Alexander polynomials and Poincaré series of sets of ideals

Earlier the authors considered and, in some cases, computed Poincaré series for two sorts of multi-index filtrations on the ring of germs of functions on a complex (normal) surface singularity (in particular, on the complex plane). A filtration of the first class was defined by a curve (with several branches) on the surface singularity. A filtration of the second class (called divisorial) was defined by a set of components of the exceptional divisor of a modification of the surface singularity. Here we define and compute in some cases the Poincaré series corresponding to a set of ideals in the ring of germs of functions on a surface singularity. For the complex plane, this notion unites the two classes of filtrations described above.     

Poincaré series of Klein groups,coxeter polynomials,the Burau representation,and Milnor invariants

We obtain several formulas for the Poincaré series defined by B. Kostant for Klein groups (binary polyhedral groups) and some formulas for Coxeter polynomials (characteristic polynomials of monodromy in the case of singularities). Some of these formulas—the generalized Ebeling formula, the Christoffel-Darboux identity, and the combinatorial formula—are corollaries to the well-known statements on the characteristic polynomial of a graph and are analogous to formulas for orthogonal polynomials. The ratios of Poincaré series and Coxeter polynomials are represented in terms of branched continued fractions, which are q-analogs of continued fractions that arise in the theory of resolution of singularities and in the Kirby calculus. Other formulas connect the ratios of some Poincaré series and Coxeter polynomials with the Burau representation and Milnor invariants of string links. The results obtained by S.M. Gusein-Zade, F. Delgado, and A. Campillo allow one to consider these facts as statements on the Poincaré series of the rings of functions on the singularities of curves, which suggests the following conjecture: the ratio of the Poincaré series of the rings of functions for close (in the sense of adjacency or position in a series) singularities of curves is determined by the Burau representation or by the Milnor invariants of a string link, which is an intermediate object in the transformation of the knot of one singularity into the knot of the other.     

McKay correspondence for the Poincaré series of Kleinian and Fuchsian singularities

We give a uniform and, to a large extent, geometrical proof that the Poincaré series of the coordinate algebra of a Kleinian singularity and of a Fuchsian singularity of genus 0 is the quotient of the characteristic polynomials of two Coxeter elements. These Coxeter elements are interpreted geometrically, using 2-Calabi-Yau triangulated categories and spherical twist functors.     

Denominators for the Poincaré series of invariants of small matrices

We determine explicit denominators for the Poincaré series of (a) the invariants ofm genericN ×N matrices, and (b) the ring generated bym genericN ×N matrices and their traces, forN≤4. ForN≤3 we prove (and forN=4 we conjecture) that the denominators we obtain are of minimum degree. We also give explicit rational fractions for both series for small values ofm andN. Research supported by NSF grants DMS-9622062 and DMS-9700787.     

On the interlacing of the zeros of Poincaré series

The Ramanujan Journal - Rankin proved that the Poincaré series for $$mathbf{SL}(2,{{mathbb {Z}}})$$ that are not cusp forms have all their zeros on the unit circle in the standard...     

Denominators for the Poincaré series of invariants of matrices with involution

The goal of this paper is to study some Poincaré series associated to the invariants of the symplectic and odd orthogonal groups. These series turn out to be rational functions and our main results will describe the denominators. This work will generalize some known results on the invariants of the general linear groups. In addition to whatever intrinsic interest we hope our results may have, the subject involves an interesting interplay of invariant theory and complex variables. The first author gratefully acknowledges Support from DePaul University Research Council. The second author was supported in part by the Israel Science Foundation, founded by the Israel Academy of Sciences and Humanities, and by an Internal Research Grant from Bar-Ilan University.     

Non-vanishing of Taylor coefficients and Poincaré series

We prove recursive formulas for the Taylor coefficients of cusp forms, such as Ramanujan’s Delta function, at points in the upper half-plane. This allows us to show the non-vanishing of all Taylor coefficients of Delta at CM points of small discriminant as well as the non-vanishing of certain Poincaré series. At a “generic” point, all Taylor coefficients are shown to be non-zero. Some conjectures on the Taylor coefficients of Delta at CM points are stated.     

Equivariant Poincaré series of filtrations and topology

Earlier, for an action of a finite group G on a germ of an analytic variety, an equivariant G-Poincaré series of a multi-index filtration in the ring of germs of functions on the variety was defined as an element of the Grothendieck ring of G-sets with an additional structure. We discuss to which extent the G-Poincaré series of a filtration defined by a set of curve or divisorial valuations on the ring of germs of analytic functions in two variables determines the (equivariant) topology of the curve or of the set of divisors.     

Poincaré series and monodromy of the simple and unimodal boundary singularities

A boundary singularity is a singularity of a function on a manifold with boundary. The simple and unimodal boundary singularities were classified by V.I. Arnold and V.I. Matov. The McKay correspondence can be generalized to the simple boundary singularities. We consider the monodromy of the simple, parabolic, and exceptional unimodal boundary singularities. We show that the characteristic polynomial of the monodromy is related to the Poincaré series of the coordinate algebra of the ambient singularity.     

Nonvanishing of Siegel Poincaré series

We prove that, under suitable conditions, certain Siegel Poincaré series of exponential type of even integer weight and degree 2 do not vanish identically. We also find estimates for twisted Kloosterman sums and Salié sums in all generality.     

Poincaré series of Drinfeld type

Let K be the rational function field $\mathbb{F}_q (t)$ . We construct Poincaré series on the Bruhat-Tits tree of GL₂ over K _∞ and show that they generate the space of automorphic cusp forms of Drinfeld type.     

Poincaré duality for p-adic Lie groups

We establish Poincaré duality for continuous group cohomology of p-adic Lie groups with rational coefficients and compare integral structures under this duality.     

Generalised Poincaré series and embedded resolution of curves

The purpose of this paper is to extend the notions of generalised Poincaré series and divisorial generalised Poincaré series (of motivic nature) introduced by Campillo, Delgado and Gusein–Zade for complex curve singularities to curves defined over perfect fields, as well as to express them in terms of an embedded resolution of curves.     

Generalised Poincaré series and embedded resolution of curves

The purpose of this paper is to extend the notions of generalised Poincaré series and divisorial generalised Poincaré series (of motivic nature) introduced by Campillo, Delgado and Gusein–Zade for complex curve singularities to curves defined over perfect fields, as well as to express them in terms of an embedded resolution of curves.     

Analogue of the Cayley–Sylvester formula and the Poincaré series for the algebra of invariants of n-ary form

The number of linearly independent homogeneous invariants of degree k for the n-ary form of order d is calculated. Also, a formula for the Poincaré series for the algebra of invariants of the n-ary form is found.     

Numerical experiments on the accuracy of the Chebyshev–Frobenius companion matrix method for finding the zeros of a truncated series of Chebyshev polynomials

For a function f(x)$f(x)$ that is smooth on the interval x∈[a,b]$x\in [a,b]$ but otherwise arbitrary, the real-valued roots on the interval can always be found by the following two-part procedure. First, expand f(x)$f(x)$ as a Chebyshev polynomial series on the interval and truncate for sufficiently large N. Second, find the zeros of the truncated Chebyshev series. The roots of an arbitrary polynomial of degree N  , when written in the form of a truncated Chebyshev series, are the eigenvalues of an N×N$N\times N$ matrix whose elements are simple, explicit functions of the coefficients of the Chebyshev series. This matrix is a generalization of the Frobenius companion matrix. We show by experimenting with random polynomials, Wilkinson's notoriously ill-conditioned polynomial, and polynomials with high-order roots that the Chebyshev companion matrix method is remarkably accurate for finding zeros on the target interval, yielding roots close to full machine precision. We also show that it is easy and cheap to apply Newton's iteration directly to the Chebyshev series so as to refine the roots to full machine precision, using the companion matrix eigenvalues as the starting point. Lastly, we derive a couple of theorems. The first shows that simple roots are stable under small perturbations of magnitude ε$\epsilon$ to a Chebyshev coefficient: the shift in the root _x*$x_{*}$ is bounded by ε/df/dx(_x*)+O(ε²)$\epsilon /\mathrm{d}f/\mathrm{d}x(x_{*})+\mathrm{O}({\epsilon }^2)$ for sufficiently small ε$\epsilon$. Second, we show that polynomials with definite parity (only even or only odd powers of x) can be solved by a companion matrix whose size is one less than the number of nonzero coefficients, a vast cost-saving.     

Pro-p completions of Poincaré duality groups

We consider some sufficient conditions for the pro-p completion of an orientable Poincaré duality group of dimension n ≥ 3 to be a virtually pro-p Poincaré duality group of dimension at most n ? 2.     

Szegő kernels and Poincaré series

Let \(M = {{\widetilde M} \mathord{\left/ {\vphantom {{\widetilde M} \Gamma }} \right. \kern-\nulldelimiterspace} \Gamma }\) be a Kähler manifold, where Γ ~ π₁(M) and \(\widetilde M\) is the universal Kähler cover. Let (L, h) → M be a positive hermitian holomorphic line bundle. We first prove that the L² Szeg? projector \({\widetilde \Pi _N}\) for L²-holomorphic sections on the lifted bundle \({\widetilde L^N}\) is related to the Szeg? projector for H⁰(M, L^N) by \({\widehat \Pi _N}\left( {x,y} \right) = \sum\nolimits_{\gamma \in \Gamma } {{{\widetilde {\widehat \Pi }}_N}} \left( {\gamma \cdot x,y} \right)\). We then apply this result to give a simple proof of Napier’s theorem on the holomorphic convexity of \(\widetilde M\) with respect to \({\widetilde L^N}\) and to surjectivity of Poincaré series.