We consider a linear Diophantine equation of the form x\a\ + ...+Jt"0" -N t where n is a fixed integer greater than one, 0 < a\ < ... < a n is a fixed set of integers such that (a\, ...,a n ) = 1. We denote by/(W) the number of solutions in non-negative integers. It is well known that/Cx) = P(x) + A(JC), where P(x) is a polynomial in χ of degree η -1 and Δ(*) is a periodic function with period a\ ...a". We apply an elementary approach to the problem of calculating Δ(*), and utilize roots of unity arguments in constructing this periodic function. For/(N), an explicit expression is obtained for arbitrary Λ; this expression includes complicated sums containing the roots of unity. In the case π = 2, this approach leads to a computable explicit expression for/(jc). We note that previously the expression for Δ(*) has not been known.
In this note, we study the action of finite groups of symplectic automorphisms on K3 surfaces which yield quotients birational to generalized Kummer surfaces. For each possible group, we determine the Picard number of the K3 surface admitting such an action and for singular K3 surfaces we show the uniqueness of the associated abelian surface.
The local behaviour of the leaves of a singular holomorphic foliation is analyzed around the singularity. It is shown that certain integers classify the singular foliation germs. Maps from simpler classes to more complicated classes are constructed and it is shown that every foliation can be expressed in terms of simpler ones, under some mild assumptions.
We give explicit formulas, without using the Poisson integral, for the functions that are C-harmonic on the unit disk and restrict to a prescribed polynomial on the boundary.
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