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1. ON THE CONNECTION BETWEEN DIFFERENTIAL POLYNOMIAL RINGS AND POLYNOMIAL RINGS OVER NIL RINGS

   https://doi.org/10.1017/S0004972719000923  

   CATALANO, LOUISA; CHANG-LEE, MEGAN (June 2020, Bulletin of the Australian Mathematical Society)

   In this paper, we study some connections between the polynomial ring $R[y]$ and the differential polynomial ring $R[x;D]$ . In particular, we answer a question posed by Smoktunowicz, which asks whether $R[y]$ is nil when $R[x;D]$ is nil, provided that $$R$$ is an algebra over a field of positive characteristic and $$D$$ is a locally nilpotent derivation. 

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2. Rounding in the Rings

   Liu, Feng-Hao; Wang, Zhedong (August 2020, Crypto)

   null (Ed.)

   In this work, we conduct a comprehensive study on establishing hardness reductions for (Module) Learning with Rounding over rings (RLWR). Towards this, we present an algebraic framework of LWR, inspired by a recent work of Peikert and Pepin (TCC ’19). Then we show a search-to-decision reduction for Ring-LWR, generalizing a result in the plain LWR setting by Bogdanov et al. (TCC ’15). Finally, we show a reduction from Ring-LWE to Module Ring-LWR (even for leaky secrets), generalizing the plain LWE to LWR reduction by Alwen et al. (Crypto ’13). One of our central techniques is a new ring leftover hash lemma, which might be of independent interests. 

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3. Measuring Photon Rings with the ngEHT

   https://doi.org/10.3390/galaxies10060111  

   Tiede, Paul; Johnson, Michael D.; Pesce, Dominic W.; Palumbo, Daniel C.; Chang, Dominic O.; Galison, Peter (December 2022, Galaxies)

   General relativity predicts that images of optically thin accretion flows around black holes should generically have a “photon ring”, composed of a series of increasingly sharp subrings that correspond to increasingly strongly lensed emission near the black hole. Because the effects of lensing are determined by the spacetime curvature, the photon ring provides a pathway to precise measurements of the black hole properties and tests of the Kerr metric. We explore the prospects for detecting and measuring the photon ring using very long baseline interferometry (VLBI) with the Event Horizon Telescope (EHT) and the next-generation EHT (ngEHT). We present a series of tests using idealized self-fits to simple geometrical models and show that the EHT observations in 2017 and 2022 lack the angular resolution and sensitivity to detect the photon ring, while the improved coverage and angular resolution of ngEHT at 230 GHz and 345 GHz is sufficient for these models. We then analyze detection prospects using more realistic images from general relativistic magnetohydrodynamic simulations by applying “hybrid imaging”, which simultaneously models two components: a flexible raster image (to capture the direct emission) and a ring component. Using the Bayesian VLBI modeling package Comrade.jl, we show that the results of hybrid imaging must be interpreted with extreme caution for both photon ring detection and measurement—hybrid imaging readily produces false positives for a photon ring, and its ring measurements do not directly correspond to the properties of the photon ring. 

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4. Equivariant resolutions over Veronese rings

   https://doi.org/10.1112/jlms.12848  

   Almousa, Ayah; Perlman, Michael; Pevzner, Alexandra; Reiner, Victor; VandeBogert, Keller (December 2023, Journal of the London Mathematical Society)

   Abstract Working in a polynomial ring , where is an arbitrary commutative ring with 1, we consider the th Veronese subalgebras , as well as natural ‐submodules inside . We develop and use characteristic‐free theory of Schur functors associated to ribbon skew diagrams as a tool to construct simple ‐equivariant minimal free ‐resolutions for the quotient ring and for these modules . These also lead to elegant descriptions of for all and for any pair of these modules . 

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5. Symmetric group fixed quotients of polynomial rings

   https://doi.org/10.1016/j.jpaa.2023.107537  

   Pevzner, Alexandra (April 2024, Journal of Pure and Applied Algebra)

   Given a representation of a finite group G over some commutative base ring k, the cofixed space is the largest quotient of the representation on which the group acts trivially. If G acts by k-algebra automorphisms, then the cofixed space is a module over the ring of G-invariants. When the order of G is not invertible in the base ring, little is known about this module structure. We study the cofixed space in the case that G is the symmetric group on n letters acting on a polynomial ring by permuting its variables. When k has characteristic 0, the cofixed space is isomorphic to an ideal of the ring of symmetric polynomials in n variables. Localizing k at a prime integer p while letting n vary reveals striking behavior in these ideals. As n grows, the ideals stay stable in a sense, then jump in complexity each time n reaches a multiple of p. 

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