single complex…

In this paper, we consider a weighted version of one-dimensional discrete Hardy inequalities with power weights of the form $n^\alpha$. We prove the inequality when $\alpha$ is an even natural number with the sharp constant and remainder…

We generalize previous results on N=1, (3+1)-dimensional superconformal block quiver gauge theories. It is known that the necessary conditions for a theory to be superconformal, i.e. that the beta and gamma functions vanish in addition to…

As an application of the method of Thue-Siegel, we will resolve a conjecture of Walsh to the effect that the Diophantine equation $aX^{4} - bY^2=1$, for fixed positive integers $a$ and $b$, possesses at most two solutions in positive…

Let $\sigma_{i}(x_{1},\ldots, x_{n})=\sum_{1\leq k_{1}<k_{2}<\ldots <k_{i}\leq n}x_{k_{1}}\ldots x_{k_{i}}$ be the $i$-th elementary symmetric polynomial. In this note we generalize and extend the results obtained in a recent work of Zhang…

We generalize the classical theorem by Jarnik and Besicovitch on the irrationality exponents of real numbers and Hausdorff dimension. Let a be any real number greater than or equal to 2 and let b be any non-negative real less than or equal…

We prove a result on approximations to a real number $\theta$ by algebraic numbers of degree $\le 2$ in the case when we have information about the uniform Diophantine exponent $\hat{\omega}$ for the linear form $x_0 +\theta…

Let $Q$ be a positive-definite quaternary quadratic form with prime discriminant. We give an explicit lower bound on the number of representations of a positive integer $n$ by $Q$. This problem is connected with deriving an upper bound on…

In this extended abstract we deal with the relations between the numerical/diophantine approximation and the symbolic/algebraic geometry approachs to solving of multivariate diophentine polynomial systems, obtaining several consecuences…

In this paper, we consider the following diagonal quadratic forms \begin{equation*} a_1x_1^2 + a_2x_2^2 + \cdots + a_{\ell}x_{\ell}^2, \end{equation*} where $\ell\ge 5$ is an odd integer and $a_i\ge 1$ are integers. By using the extended…