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1. Extensions of polynomial plank covering theorems

   https://doi.org/10.1112/blms.12979  

   Glazyrin, Alexey; Karasev, Roman; Polyanskii, Alexander (December 2023, Bulletin of the London Mathematical Society)

   Abstract We prove a complex polynomial plank covering theorem for not necessarily homogeneous polynomials. As the consequence of this result, we extend the complex plank theorem of Ball to the case of planks that are not necessarily centrally symmetric and not necessarily round. We also prove a weaker version of the spherical polynomial plank covering conjecture for planks of different widths. 

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2. Covering by Planks and Avoiding Zeros of Polynomials

   https://doi.org/10.1093/imrn/rnac259  

   Glazyrin, Alexey; Karasev, Roman; Polyanskii, Alexandr (October 2022, International Mathematics Research Notices)

   Abstract We note that the recent polynomial proofs of the spherical and complex plank covering problems by Zhao and Ortega-Moreno give some general information on zeros of real and complex polynomials restricted to the unit sphere. As a corollary of these results, we establish several generalizations of the celebrated Bang plank covering theorem. We prove a tight polynomial analog of the Bang theorem for the Euclidean ball and an even stronger polynomial version for the complex projective space. Specifically, for the ball, we show that for every real nonzero $$d$$-variate polynomial $$P$$ of degree $$n$$, there exists a point in the unit $$d$$-dimensional ball at distance at least $1/n$ from the zero set of the polynomial $$P$$. Using the polynomial approach, we also prove the strengthening of the Fejes Tóth zone conjecture on covering a sphere by spherical segments, closed parts of the sphere between two parallel hyperplanes. In particular, we show that the sum of angular widths of spherical segments covering the whole sphere is at least $$\pi $$. 

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3. Short interval results for powerfree polynomials over finite fields

   https://doi.org/10.1142/S1793042124500453  

   Kumchev, Angel V; McNew, Nathan G; Park, Ariana (April 2024, International Journal of Number Theory)

   Let [Formula: see text] be an integer and [Formula: see text] be a finite field with [Formula: see text] elements. We prove several results on the distribution in short intervals of polynomials in [Formula: see text] that are not divisible by the [Formula: see text]th power of any non-constant polynomial. Our main result generalizes a recent theorem by Carmon and Entin on the distribution of squarefree polynomials to all [Formula: see text]. We also develop polynomial versions of the classical techniques used to study gaps between [Formula: see text]-free integers in [Formula: see text]. We apply these techniques to obtain analogs in [Formula: see text] of some classical theorems on the distribution of [Formula: see text]-free integers. The latter results complement the main theorem in the case when the degrees of the polynomials are of moderate size. 

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4. Harmonic spinors on the Davis hyperbolic 4-manifold

   https://doi.org/10.1142/S1793525320500247  

   Ratcliffe, John G.; Ruberman, Daniel; Tschantz, Steven T. (September 2021, Journal of Topology and Analysis)

   In this paper, we use the [Formula: see text]-spin theorem to show that the Davis hyperbolic 4-manifold admits harmonic spinors. This is the first example of a closed hyperbolic [Formula: see text]-manifold that admits harmonic spinors. We also explicitly describe the spinor bundle of a spin hyperbolic 2- or 4-manifold and show how to calculated the subtle sign terms in the [Formula: see text]-spin theorem for an isometry, with isolated fixed points, of a closed spin hyperbolic 2- or 4-manifold. 

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5. Conformally Prescribed Scalar Curvature on Orbifolds

   https://doi.org/10.1007/s00220-022-04542-3  

   Ju, Tao; Viaclovsky, Jeff (November 2022, Communications in Mathematical Physics)

   Abstract We study the prescribed scalar curvature problem in a conformal class on orbifolds with isolated singularities. We prove a compactness theorem in dimension 4, and an existence theorem which holds in dimensions$$n \ge 4$$ $n\ge 4$. This problem is more subtle than the manifold case since the positive mass theorem does not hold for ALE metrics in general. We also determine the$$\textrm{U}(2)$$ $\text{U}\left(2\right)$-invariant Leray–Schauder degree for a family of negative-mass orbifolds found by LeBrun. 

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