Ring Of Formal Power Series Research Articles

Invariant star products on nondegenerate triangular Lie bialgebras over formal power series rings

We obtain a bijection between the set of equivalent classes of invariant star products on a non-degenerate triangular finite dimensional Lie bialgebra over the formal power series ring and the set , working in the framework developed by Etingof–Kazhdan for the quantization of Lie bialgebras. Two of the corresponding triangular Hopf algebras over the ring are isomorphic if and only if the invariant star products defining them are equivalent. Therefore, when t = ℏ, we obtain a set of triangular Hopf quantized universal enveloping algebras which can also be seen as quantizations of the deformation algebra . Additionally, two of them are isomorphic if and only if the above invariant star products are equivalent.

Algebraic series and valuation rings over nonclosed fields

Suppose that k is an arbitrary field. Let k [ [ x 1 , … , x n ] ] be the ring of formal power series in n variables with coefficients in k . Let k ¯ be the algebraic closure of k and σ ∈ k ¯ [ [ x 1 , … , x n ] ] . We give a simple necessary and sufficient condition for σ to be algebraic over the quotient field of k [ [ x 1 , … , x n ] ] . We also characterize valuation rings V dominating an excellent Noetherian local domain R of dimension 2, and such that the rank increases after passing to the completion of a birational extension of R . This is a generalization of the characterization given by M. Spivakovsky [Valuations in function fields of surfaces, Amer. J. Math. 112 (1990) 107–156] in the case when the residue field of R is algebraically closed.

Higher correlation inequalities

We prove a correlation inequality for n increasing functions on a distributive lattice, which for n = 2 reduces to a special case of the FKG inequality. The key new idea is to reformulate the inequalities for all n into a single positivity statement in the ring of formal power series. We also conjecture that our results hold in greater generality.

Moyal algebra: relevant properties, projective limits and applications in noncommutative field theory

From the definition of the Moyal ∗-product in terms of projective limits of the ring of polynomials of vector fields, the Moyal configuration space of Schwartzian functions, equipped with the ∗-product, is built as a formal power series ring with elements assimilated to free indeterminates. We then define the projector on the ideal depending on a fixed indeterminate, which allows to use the definition of algebraic derivations with respect to any order of field derivative. As a consequence and in a direct manner, Euler-Lagrange equations of motion, in the framework of both the noncommutative scalar and gauge induced Dirac fields, are deduced from the nonlocal Lagrange function. A connection of this theory to a generalized Ostrogradski's formalism is also discussed here.

Calculation of a formal moment generating function by using a differential operator

A differential form in a formal moment generating function is given by the decomposition of powers in terms of the Hermite polynomials. This paper shows that this differential form for calculating the expectation of normal and χ 2 distributions has the benefit of avoiding divergence for Edgeworth type approximations from the viewpoint of a formal power series ring. A symbolic computational algorithm is also discussed, within the distribution theory of statistics.

Examples of miniversal deformations of infinity algebras

A classical problem in algebraic deformation theory is whether an infinitesimal deformation can be extended to a formal deformation. The answer to this question is usually given in terms of Massey powers. If all Massey powers of the cohomology class determined by the infinitesimal deformation vanish, then the de- formation extends to a formal one. We consider another approach to this problem, by constructing a miniversal deformation of the algebra. One advantage of this approach is that it answers not only the question of existence, but gives a construction of an extension as well. In this paper, we study some examples of miniversal deformations of infinity algebras, and use these examples to illustrate how to use a miniversal deformation to determine when an infinitesimal deforma- tion extends to a formal deformation. Actually, using a miniversal deformation, one can construct such an extension explicitly. Also, the obstruction to an extension can be computed by this method. An infinitesimal deformation extends to a formal one precisely when the unique morphism from the base of the universal infinitesimal defor- mation to the base of the given deformation inducing the infinitesimal deformation can be lifted to a morphism from the miniversal defor- mation to the formal power series ring in the parameter of the defor- mation. A nice property of this algebraic approach is that it answers more than the question of existence; in fact, it gives a construction of an extension of the infinitesimal deformation to a formal deformation. Moreover, since the question is reduced to studying the morphisms of the base of the miniversal deformation to a formal power series ring, the

Cyclically valued rings and formal power series

Rings of formal power series k[[C]] with exponents in a cyclically ordered group C were defined in [2]. Now, there exists a on k[[C]]: for every a in k[[C]] and c in C, we let v(c, a) be the first element of the support of a which is greater than or equal to c. Structures with such a valuation can be called cyclically valued rings. Others examples of cyclically valued rings are obtained by twisting the multiplication in k[[C]]. We prove that a cyclically valued ring is a subring of a power series ring k[[C, θ]] with twisted multiplication if and only if there exist invertible monomials of every degree, and the support of every element is well-ordered. We also give a criterion for being isomorphic to a power series ring with twisted multiplication. Next, by the way of quotients of cyclic valuations, it follows that any power series ring k[[C, θ]] with twisted multiplication is isomorphic to a R'[[C',θ']], where C' is a subgroup of the cyclically ordered group of all roots of 1 in the field of complex numbers, and R'≃ k[[H,θ]], with H a totally ordered group. We define a valuation υ(∈,·) which is closer to the usual valuations because, with the topology defined by υ(a,·), a cyclically valued ring is a topological ring if and only if a = ∈ and the cyclically ordered group is indeed a totally ordered one.

OVERCONVERGENT REAL CLOSED QUANTIFIER ELIMINATION

Let K be the (real closed) field of Puiseux series in t over R endowed with the natural linear order. Then the elements of the formal power series rings R[[ξ1, . . . , ξn]] converge t-adically on [−t, t]n, and hence define functions [−t, t]n → K. Let L be the language of ordered fields, enriched with symbols for these functions. By Corollary 3.17, K is o-minimal in L. This result is obtained from a quantifier elimination theorem. The proofs use methods from non-Archimedean analysis.

Classification of the ring of Witt vectors and the necklace ring associated with the formal group law [formula omitted

In this paper, we classify the ring of Witt vectors and the necklace ring associated with the formal group law X + Y − q X Y up to strict-isomorphism as q varies over the set of integers. We also introduce a q-deformation of the Grothendieck ring of formal power series with constant term 1 over rings in which q ⋅ 1 is a unit. Various bijections related to Witt vectors, necklaces, and formal power series are given. As an application, we provide a q-analogue of the celebrated cyclotomic identity and its bijective proof.

On Polynomials Over Rickart Rings and Their Generalizations I

This paper continues the investigation of polynomials and formal power series over a ring with various annihilator conditions which were originally used by Rickart and Kaplansky to abstract the algebraic properties of von Neumann algebras. Results of Armendariz on polynomial rings over a PP ring are extended to analogous annihilator conditions in nearrings of polynomials and nearrings of formal power series. These results are somewhat striking since, in contrast to the polynomial ring case, the nearring of polynomials or formal power series has substitution for its "multiplication" operation. These investigations provide an alternative viewpoint in illustrating the structure of polynomials and formal power series. Extensions of Rickart rings to formal power series rings are also discussed.

Power series rings and projectivity

We show that a formal power series ring A[[X]] over a noetherian ring A is not a projective module unless A is artinian. However, if (A, ) is any local ring, then A[[X]] behaves like a projective module in the sense that Ext p A (A[[X]], M)=0 for all -adically complete A-modules. The latter result is shown more generally for any flat A-module B instead of A[[X]]. We apply the results to the (analytic) Hochschild cohomology over complete noetherian rings.

On the solutions of the translation equation in rings of formal power series

We study in this paper solutions of the translation equation in rings of formal power series K[X] where K ∈R, C (so called one-parameter groups or flows), and even, more generally, homomorphisms Ф from an abelian group (G, +) into the group Г(K) of invertible power series in K[X]. This problem can equivalently be formulated as the question of constructing homomorphisms Ф from (G, +) into the differential group Г1∞ describing the chain rules of higher order of C∞ functions with fixed point 0. In this paper we present the general form of these homomorphisms Ф : G → Г(K) (or L1∞),Ф = (fn n≤1,forwhich f1 = l, f2 = ... = fp+l =0,fp+2 ≠ 0 for fixed, but arbitrary p ≤ 0 (see Theorem 5, Corollary 6 and Theorem 6). This representation uses a sequence (w n p )n≥p+2 of universal polynomials in fp+2 and a sequence of parameters, which determines the individual one-parameter group. Instead of (w n p )n≥p+2 we may also use another sequence (L n p )n≥p+2 of universal polynomials, and we describe the connection between these forms of the solutions. The problem for homomorphisms Ф with f1≠1 will be treated in a separate paper.

Commutator automorphisms of formal power series rings

For a big class of commutative rings R R , every continuous R R -automorphism of R [ [ X 1 , … , X n ] ] R[[X_1,\ldots ,X_n]] with the linear part the identity is in the commutator subgroup of Aut ⁡ ( R [ [ X 1 , … , X n ] ] ) \operatorname {Aut}(R[[X_1,\ldots ,X_n]]) . An explicit bound for the number of commutators involved and a K K -theoretic interpretation of this result are provided.

Resolution of curve and surface singularities in characteristic zero

Preface. Note to the Reader. Terminology. I: Valuation Theory. 1. Marot Rings. 2. Manis Valuation Rings. 3. Valuation Rings and Valuations. 4. The Approximate Theorem for Independent Valuations. 5. Extensions of Valuations. 6. Extending Valuations to Algebraic Overfields. 7. Extensions of Discrete Valuations. 8. Ramification Theory of Valuations. 9. Extending Valuations to Non-Algebraic Overfields. 10. Valuations of Algebraic Function Fields. 11. Valuations Dominating a Local Domain. II: One-Dimensional Semilocal Cohen-Macaulay Rings. 1. Transversal Elements. 2. Integral Closure of One-Dimensional Semilocal Cohen-Macaulay Rings. 3. One-Dimensional Analytically Unramified and Analytically Irreducible CM-Rings. 4. Blowing up Ideals. 5. Infinitely Near Rings. III: Differential Modules and Ramification. 1. Introduction. 2. Norms and Traces. 3. Formally Unramified and Ramified Extensions. 4. Unramified Extensions and Discriminants. 5. Ramification for Quasilocal Rings. 6. Integral Closure and Completion. IV: Formal and Convergent Power Series Rings. 1. Formal Power Series Rings. 2. Convergent Power Series Rings. 3. Weierstrass Preparation Theorem. 4. The Category of Formal and Analytic Algebras. 5. Extensions of Formal and Analytic Algebras. V: Quasiordinary Singularities. 1. Fractionary Power Series. 2. The Jung-Abhyankar Theorem: Formal Case. 3. The Jung-Abhyankar Theorem: Analytic Case. 4. Quasiordinary Power Series.5. A Generalized Newton Algorithm. 6. Strictly Generated Semigroups. VI: The Singularity Zq = XYp. 1. Hirzebruch-Jung Singularities. 2. Semigroups and Semigroup Rings. 3. Continued Factions. 4. Two-Dimensional Cones. 5. Resolution of Singularities. VII: Two-Dimensional Regular Local Rings. 1. Ideal Transform. 2. Quadratic Transforms and Ideal Transforms. 3. Complete Ideals. 4. Factorization of Complete Ideals. 5. The Predecessors of a Simple Ideal. 6. Uniformization. 7. Resolution of Surface Singularities II: Blowing up and Normalizing. Appendices. A: Results from Classical Algebraic Geometry. 1. Generalities. 2. Affine and Finite Morphisms. 3. Products. 4. Proper Morphisms. 5. Algebraic Cones and Projective Varieties. 6. Regular and Singular points. 7. Normalization of a Variety. 8. Desingularization of a Variety. 9. Dimension of Fibres. 10. Quasifinite Morphisms and Ramification. 11. Divisors. 12. Some Results on Projections. 13. Blowing up. 14. Blowing up: the Local Rings. B: Miscellaneous Results. 1. Ordered Abelian Groups. 2. Localization. 3. Integral Extensions. 4. Some Results on Graded Rings and Modules. 5. Properties of the Rees Ring. 6. Integral Closure of Ideals. 7. Decomposition Group and Inertia Group. 8. Decomposable Rings. 9.

Hypersurfaces of bounded Cohen–Macaulay type

Let R = k [ [ x 0 , … , x d ] ] / ( f ) , where k is a field and f is a non-zero non-unit of the formal power series ring k [ [ x 0 , … , x d ] ] . We investigate the question of which rings of this form have bounded Cohen–Macaulay type, that is, have a bound on the multiplicities of the indecomposable maximal Cohen–Macaulay modules. As with finite Cohen–Macaulay type, if the characteristic is different from two, the question reduces to the one-dimensional case: The ring R has bounded Cohen–Macaulay type if and only if R ≅ k [ [ x 0 , … , x d ] ] / ( g + x 2 2 + ⋯ + x d 2 ) , where g ∈ k [ [ x 0 , x 1 ] ] and k [ [ x 0 , x 1 ] ] / ( g ) has bounded Cohen–Macaulay type. We determine which rings of the form k [ [ x 0 , x 1 ] ] / ( g ) have bounded Cohen–Macaulay type.

Formal power series with cyclically ordered exponents

We define and study a notion of ring of formal power series with exponents in a cyclically ordered group. Such a ring is a quotient of various subrings of classical formal power series rings. It carries a two variable valuation function. In the particular case where the cyclically ordered group is actually totally ordered, our notion of formal power series is equivalent to the classical one in a language enriched with a predicate interpreted by the set of all monomials.

Reduction Mod p of Standard Bases

ABSTRACT We investigate the behavior of standard bases (in the sense of Hironaka and Grauert) for ideals in rings of formal power series over commutative rings with respect to specializations of the coefficients. For instance, we show that any ideal I of the ring of formal power series A[[X]] = A[[X 1 ,…, X N ]] with coefficients in a Noetherian ring A admits a standard basis whose image under every specialization of A onto a field is a standard basis of the image of I. Applications include a modular criterion for ideal membership in ℤ[[X]] and a constructibility result for ideal membership in K[[X]], where K is a field. # Communicated by A. Prestel. † To Volker Weispfenning, on his 60th birthday.

Generators of $D$–modules in positive characteristic

Let R = k[x1, . . . , xd] or R = k[[x1, . . . , xd]] be either a polynomial or a formal power series ring in a finite number of variables over a field k of characteristic p > 0 and let DR|k be the ring of klinear differential operators of R. In this paper we prove that if f is a non-zero element of R then Rf , obtained from R by inverting f , is generated as a DR|k–module by 1 f . This is an amazing fact considering that the corresponding characteristic zero statement is very false. In fact we prove an analog of this result for a considerably wider class of rings R and a considerably wider class of DR|k-modules.

When is [formula omitted] Noetherian?

Let A ⊆ B be an extension of commutative rings with identity, X an analytic indeterminate over B, and R : = A + X B [ [ X ] ] , the subring of the formal power series ring B [ [ X ] ] , consisting of the series with constant terms in A. In this Note we study when the ring R is Noetherian. We prove that R is Noetherian if and only if A is Noetherian and B is a finitely generated A-module. To cite this article: S. Hizem, A. Benhissi, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

Cranks and dissections in Ramanujan's lost notebook

In his lost notebook, Ramanujan offers several results related to the crank, the existence of which was first conjectured by F. J. Dyson and later established by G.E. Andrews and F.G. Garvan. Using an obscure identity found on p. 59 of the lost notebook, we provide uniform proofs of several congruences in the ring of formal power series for the generating function F ( q ) of cranks. All are found, sometimes in abbreviated form, in the lost notebook, and imply dissections of F ( q ) . Consequences of our work are interesting new q-series identities and congruences in the spirit of Atkin and Swinnerton-Dyer.