Related papers: A theorem of complete reducibility for exponential…

We generalize the theory of logarithmic derivations through a self-contained study of modules here dubbed tangential idealizers. We establish reflexiveness criteria for such modules, provided the ring is a factorial domain. As a main…

In this note we prove that the factorization theorem for dominated polynomials previously proved by the authors is equivalent to an alternative factorization scheme that uses classical linear techniques and a linearization process. However,…

We axiomatize the first-order theories of exponential integer parts of real-closed exponential fields in a language with $2^x$, in a language with a predicate for powers of 2, and in the basic language of ordered rings. In particular, the…

The ring of integer-valued polynomials over a given subset $S$ of $\Z$ (or $ \mathrm{Int}(S,\Z ))$ is defined as the set of polynomials in $\Q[x]$ which maps $S$ to $\Z$. In factorization theory, it is crucial to check the irreducibility of…

This paper establishes an analogue of the special chain theorem for the embedding dimension of polynomial rings, with direct application on the (embedding) codimension. In particular, we recover a classic result on the transfer of…

We extend the Ax-Katz theorem for a single polynomial from finite fields to the rings Z_m with m composite. This extension not only yields the analogous result, but gives significantly higher divisibility bounds. We conjecture what computer…

We prove first-order definability of the prime subring inside polynomial rings, whose coefficient rings are (commutative unital) reduced and indecomposable. This is achieved by means of a uniform formula in the language of rings with…

It is known that Plotkin's reduction theorem is very important for his theory of universal algebraic geometry [arXiv:math. GM/0210187], [arXiv:math. GM/0210194]. It turns out that this theorem can be generalized to arbitrary categories…

Using Newton polygons, a key factorization result for polynomials over discrete valuation domains is proved, which in particular yields new irreducibility criteria including a generalization of the classical irreducibility criterion of…

Elementary proofs of unique factorization in rings of arithmetic functions using a simple variant of Euclid's proof for the fundamental theorem of arithmetic.

Recently Dritschel proves that any positive multivariate Laurent polynomial can be factorized into a sum of square magnitudes of polynomials. We first give another proof of the Dritschel theorem. Our proof is based on the univariate matrix…

The article proved the upper bound of leading coefficient of characteristic polynomial of graded ideal in a ring of generalized polynomials.

The Bernstein-Sato polynomial is an important invariant of an element or an ideal in a polynomial ring or power series ring of characteristic zero, with interesting connections to various algebraic and topological aspects of the…

Let $K$ be a field of characteristic zero. We deal with the algebraic closure of the field of fractions of the ring of formal power series $K[[x_1,\ldots,x_r]]$, $r\geq 2$. More precisely, we view the latter as a subfield of an iterated…

Terao's factorization theorem shows that if an arrangement is free, then its characteristic polynomial factors into the product of linear polynomials over the integer ring. This is not a necessary condition, but there are not so many…

This book is a rigorous and conceptually oriented introduction to ring theory. The emphasis is on structural understanding rather than encyclopedic coverage: rings are studied through ideals, homomorphisms, quotients, and universal…

We generalize Carlitz' result on the number of self reciprocal monic irreducible polynomials over finite fields by showing that similar explicit formula hold for the number of irreducible polynomials obtained by a fixed quadratic…

We address some questions concerning indecomposable polynomials and their spectrum. How does the spectrum behave via reduction or specialization, or via a more general ring morphism? Are the indecomposability properties equivalent over a…

Let $R$ be a commutative ring with identity. An element $r \in R$ is said to be absolutely irreducible in $R$ if for all natural numbers $n>1$, $r^n$ has essentially only one factorization namely $r^n = r \cdots r$. If $r \in R$ is…

Rational transformations of polynomials are extensively studied in the context of finite fields, especially for the construction of irreducible polynomials. In this paper, we consider the factorization of rational transformations with…