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1. Transversality of holomorphic maps into hyperquadrics

   https://doi.org/10.1007/s00208-025-03134-5  

   Huang, Xiaojun; Zhu, Weixia (March 2025, Mathematische Annalen)

   Abstract We study holomorphic mapsFfrom a smooth Levi non-degenerate real hypersurface$$ M_{\ell }\subset {\mathbb {C}}^n $$ $M_{\ell }\subset C^n$into a hyperquadric$$ {\mathbb {H}}_{\ell '}^N $$ $H_{{\ell }^{\prime }}^N$with signatures$$ \ell \le (n-1)/2 $$ $\ell \le (n-1)/2$and$$ \ell '\le (N-1)/2,$$ ${\ell }^{\prime }\le (N-1)/2,$respectively. Assuming that$$ N - n < n - 1,$$ $N-n<n-1,$we prove that if$$ \ell = \ell ',$$ $\ell ={\ell }^{\prime },$thenFis either CR transversal to$$ {\mathbb {H}}_{\ell }^N $$ $H_{\ell }^N$at every point of$$ M_{\ell },$$ $M_{\ell },$or it maps a neighborhood of$$ M_{\ell } $$ $M_{\ell }$in$$ {\mathbb {C}}^n $$ $C^n$into$$ {\mathbb {H}}_{\ell }^N.$$ $H_{\ell }^N.$Furthermore, in the case where$$ \ell ' > \ell ,$$ ${\ell }^{\prime }>\ell ,$we show that ifFis not CR transversal at$$0\in M_\ell ,$$ $0\in M_{\ell },$then it must be transversally flat. The latter is best possible. 

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2. Misiurewicz polynomials and dynamical units, part II

   https://doi.org/10.1007/s40993-024-00539-0  

   Benedetto, Robert_L; Goksel, Vefa (June 2024, Research in Number Theory)

   Abstract Fix an integer$$d\ge 2$$ $d\ge 2$. The parameters$$c_0\in \overline{\mathbb {Q}}$$ $c_0\in \overline{Q}$for which the unicritical polynomial$$f_{d,c}(z)=z^d+c\in \mathbb {C}[z]$$ $f_{d,c}\left(z\right)=z^d+c\in C\left[z\right]$has finite postcritical orbit, also known asMisiurewiczparameters, play a significant role in complex dynamics. Recent work of Buff, Epstein, and Koch proved the first known cases of a long-standing dynamical conjecture of Milnor using their arithmetic properties, about which relatively little is otherwise known. Continuing our work from a companion paper, we address further arithmetic properties of Misiurewicz parameters, especially the nature of the algebraic integers obtained by evaluating the polynomial defining one such parameter at a different Misiurewicz parameter. In the most challenging such combinations, we describe a connection between such algebraic integers and the multipliers of associated periodic points. As part of our considerations, we also introduce a new class of polynomials we callp-special, which may be of independent number theoretic interest. 

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3. Sparse representation of vectors in lattices and semigroups

   https://doi.org/10.1007/s10107-021-01657-8  

   Aliev, Iskander; Averkov, Gennadiy; De_Loera, Jesús_A; Oertel, Timm (May 2021, Mathematical Programming)

   Abstract We study the sparsity of the solutions to systems of linear Diophantine equations with and without non-negativity constraints. The sparsity of a solution vector is the number of its nonzero entries, which is referred to as the$$\ell _0$$ ${\ell }_0$-norm of the vector. Our main results are new improved bounds on the minimal$$\ell _0$$ ${\ell }_0$-norm of solutions to systems$$A\varvec{x}=\varvec{b}$$ $Ax=b$, where$$A\in \mathbb {Z}^{m\times n}$$ $A\in Z^{m\times n}$,$${\varvec{b}}\in \mathbb {Z}^m$$ $b\in Z^m$and$$\varvec{x}$$ $x$is either a general integer vector (lattice case) or a non-negative integer vector (semigroup case). In certain cases, we give polynomial time algorithms for computing solutions with$$\ell _0$$ ${\ell }_0$-norm satisfying the obtained bounds. We show that our bounds are tight. Our bounds can be seen as functions naturally generalizing the rank of a matrix over$$\mathbb {R}$$ $R$, to other subdomains such as$$\mathbb {Z}$$ $Z$. We show that these new rank-like functions are all NP-hard to compute in general, but polynomial-time computable for fixed number of variables. 

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4. Ruijsenaars wavefunctions as modular group matrix coefficients

   https://doi.org/10.1007/s11005-024-01881-1  

   Di_Francesco, Philippe; Kedem, Rinat; Khoroshkin, Sergey; Schrader, Gus; Shapiro, Alexander (November 2024, Letters in Mathematical Physics)

   Abstract We give a description of the Hallnäs–Ruijsenaars eigenfunctions of the 2-particle hyperbolic Ruijsenaars system as matrix coefficients for the order 4 element$$S\in SL(2,{\mathbb {Z}})$$ $S\in SL(2,Z)$acting on the Hilbert space ofGL(2) quantum Teichmüller theory on the punctured torus. TheGL(2) Macdonald polynomials are then obtained as special values of the analytic continuation of these matrix coefficients. The main tool used in the proof is the cluster structure on the moduli space of framedGL(2)-local systems on the punctured torus, and an$$SL(2,{\mathbb {Z}})$$ $SL(2,Z)$-equivariant embedding of theGL(2) spherical DAHA into the quantized coordinate ring of the corresponding cluster Poisson variety. 

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5. Measurement of the mass dependence of the transverse momentum of lepton pairs in Drell–Yan production in proton–proton collisions at $$\sqrt{s} = 13\,\text {Te\hspace{-.08em}V} $$

   https://doi.org/10.1140/epjc/s10052-023-11631-7  

   Tumasyan, A.; Adam, W.; Andrejkovic, J. W.; Bergauer, T.; Chatterjee, S.; Dragicevic, M.; Valle, A. Escalante; Frühwirth, R.; Jeitler, M.; Krammer, N.; et al (July 2023, The European Physical Journal C)

   Abstract The double differential cross sections of the Drell–Yan lepton pair ($$\ell ^+\ell ^-$$ ${\ell }^{+}{\ell }^{-}$, dielectron or dimuon) production are measured as functions of the invariant mass$$m_{\ell \ell }$$ $m_{\ell \ell }$, transverse momentum$$p_{\textrm{T}} (\ell \ell )$$ $p_{\text{T}}\left(\ell \ell \right)$, and$$\varphi ^{*}_{\eta }$$ ${\phi }_{\eta }^{\ast }$. The$$\varphi ^{*}_{\eta }$$ ${\phi }_{\eta }^{\ast }$observable, derived from angular measurements of the leptons and highly correlated with$$p_{\textrm{T}} (\ell \ell )$$ $p_{\text{T}}\left(\ell \ell \right)$, is used to probe the low-$$p_{\textrm{T}} (\ell \ell )$$ $p_{\text{T}}\left(\ell \ell \right)$region in a complementary way. Dilepton masses up to 1$$\,\text {Te\hspace{-.08em}V}$$ $\text{Te}\text{V}$are investigated. Additionally, a measurement is performed requiring at least one jet in the final state. To benefit from partial cancellation of the systematic uncertainty, the ratios of the differential cross sections for various$$m_{\ell \ell }$$ $m_{\ell \ell }$ranges to those in the Z mass peak interval are presented. The collected data correspond to an integrated luminosity of 36.3$$\,\text {fb}^{-1}$$ ${\text{fb}}^{-1}$of proton–proton collisions recorded with the CMS detector at the LHC at a centre-of-mass energy of 13$$\,\text {Te\hspace{-.08em}V}$$ $\text{Te}\text{V}$. Measurements are compared with predictions based on perturbative quantum chromodynamics, including soft-gluon resummation. 

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