Daniel Birmajer

Arithmetic in the Ring of Formal Power Series with Integer Coefficients

of polynomials R[x] over R, namely the ring ^[[x]] of formal powers series in one variable over R, is hardly ever mentioned in such a course. In most cases, it is relegated to the homework problems (or to the exercises in the textbooks), and one learns that, like R[x], R[[x]] is an integral domain provided that R is an integral domain. More surprising is to learn that, in contrast to the situation of polynomials, in R[[x]] there are many invertible elements: while the only units in R[x] are the units of R, a necessary and sufficient condition for a power series to be invertible is that its constant term be invertible in R. This fact makes the study of arithmetic in /?[[*]] simple when R is a field: the only prime element is the variable x. As might be expected, the study of prime factorization in Z[[x]] is much more interesting (and complicated), but to the best of our knowledge it is not treated in detail in the available literature. After some basic considerations, it is apparent that the question of deciding whether or not an integral power series is prime is a difficult one, and it seems worthwhile to develop criteria to determine irreducibility in Z[[x]] similar to Eisensteins criterion for polynomials. In this note we propose an easy argument that provides us with an infinite class of irreducible power series over Z. As in the case of Eisensteins criterion in 7L\x\, our criteria give only sufficient conditions, and the question of whether or not a given power series is irreducible remains open in a vast array of cases, including quadratic polynomials. It is important to note that irreducibility in Z[x] and in Z[[x]] are, in general, un related. For instance, 6 + x + x2 is irreducible in Z[x] but can be factored in Z[[x]], while 2 + Ix + 3x2 is irreducible in Z[[x]] but equals (2 + x)(l +3x) as a polyno mial (observe that this is not a proper factorization in Z[[x]] since 1 + 3x is invert

FACTORIZATION OF QUADRATIC POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER ℤ

We establish necessary and sufficient conditions for a quadratic polynomial to be irreducible in the ring

Colored partitions of a convex polygon by noncrossing diagonals

Z[[x]]

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER ℤ

of formal power series with integer coefficients. For

A family of Bell transformations

n,m\ge 1

On rational Dyck paths and the enumeration of factor-free Dyck words

and

On factor-free Dyck words with half-integer slope

p

Polynomial detection of matrix subalgebras

prime, we show that

Some convolution identities and an inverse relation involving partial Bell polynomials

p^n+p^m\beta x+\alpha x^2

Linear Recurrence Sequences and Their Convolutions via Bell Polynomials

is reducible in