On algebraic functions satisfying a class of functional equations. (Short Communication).
What should be assumed about the integral polynomials in order that the solvability of the congruence for sufficiently large primes p implies the solvability of the equation in integers x? We provide some explicit characterizations for the cases when are binomials or have cyclic splitting fields.
We prove that almost all positive even integers can be represented as with for . As a consequence, we show that each sufficiently large odd integer can be written as with for .
Let be a complex number, be a positive rational integer and , where denotes the set of polynomials with rational integer coefficients of absolute value . We determine in this note the maximum of the quantities when runs through the interval . We also show that if is a non-real number of modulus , then is a complex Pisot number if and only if for all .