skip to main content


Title: Desingularization and p-Curvature of Recurrence Operators
Linear recurrence operators in characteristic p are classified by their p-curvature. For a recurrence operator L, denote by χ(L) the characteristic polynomial of its p-curvature. We can obtain information about the factorization of L by factoring χ(L). The main theorem of this paper gives an unexpected relation between χ(L) and the true singularities of L. An application is to speed up a fast algorithm for computing χ(L) by desingularizing L first. Another contribution of this paper is faster desingularization.  more » « less
Award ID(s):
2007959
NSF-PAR ID:
10385968
Author(s) / Creator(s):
;
Date Published:
Journal Name:
ISSAC'2022
Page Range / eLocation ID:
111 to 118
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Abstract The zeta function of a curve C over a finite field may be expressed in terms of the characteristic polynomial of a unitary matrix Θ C . We develop and present a new technique to compute the expected value of tr(Θ C n ) for various moduli spaces of curves of genus g over a fixed finite field in the limit as g is large, generalising and extending the work of Rudnick [ Rud10 ] and Chinis [ Chi16 ]. This is achieved by using function field zeta functions, explicit formulae, and the densities of prime polynomials with prescribed ramification types at certain places as given in [ BDF + 16 ] and [ Zha ]. We extend [ BDF + 16 ] by describing explicit dependence on the place and give an explicit proof of the Lindelöf bound for function field Dirichlet L -functions L (1/2 + it , χ). As applications, we compute the one-level density for hyperelliptic curves, cyclic ℓ-covers, and cubic non-Galois covers. 
    more » « less
  2. Abstract The goal of this paper is to generalise, refine and improve results on large intersections from [2, 8]. We show that if G is a countable discrete abelian group and $\varphi , \psi : G \to G$ are homomorphisms, such that at least two of the three subgroups $\varphi (G)$ , $\psi (G)$ and $(\psi -\varphi )(G)$ have finite index in G , then $\{\varphi , \psi \}$ has the large intersections property . That is, for any ergodic measure preserving system $\textbf {X}=(X,\mathcal {X},\mu ,(T_g)_{g\in G})$ , any $A\in \mathcal {X}$ and any $\varepsilon>0$ , the set $$ \begin{align*} \{g\in G : \mu(A\cap T_{\varphi(g)}^{-1} A \cap T_{\psi(g)}^{-1} A)>\mu(A)^3-\varepsilon\} \end{align*} $$ is syndetic (Theorem 1.11). Moreover, in the special case where $\varphi (g)=ag$ and $\psi (g)=bg$ for $a,b\in \mathbb {Z}$ , we show that we only need one of the groups $aG$ , $bG$ or $(b-a)G$ to be of finite index in G (Theorem 1.13), and we show that the property fails, in general, if all three groups are of infinite index (Theorem 1.14). One particularly interesting case is where $G=(\mathbb {Q}_{>0},\cdot )$ and $\varphi (g)=g$ , $\psi (g)=g^2$ , which leads to a multiplicative version of the Khintchine-type recurrence result in [8]. We also completely characterise the pairs of homomorphisms $\varphi ,\psi $ that have the large intersections property when $G = {{\mathbb Z}}^2$ . The proofs of our main results rely on analysis of the structure of the universal characteristic factor for the multiple ergodic averages $$ \begin{align*} \frac{1}{|\Phi_N|} \sum_{g\in \Phi_N}T_{\varphi(g)}f_1\cdot T_{\psi(g)} f_2. \end{align*} $$ In the case where G is finitely generated, the characteristic factor for such averages is the Kronecker factor . In this paper, we study actions of groups that are not necessarily finitely generated, showing, in particular, that, by passing to an extension of $\textbf {X}$ , one can describe the characteristic factor in terms of the Conze–Lesigne factor and the $\sigma $ -algebras of $\varphi (G)$ and $\psi (G)$ invariant functions (Theorem 4.10). 
    more » « less
  3. Abstract

    LetEbe an elliptic curve over$${{\mathbb {Q}}}$$Q. We conjecture asymptotic estimates for the number of vanishings of$$L(E,1,\chi )$$L(E,1,χ)as$$\chi $$χvaries over all primitive Dirichlet characters of orders 4 and 6, subject to a mild hypothesis onE. Our conjectures about these families come from conjectures about random unitary matrices as predicted by the philosophy of Katz-Sarnak. We support our conjectures with numerical evidence. Compared to earlier work by David, Fearnley and Kisilevsky that formulated analogous conjectures for characters of any odd prime order, in the composite order case, we need to justify our use of random matrix theory heuristics by analyzing the equidistribution of the squares of normalized Gauss sums. To do this, we introduce the notion of totally order$$\ell $$characters to quantify how quickly the quartic and sextic Gauss sums become equidistributed. Surprisingly, the rate of equidistribution in the full family of quartic (resp., sextic) characters is much slower than in the sub-family of totally quartic (resp., sextic) characters. We provide a conceptual explanation for this phenomenon by observing that the full family of order$$\ell $$twisted elliptic curveL-functions, with$$\ell $$even and composite, is a mixed family with both unitary and orthogonal aspects.

     
    more » « less
  4. ABSTRACT

    Many astrophysical environments, from star clusters and globular clusters to the discs of active galactic nuclei, are characterized by frequent interactions between stars and the compact objects that they leave behind. Here, using a suite of 3D hydrodynamics simulations, we explore the outcome of close interactions between $1\, \mathrm{M}_{\odot }$ stars and binary black holes (BBHs) in the gravitational wave regime, resulting in a tidal disruption event (TDE) or a pure scattering, focusing on the accretion rates, the back reaction on the BH binary orbital parameters, and the increase in the binary BH effective spin. We find that TDEs can make a significant impact on the binary orbit, which is often different from that of a pure scattering. Binaries experiencing a prograde (retrograde) TDE tend to be widened (hardened) by up to $\simeq 20{{\ \rm per\ cent}}$. Initially circular binaries become more eccentric by $\lesssim 10{{\ \rm per\ cent}}$ by a prograde or retrograde TDE, whereas the eccentricity of initially eccentric binaries increases (decreases) by a retrograde (prograde) TDE by $\lesssim 5{{\ \rm per\ cent}}$. Overall, a single TDE can generally result in changes of the gravitational-wave-driven merger time-scale by order unity. The accretion rates of both black holes are very highly super-Eddington, showing modulations (preferentially for retrograde TDEs) on a time-scale of the orbital period, which can be a characteristic feature of BBH-driven TDEs. Prograde TDEs result in the effective spin parameter χ to vary by ≲0.02, while χ ≳ −0.005 for retrograde TDEs.

     
    more » « less
  5. Molecular dynamics (MD) is a powerful tool for studying intrinsically disordered proteins, however, its reliability depends on the accuracy of the force field. We assess Amber ff19SB, Amber ff14SB, OPLS-AA/M, and CHARMM36m with respect to their capacity to capture intrinsic conformational dynamics of 14 guest residues x (=G, A, L, V, I, F, Y, D P , E P , R, C, N, S, T) in GxG peptides in water. The MD-derived Ramachandran distribution of each guest residue is used to calculate 5 J-coupling constants and amide I′ band profiles to facilitate a comparison to spectroscopic data through reduced χ 2 functions. We show that the Gaussian model, optimized to best fit the experimental data, outperforms all MD force fields by an order of magnitude. The weaknesses of the MD force fields are: (i) insufficient variability of the polyproline II (pPII) population among the guest residues; (ii) oversampling of antiparallel at the expense of transitional β-strand region; (iii) inadequate sampling of turn-forming conformations for ionizable and polar residues; and (iv) insufficient guest residue-specificity of the Ramachandran distributions. Whereas Amber ff19SB performs worse than the other three force fields with respect to χ 2 values, it accounts for residue-specific pPII content better than the other three force fields. Additional testing of residue-specific RSFF1 and Amber ff14SB combined with TIP4P/2005 on six guest residues x (=A, I, F, D P , R, S) reveals that residue specificity derived from protein coil libraries or an improved water model alone do not result in significantly lower χ 2 values. 
    more » « less