The Poincaré series of multiplier ideals of a simple complete ideal in a local ring of a smooth surface

For a simple complete ideal ℘ of a local ring at a closed point on a smooth complex algebraic surface, we introduce an algebraic object, named Poincaré series P_℘, that gathers in a unified way the jumping numbers and the dimensions of the vector space quotients given by consecutive multiplier ideals attached to ℘. This paper is devoted to prove that P_℘ is a rational function giving an explicit expression for it.     

Generalizations of a lemma of Freudenburg

Let k be an algebraically closed field of characteristic zero and ℘ a prime ideal in k[X]?k[X₁,…,X_n]. Let g∈k[X] and d?1. If for all 1?|α|?d the derivatives ∂^αg belong to ℘, then there exists c∈k such that g−c∈℘(d+1), the d+1th symbolic power of ℘. In particular, if ℘ is a complete intersection it follows that g−c∈℘d+1.     

A Grothendieck module with applications to Poincaré rationality

In this paper, we define a Grothendieck module associated to a Noetherian ring A. This structure is designed to encode relations between A-modules which can be responsible for the relations among Betti numbers and therefore rationality of the Poincaré series. We will define the Grothendieck module, demonstrate that the condition of being torsion in the Grothendieck module implies rationality of the Poincaré series, and provide examples. The paper concludes with an example which demonstrates that the condition of being torsion in the Grothendieck module is strictly stronger than having rational Poincaré series.     

Generating sequences and Poincaré series for a finite set of plane divisorial valuations

Let V be a finite set of divisorial valuations centered at a 2-dimensional regular local ring R. In this paper we study its structure by means of the semigroup of values, SV, and the multi-index graded algebra defined by V, grVR. We prove that SV is finitely generated and we compute its minimal set of generators following the study of reduced curve singularities. Moreover, we prove a unique decomposition theorem for the elements of the semigroup. The comparison between valuations in V, the approximation of a reduced plane curve singularity C by families of sets V(k) of divisorial valuations, and the relationship between the value semigroup of C and the semigroups of the sets V(k), allow us to obtain the (finite) minimal generating sequences for C as well as for V.We also analyze the structure of the homogeneous components of grVR. The study of their dimensions allows us to relate the Poincaré series for V and for a general curve C of V. Since the last series coincides with the Alexander polynomial of the singularity, we can deduce a formula of A'Campo type for the Poincaré series of V. Moreover, the Poincaré series of C could be seen as the limit of the series of V(k), k?0.     

Minimum number of ideal generators for a linear center perturbed by homogeneous polynomials

Using the algorithm presented in [J. Giné, X. Santallusia, On the Poincaré-Liapunov constants and the Poincaré series, Appl. Math. (Warsaw) 28 (1) (2001) 17-30] the Poincaré-Liapunov constants are calculated for polynomial systems of the form , , where P_n and Q_n are homogeneous polynomials of degree n. The objective of this work is to calculate the minimum number of ideal generators i.e., the number of functionally independent Poincaré-Liapunov constants, through the study of the highest fine focus order for n=4 and n=5 and compare it with the results that give the conjecture presented in [J. Giné, On the number of algebraically independent Poincaré-Liapunov constants, Appl. Math. Comput. 188 (2) (2007) 1870-1877]. Moreover, the computational problems which appear in the computation of the Poincaré-Liapunov constants and the determination of the number of functionally independent ones are also discussed.     

On the multigraded Hilbert and Poincaré-Betti series and the Golod property of monomial rings

In this paper we study the multigraded Hilbert and Poincaré-Betti series of A=S/a, where S is the ring of polynomials in n indeterminates divided by the monomial ideal a. There is a conjecture about the multigraded Poincaré-Betti series by Charalambous and Reeves which they proved in the case where the Taylor resolution is minimal. We introduce a conjecture about the minimal A-free resolution of the residue class field and show that this conjecture implies the conjecture of Charalambous and Reeves and, in addition, gives a formula for the Hilbert series. Using Algebraic Discrete Morse theory, we prove that the homology of the Koszul complex of A with respect to x₁,…,x_n is isomorphic to a graded commutative ring of polynomials over certain sets in the Taylor resolution divided by an ideal r of relations. This leads to a proof of our conjecture for some classes of algebras A. We also give an approach for the proof of our conjecture via Algebraic Discrete Morse theory in the general case.The conjecture implies that A is Golod if and only if the product (i.e. the first Massey operation) on the Koszul homology is trivial. Under the assumption of the conjecture we finally prove that a very simple purely combinatorial condition on the minimal monomial generating system of a implies Golodness for A.     

Poincaré series of modules over compressed Gorenstein local rings

Given two positive integers e and s we consider Gorenstein Artinian local rings R   whose maximal ideal m$\mathfrak{m}$ satisfies m^s≠0=m^s+1${\mathfrak{m}}^s\ne 0={\mathfrak{m}}^{s+1}$ and _rankR/m(m/m²)=e${\mathrm{rank}}_{R/\mathfrak{m}}(\mathfrak{m}/{\mathfrak{m}}^2)=e$. We say that R is a compressed Gorenstein local ring   when it has maximal length among such rings. It is known that generic Gorenstein Artinian algebras are compressed. If s≠3$s\ne 3$, we prove that the Poincaré series of all finitely generated modules over a compressed Gorenstein local ring are rational, sharing a common denominator. A formula for the denominator is given. When s is even this formula depends only on the integers e and s  . Note that for s=3$s=3$ examples of compressed Gorenstein local rings with transcendental Poincaré series exist, due to Bøgvad.     

A study of type I intermittency of a circular differential equation under a discontinuous right-hand side

In this paper we study a circular differential equation under a discontinuous periodic input, developing a quadratic differential equations system on S¹ and a linear differential equations system in the Minkowski space M³. The symmetry groups of these two systems are, respectively, PSOo(2,1) and SOo(2,1). The Poincaré circle map is constructed exactly, and a critical value αc of the parameter is identified. Depending on α of the input amplitude the equation may exhibit periodic, subharmonic or quasiperiodic motions. When α varies from α>αc to α<αc, there undergoes an inverse tangent bifurcation; consequently, the resultant Poincaré circle map offers one route to the quasiperiodicity via a type I intermittency. In the parameter range of α<αc the orbit generated by the Poincaré circle map is either m-periodic or quasiperiodic when n/m is a rational or an irrational number.     

Sub-elliptic global high order Poincaré inequalities in stratified Lie groups and applications

Sharp Poincaré inequalities on balls or chain type bounded domains have been extensively studied both in classical Euclidean space and Carnot-Carathéodory spaces associated with sub-elliptic vector fields (e.g., vector fields satisfying Hörmander's condition). In this paper, we investigate the validity of sharp global Poincaré inequalities of both first order and higher order on the entire nilpotent stratified Lie groups or on unbounded extension domains in such groups. We will show that simultaneous sharp global Poincaré inequalities also hold and weighted versions of such results remain to be true. More precisely, let G be a nilpotent stratified Lie group and f be in the localized non-isotropic Sobolev space , where 1?p<Q/m and Q is the homogeneous dimension of the Lie group G. Suppose that the mth sub-elliptic derivatives of f is globally Lp integrable; i.e., is finite (but assume that lower order sub-elliptic derivatives are only locally Lp integrable). We denote the space of such functions as Bm,p(G). We prove a high order Poincaré inequality for f minus a polynomial of order m−1 over the entire space G or unbounded extension domains. As applications, we will prove a density theorem stating that smooth functions with compact support are dense in Bm,p(G) modulus a finite-dimensional subspace.     

P-orderings: a metric viewpoint and the non-existence of simultaneous orderings

For a prime ideal ℘ and a subset S of a Dedekind ring R, a ℘-ordering of S is a sequence of elements of S with a certain minimizing property. These ℘-orderings were introduced in Bhargava (J. Reine Angew. Math., 490 (1997) 101) to generalize the usual factorial function and many classical results were thereby extended, including results about integer-valued polynomials. We consider ℘-orderings from the viewpoint of the ℘-adic metric on R. We find that the ℘-sequences of S depend only on the closure of S in . When R is a Dedekind domain and R′ is the integral closure of R in a finite extension of the fraction field of R, we relate the ℘-sequences of R and R′. Lastly, we investigate orderings that are simultaneously ℘-orderings for all prime ideals ℘⊂R, and show that such simultaneous orderings do not exist for imaginary quadratic number rings.     

Weak curvature conditions and functional inequalities

We give sufficient conditions for a measured length space (X,d,ν) to admit local and global Poincaré inequalities, along with a Sobolev inequality. We first introduce a condition DM on (X,d,ν), defined in terms of transport of measures. We show that DM, together with a doubling condition on ν, implies a scale-invariant local Poincaré inequality. We show that if (X,d,ν) has nonnegative N-Ricci curvature and has unique minimizing geodesics between almost all pairs of points then it satisfies DM, with constant N². The condition DM is preserved by measured Gromov-Hausdorff limits. We then prove a Sobolev inequality for measured length spaces with N-Ricci curvature bounded below by K>0. Finally we derive a sharp global Poincaré inequality.     

A kernel for automorphic L-functions

The full multiple Dirichlet series of an automorphic cusp form is defined, in classical language, as a Dirichlet series of several complex variables over all the Fourier coefficients of the cusp form. It is different from the L-function of Godement and Jacquet, which is defined as a Dirichlet series in one complex variable over a one-dimensional array of the Fourier coefficients. In GL(2) and GL(3), the two notions are simply related. In this paper, we construct a kernel function that gives the full multiple Dirichlet series of automorphic cusp forms on GL(n,R). The kernel function is a new Poincaré series. Specifically, the inner product of a cusp form with this Poincaré series is the product of the full multiple Dirichlet series of the form times a function that is essentially the Mellin transform of Jacquet's Whittaker function. In the proof, the full multiple Dirichlet series is produced by applying the Lipschitz summation formula several times and by an integral which collapses the sum over SL(n−1,Z) in the Fourier expansion of the cusp form.     

Splitting theorem, Poincaré-Hopf theorem and jumping nonlinear problems

In this paper, we establish a Gromoll-Meyer splitting theorem and a shifting theorem for J∈C^2-0(E,R) and by using the finite-dimensional approximation, mollifiers and Morse theory we generalize the Poincaré-Hopf theorem to J∈C¹(E,R) case. By combining the Poincaré-Hopf theorem and the splitting theorem, we study the existence of multiple solutions for jumping nonlinear elliptic equations.     

The tangential end fibration of an aspherical Poincaré complex

We construct a sphere fibration over a finite aspherical Poincaré complex X, which we call the tangential end fibration, under the condition that the universal cover of X is forward tame and simply connected at infinity. We show that it is tangent to X if the formal dimension of X is even or, when the formal dimension is odd, if the diagonal X→X×X admits a Poincaré embedding structure.     

Poincaré duality algebras mod two

We study Poincaré duality algebras over the field F₂ of two elements. After introducing a connected sum operation for such algebras we compute the corresponding Grothendieck group of surface algebras (i.e., Poincaré algebras of formal dimension 2). We show that the corresponding group for 3-folds (i.e., algebras of formal dimension 3) is not finitely generated, but does have a Krull-Schmidt property.We then examine the isomorphism classes of 3-folds with at most three generators of degree 3, provide a complete classification, settle which such occur as the cohomology of a smooth 3-manifold, and list separating invariants.The closing section and Appendix A provide several different means of constructing connected sum indecomposable 3-folds.     

Semisymmetric graphs

Let be a regular covering projection of connected graphs with the group of covering transformations isomorphic to N. If N is an elementary abelian p-group, then the projection ℘_N is called p-elementary abelian. The projection ℘_N is vertex-transitive (edge-transitive) if some vertex-transitive (edge-transitive) subgroup of the automorphism group of X lifts along ℘_N, and semisymmetric if it is edge- but not vertex-transitive. The projection ℘_N is minimal semisymmetric if it cannot be written as a composition ℘_N=℘℘_M of two (nontrivial) regular covering projections, where ℘_M is semisymmetric.Malni? et al. [Semisymmetric elementary abelian covers of the Möbius-Kantor graph, Discrete Math. 307 (2007) 2156-2175] determined all pairwise nonisomorphic minimal semisymmetric elementary abelian regular covering projections of the Möbius-Kantor graph, the Generalized Petersen graph GP(8,3), by explicitly giving the corresponding voltage rules generating the covering projections. It was remarked at the end of the above paper that the covering graphs arising from these covering projections need not themselves be semisymmetric (a graph with regular valency is said to be semisymmetric if its automorphism group is edge- but not vertex-transitive). In this paper it is shown that all these covering graphs are indeed semisymmetric.     

On a class of equilibrium problems in the real axis

In a series of seminal papers, Thomas J. Stieltjes (1856-1894) gave an elegant electrostatic interpretation for the zeros of classical families of orthogonal polynomials, such as Jacobi, Hermite and Laguerre polynomials. More generally, he extended this approach to the zeros of polynomial solutions of certain second-order linear differential equations (Lamé equations), the so-called Heine-Stieltjes polynomials.In this paper, a class of electrostatic equilibrium problems in R, where the free unit charges x₁,…,x_n∈R are in presence of a finite family of “attractors” (i.e., negative charges) z₁,…,z_m∈C?R, is considered and its connection with certain class of Lamé-type equations is shown. In addition, we study the situation when both n→∞ and m→∞, by analyzing the corresponding (continuous) equilibrium problem in presence of a certain class of external fields.     

A solution to de Groot’s absolute cone conjecture

A compactum X is an ‘absolute cone’ if, for each of its points x, the space X is homeomorphic to a cone with x corresponding to the cone point. In 1971, J. de Groot conjectured that each n-dimensional absolute cone is an n-cell. In this paper, we give a complete solution to that conjecture. In particular, we show that the conjecture is true for n≤3 and false for n≥5. For n=4, the absolute cone conjecture is true if and only if the 3-dimensional Poincaré Conjecture is true.     

Braided algebras and their applications to Noncommutative Geometry

We introduce the notion of a braided algebra and study some examples of these. In particular, R-symmetric and R-skew-symmetric algebras of a linear space V equipped with a skew-invertible Hecke symmetry R are braided algebras. We prove the “mountain property” for the numerators and denominators of their Poincaré–Hilbert series (which are always rational functions).