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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.
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NSF-PAR ID:
10385968
Journal Name:
ISSAC'2022
Page Range or eLocation-ID:
111 to 118
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 holesmore »
5. Abstract The elliptic algebras in the title are connected graded $\mathbb {C}$ -algebras, denoted $Q_{n,k}(E,\tau )$ , depending on a pair of relatively prime integers $n>k\ge 1$ , an elliptic curve E and a point $\tau \in E$ . This paper examines a canonical homomorphism from $Q_{n,k}(E,\tau )$ to the twisted homogeneous coordinate ring $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ on the characteristic variety $X_{n/k}$ for $Q_{n,k}(E,\tau )$ . When $X_{n/k}$ is isomorphic to $E^g$ or the symmetric power $S^gE$ , we show that the homomorphism $Q_{n,k}(E,\tau ) \to B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ is surjective, the relations for $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ are generated in degrees $\le 3$ and the noncommutative scheme $\mathrm {Proj}_{nc}(Q_{n,k}(E,\tau ))$ has a closed subvariety that is isomorphic to $E^g$ or $S^gE$ , respectively. When $X_{n/k}=E^g$ and $\tau =0$ , the results about $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ show that the morphism $\Phi _{|\mathcal {L}_{n/k}|}:E^g \to \mathbb {P}^{n-1}$ embeds $E^g$ as a projectively normal subvariety that is a scheme-theoretic intersection of quadric and cubic hypersurfaces.