Ring Of Formal Power Series Research Articles

Quasi-Ordinary Singularities: Tree Model, Discriminant, and Irreducibility

Let $f(Y)\in K[[X_1,\dots,X_d]][Y]$ be a quasi-ordinary Weierstrass polynomial with coefficients in the ring of formal power series over an algebraically closed field of characteristic zero. In this paper we study the discriminant $D_f$ of $f(Y)-V$, where $V$ is a new variable. We show that the Newton polytope of $D_f$ depends only on contacts between the roots of $f(Y)$. Then we prove that $f(Y)$ is irreducible if and only if the Newton polytope of $D_f$ satisfies some arithmetic conditions. Finally we generalize these results to quasi-ordinary power series.

Natural numerosities of sets of tuples

We consider a notion of “numerosity” for sets of tuples of natural numbers that satisfies the five common notions of Euclid’s Elements, so it can agree with cardinality only for finite sets. By suitably axiomatizing such a notion, we show that, contrasting to cardinal arithmetic, the natural “Cantorian” definitions of order relation and arithmetical operations provide a very good algebraic structure. In fact, numerosities can be taken as the non-negative part of a discretely ordered ring, namely the quotient of a formal power series ring modulo a suitable (“gauge”) ideal. In particular, special numerosities, called “natural”, can be identified with the semiring of hypernatural numbers of appropriate ultrapowers of N \mathbb {N} .

Elliptic Algebra $U_{q,p}(\widehat {\mathfrak {g}})$ and Quantum Z-algebras

A new definition of the elliptic algebra $U_{q,p}(\widehat {\mathfrak {g}})$ associated with an untwisted affine Lie algebra $\widehat {\mathfrak {g}}$ is given as a topological algebra over the ring of formal power series in p. We also introduce a quantum dynamical analogue of Lepowsky-Wilson’s Z-algebras. The Z-algebra governs the irreducibility of the infinite dimensional $U_{q,p}({\widehat {\mathfrak {g}}})$ -modules. Some level-1 examples indicate a direct connection of the irreducible $U_{q,p}(\widehat {\mathfrak {g}})$ -modules to those of the W-algebras associated with the coset $\widehat {\mathfrak {g}}\oplus \widehat {\mathfrak {g}}\supset (\widehat {\mathfrak {g}})_{{diag}}$ with level (r − g − 1, 1) (g:the dual Coxeter number), which includes Fateev-Lukyanov’s W B l -algebra.

Rings of Formal Power Series in an Infinite Set of Indeterminates

Let α be an infinite cardinal number, Λ be an index set of cardinality > α, and {X λ}λ∈Λ be a set of indeterminates over an integral domain D. It is well known that there are three ways of defining the ring of formal power series in {X λ}λ∈Λ over D, say, D[[{X λ}]] i for i = 1, 2, 3. In this paper, we let D[[{X λ}]]α = ∪ {D[[{X λ}λ∈Γ]]3 | Γ ⊆ Λ and |Γ| ≤ α}, and we then show that D[[{X λ}]]α is an integral domain such that D[[{X λ}]]2 ⊊ D[[{X λ}]]α ⊊ D[[{X λ}]]3. We also prove that (1) D is a Krull domain if and only if D[[{X λ}]]α is a Krull domain and (2) D[[{X λ}]]α is a unique factorization domain (UFD) (resp., π-domain) if and only if D[[X 1,…, X n ]] is a UFD (resp., π-domain) for every integer n ≥ 1.

Test ideals of non-principal ideals: Computations, jumping numbers, alterations and division theorems

Given an ideal a⊆R in a (log) Q-Gorenstein F-finite ring of characteristic p>0, we study and provide a new perspective on the test ideal τ(R,at) for a real number t>0. Generalizing a number of known results from the principal case, we show how to effectively compute the test ideal and also describe τ(R,at) using (regular) alterations with a formula analogous to that of multiplier ideals in characteristic zero. We further prove that the F-jumping numbers of τ(R,at) as t varies are rational and have no limit points, including the important case where R is a formal power series ring. Additionally, we obtain a global division theorem for test ideals related to results of Ein and Lazarsfeld from characteristic zero, and also recover a new proof of Skoda's theorem for test ideals which directly mimics the proof for multiplier ideals.

EXTENSIONS OF HOMOGENEOUS SEMILOCAL RINGS I

A ring R with Jacobson radical J(R) is a homogeneous semilocal ring if R/J(R) is simple artinian. In this paper, we study the transfer of the property of being homogeneous semilocal from a ring R to the formal power series ring R[[x]], the skew formal power series ring R[[x, α]] and the Hurwitz series ring HR. The results of the paper generalize those proved for commutative local rings. We also consider finite centralizing extensions proving that if the ring of matrices Mn(R) is a homogeneous semilocal ring, then so is R. More generally, if e is an idempotent of a homogeneous semilocal ring S, then eSe is homogeneous semilocal.

Subgroupoids of some direct products and formal power series

We consider, in rings of formal power series, the series whose coefficients, in a certain sense, “tends to infinity”. In this manner, we determine subrings of the rings as above, which have several derivations. Some of these subrings are composition rings also. At first, we study direct products of groupoids.

A Constructive Study of the Module Structure of Rings of Partial Differential Operators

The purpose of this paper is to develop constructive versions of Stafford's theorems on the module structure of Weyl algebras A n (k) (i.e., the rings of partial differential operators with polynomial coefficients) over a base field k of characteristic zero. More generally, based on results of Stafford and Coutinho-Holland, we develop constructive versions of Stafford's theorems for very simple domains D. The algorithmization is based on the fact that certain inhomogeneous quadratic equations admit solutions in a very simple domain. We show how to explicitly compute a unimodular element of a finitely generated left D-module of rank at least two. This result is used to constructively decompose any finitely generated left D-module into a direct sum of a free left D-module and a left D-module of rank at most one. If the latter is torsion-free, then we explicitly show that it is isomorphic to a left ideal of D which can be generated by two elements. Then, we give an algorithm which reduces the number of generators of a finitely presented left D-module with module of relations of rank at least two. In particular, any finitely generated torsion left D-module can be generated by two elements and is the homomorphic image of a projective ideal whose construction is explicitly given. Moreover, a non-torsion but non-free left D-module of rank r can be generated by r+1 elements but no fewer. These results are implemented in the Stafford package for D=A n (k) and their system-theoretical interpretations are given within a D-module approach. Finally, we prove that the above results also hold for the ring of ordinary differential operators with either formal power series or locally convergent power series coefficients and, using a result of Caro-Levcovitz, also for the ring of partial differential operators with coefficients in the field of fractions of the ring of formal power series or of the ring of locally convergent power series.

A factorization formula for power series

Given an odd prime p, we give an explicit factorization over the ring of formal power series with integer coefficients for certain reducible polynomials whose constant term is of the form $p^w$ with $w>1$. Our formulas are given in terms of partial Bell polynomials and rely on the inversion formula of Lagrange.

LOCAL SPLITTING PROPERTIES OF ENDOMORPHISM RINGS OF PROJECTIVE MODULES

This paper deals with the unit groups of the endomor- phism rings of projective modules over polynomial rings and further over formal power series rings. A normal subgroup of the unit group is dened and discussed. The local splitting properties of elements of endomorphism rings of projective modules over polynomial rings are given. So, if P is a projective module over a ring A, then (id+ xEndA(x)(P(x))) is constructed. This is a normal subgroup of the unit group of endomorphism rings of a projective module over a poly- nomial ring. If (A,m) is a local ring with dim(A) ≥ µ(m), then we show that (id+ xEndA(x)(P(x))) is a normal subgroup of SLA(x)(P(x)). Also, we will show that under these conditions the similar conclusion can be drawn for the ring of formal power series. In section 3, let P be a projective A-module. Let s1, s2 ∈ A be such that As1 + As2 = A. Then we use the splitting lemma of Quillen to show that every element of (idPs1s2 + xEnd As1s2(x) (Ps1s2 (x))) has two decompositions. Finally, we generalize the result to Theorem 3.4.

ON DIVISION OF QUASIANALYTIC FUNCTION GERMS

We establish the following criterion for divisibility in the local ring [Formula: see text] of those quasianalytic function germs at 0 ∈ ℝnwhich are definable in a polynomially bounded structure. A sufficient (and necessary) condition for the divisibility of two function germs in [Formula: see text] is that of their Taylor series at 0 ∈ ℝnin the formal power series ring.

Self-Dual Permutation Codes Over Formal Power Series Rings and Finite Principal Ideal Rings

In this paper, we study self-dual permutation codes over formal power series rings and finite principal ideal rings. We first give some results on the torsion codes associated with the linear codes over formal power series rings. These results allow for obtaining some conditions for non-existence of self-dual permutation codes over formal power series rings. Finally, we describe self-dual permutation codes over finite principal ideal rings by examining permutation codes over their component chain rings.

Necessary conditions for the depth formula over Cohen–Macaulay local rings

Let R be a Cohen–Macaulay local ring and let M and N be finitely generated R-modules. In this paper we investigate some of the necessary conditions for the depth formula depth(M)+depth(N)=depth(R)+depth(M⊗RN) to hold. We show that, under certain conditions, M and N satisfy the depth formula if and only if ToriR(M,N)=0 for all i≥1. We also examine the relationship between the depth of M⊗RN and the vanishing of Ext modules with various applications. One of them extends partially a result by Auslander on even dimensional regular local rings to complete intersections. In another application, we show that there is no nontrivial semidualizing module over Veronese subrings of the formal power series ring k[[X1,…,Xn]] over a field k.

Comultiplication Modules over Noncommutative Rings

Comultiplication Modules over Noncommutative Rings

On the characterization of cyclic codes over two classes of rings

Let R be a finite chain ring with maximal ideal 〈 γ 〉 and residue field ▪, and let γ be of nilpotency index t. To every code C of length n over R, a tower of codes C = (C: γ0) ⊆ (C: γ) ⊆ … ⊆ (C:γi) ⊆ … ⊆ (C: γt−1) can be associated with C, where for any r ɛ R, (C:r) = {e ɛ Rn | re ɛ C}. Using generator elements of the projection of such a tower of codes to the residue field ▪, we characterize cyclic codes over R. This characterization turns the condition for codes over R to be cyclic into one for codes over the residue field ▪. Furthermore, we obtain a characterization of cyclic codes over the formal power series ring of a finite chain ring.

Łojasiewicz inequality over the ring of power series in two variables

We prove a Łojasiewicz type inequality for a system of polynomial equations with coefficients in the ring of formal power series in two variables. This result is an effective version of the Strong Artin Approximation Theorem. From this result we deduce a bound of Artin functions of isolated singularities.

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER ℤ

We consider polynomials with integer coefficients and discuss their factorization properties in ℤ[[x]], the ring of formal power series over ℤ. We treat polynomials of arbitrary degree and give sufficient conditions for their reducibility as power series. Moreover, if a polynomial is reducible over ℤ[[x]], we provide an explicit factorization algorithm. For polynomials whose constant term is a prime power, our study leads to the discussion of p-adic integers.

On the algebraic K-theory of formal power series

AbstractIn this paper we extend the computation of the the typical curves of algebraic K-theory done by Lars Hesselholt and Ib Madsen to general tensor algebras. The models used allow us to determine the stages of the Taylor tower of algebraic K-theory as a functor of augmented algebras, as defined by Tom Goodwillie, when evaluated on derived tensor algebras.For R a discrete ring, and M a simplicial R-bimodule, we let R(M) denote the (derived) tensor algebra of M over R, and πR denote the ring of formal (derived) power series in M over R. We define a natural transformation of functors of simplicial R-bimodules Φ: which is closely related to Waldhausen's equivalence We show that Φ induces an equivalence on any finite stage of Goodwillie's Taylor towers of the functors at any simplicial bimodule. This is used to show that there is an equivalence of functors , and for connected bimodules, also an equivalence

Irreducibility Criterion For Quasi-Ordinary Polynomials

We give a criterion for a quasi ordinary polynomial to be locally irreducible. The criterion uses the notion of approximate roots and that of generalized Newton polygons. It generalizes the one given by Abhyankar for algebraic plane curves. Introduction Let K be an algebraically closed field of characteristic zero, and let R = K[[x1, . . . , xe]] = K[[x]] be the ring of formal power series in x1, . . . , xe over K. Let f = y+a1(x)y + . . .+an(x) be a nonzero polynomial of R[y], and suppose that f is irreducible in R[y]. Suppose that e = 1 and let g be a nonzero polynomial of R[y], then define the intersection multiplicity of f with g, denoted int(f, g), to be the x-order of the y resultant of f and g. The set of int(f, g), g ∈ R[y], defines a semigroup, denoted Γ(f). It is will known that a set of generators of Γ(f) can be computed from polynomials having the maximal contact with f (see [1] and [6]), namely, there exist g1, . . . , gh such that n, int(f, g1), . . . , int(f, gh) generate Γ(f) and for all 1 ≤ k ≤ h, the Newton-Puiseux expansion of gk coincides with that of f until a characteristic exponent of f . In [1], Abhyankar introduced a special set of polynomials called the approximate roots of f . These polynomials have the advantage that they can be calculated from the equation of f by using the Tschirnhausen transform. Suppose that e ≥ 2 and that the discriminant of f is of the form x1 1 . . . . .x Ne e .u(x1, . . . , xe), where u is a unit in K[[x]] (such a polynomial is called quasi-ordinary polynomial). By Abhyankar-Jung Theorem, the roots of f(x1, . . . , xe, y) = 0 are all in K[[x 1 n 1 , . . . , x 1 n e ]], in particular there exists a power series y(t1, . . . , te) = ∑ p cpt p1 1 . . . . .t pe e ∈ K[[t1, . . . , te]] such that f(t1 , . . . , t n e , y(t1, . . . , te)) = 0 and the other roots of f(t n 1 , . . . , t n e , y) = 0 are the conjugates of y(t1, . . . , te) with respect to the nth roots of unity in K. Given a polynomial g of R[y], we define the order of g to be the leading exponent with respect to the lexicographical order of the smallest homogeneous component of g(t1 , . . . , t n e , y(t1, . . . , te)). The set of orders of polynomials of R[y] defines a semigroup. In this paper we first prove that the canonical basis of (nZ) with the set of orders of the approximate roots of f generate the semigroup of f , then we give, using these approximate roots and the notion of generalized Newton polygons, a criterion for a quasi-ordinary polynomial to be irreducible. Note that if e = 1, then f is quasi-ordinary, in particular our results generalize those of Abhyankar (see [1] and [3]). The paper is organized as follows: in Section 1 we introduce the notion of approximate roots of a polynomial in one variable over a commutative ring with unity. In Section 2 we show how to associate a semigroup with an irreducible quasi-ordinary polynomial of R[y]. In Section 3 we introduce the notion of pseudo roots of a quasi-ordinary polynomial f then we prove that the orders of these polynomials together with the canonical basis of (nZ) give a set of generators of the semigroup of f . This result remains true if we replace the pseudo roots of f by its set of approximate roots. This is what we prove in Section 4. Sections 5 and 6 are devoted to the irreducibility criterion: in Section 5 we introduce the notion of generalized Newton polygon, and we define ∗2000 Mathematical Subject Classification: 32S25, 32S70. During the development of this work, the author visited the Department of Mathematics at the American University of Beirut, Lebanon. He would like to thank that institution for hospitality and support. He also would like to think the Center for Advanced Mathematical Sciences-CAMS for offering access to many facilities. †Universite d’Angers, Mathematiques, 49045 Angers cedex 01, France, e-mail:assi@univ-angers.fr

Rings of formal power series and a multidimensional analogue of the lagrange formula

Rings of formal power series and a multidimensional analogue of the lagrange formula