209 research outputs found
Constructions of diagonal quartic and sextic surfaces with infinitely many rational points
In this note we construct several infinite families of diagonal quartic
surfaces \begin{equation*} ax^4+by^4+cz^4+dw^4=0, \end{equation*} where
with infinitely many rational points and
satisfying the condition . In particular, we present an
infinite family of diagonal quartic surfaces defined over \Q with Picard
number equal to one and possessing infinitely many rational points. Further, we
present some sextic surfaces of type , , , or
, with infinitely many rational points.Comment: revised version will appear in International Journal of Number Theor
The Method Of Thue-Siegel For Binary Quartic Forms
We will use Thue-Siegel method, based on Pad\'e approximation via
hypergeometric functions, to give upper bounds for the number of integral
solutions to the equation as well as the inequalities , for a certain family of irreducible quartic binary forms.Comment: A version of this paper is to appear in Acta. Arit
Criterion for polynomial solutions to a class of linear differential equation of second order
We consider the differential equations y''=\lambda_0(x)y'+s_0(x)y, where
\lambda_0(x), s_0(x) are C^{\infty}-functions. We prove (i) if the differential
equation, has a polynomial solution of degree n >0, then \delta_n=\lambda_n
s_{n-1}-\lambda_{n-1}s_n=0, where \lambda_{n}=
\lambda_{n-1}^\prime+s_{n-1}+\lambda_0\lambda_{n-1}\hbox{and}\quad
s_{n}=s_{n-1}^\prime+s_0\lambda_{k-1},\quad n=1,2,.... Conversely (ii) if
\lambda_n\lambda_{n-1}\ne 0 and \delta_n=0, then the differential equation has
a polynomial solution of degree at most n. We show that the classical
differential equations of Laguerre, Hermite, Legendre, Jacobi, Chebyshev (first
and second kind), Gegenbauer, and the Hypergeometric type, etc, obey this
criterion. Further, we find the polynomial solutions for the generalized
Hermite, Laguerre, Legendre and Chebyshev differential equations.Comment: 12 page
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