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1. Improved effective Łojasiewicz inequality and applications

   https://doi.org/10.1017/fms.2024.66  

   Basu, Saugata; Mohammad-Nezhad, Ali (December 2024, Forum of Mathematics, Sigma)

   Abstract Let$$\mathrm {R}$$be a real closed field. Given a closed and bounded semialgebraic set$$A \subset \mathrm {R}^n$$and semialgebraic continuous functions$$f,g:A \rightarrow \mathrm {R}$$such that$$f^{-1}(0) \subset g^{-1}(0)$$, there exist an integer$$N> 0$$and$$c \in \mathrm {R}$$such that the inequality (Łojasiewicz inequality)$$|g(x)|^N \le c \cdot |f(x)|$$holds for all$$x \in A$$. In this paper, we consider the case whenAis defined by a quantifier-free formula with atoms of the form$$P = 0, P>0, P \in \mathcal {P}$$for some finite subset of polynomials$$\mathcal {P} \subset \mathrm {R}[X_1,\ldots ,X_n]_{\leq d}$$, and the graphs of$$f,g$$are also defined by quantifier-free formulas with atoms of the form$$Q = 0, Q>0, Q \in \mathcal {Q}$$, for some finite set$$\mathcal {Q} \subset \mathrm {R}[X_1,\ldots ,X_n,Y]_{\leq d}$$. We prove that the Łojasiewicz exponent in this case is bounded by$$(8 d)^{2(n+7)}$$. Our bound depends ondandnbut is independent of the combinatorial parameters, namely the cardinalities of$$\mathcal {P}$$and$$\mathcal {Q}$$. The previous best-known upper bound in this generality appeared inP. Solernó, Effective Łojasiewicz Inequalities in Semi-Algebraic Geometry, Applicable Algebra in Engineering, Communication and Computing (1991)and depended on the sum of degrees of the polynomials defining$$A,f,g$$and thus implicitly on the cardinalities of$$\mathcal {P}$$and$$\mathcal {Q}$$. As a consequence, we improve the current best error bounds for polynomial systems under some conditions. Finally, we prove a version of Łojasiewicz inequality in polynomially bounded o-minimal structures. We prove the existence of a common upper bound on the Łojasiewicz exponent for certain combinatorially defined infinite (but not necessarily definable) families of pairs of functions. This improves a prior result of Chris Miller (C. Miller, Expansions of the real field with power functions, Ann. Pure Appl. Logic (1994)). 

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2. Multi-linear forms, graphs, and $L^p$-improving measures in $${\Bbb F}_q^d$$

   https://doi.org/10.4153/S0008414X2300086X  

   Bhowmik, Pablo; Iosevich, Alex; Koh, Doowon; Pham, Thang (February 2025, Canadian journal of mathematics)

   Abstract The purpose of this paper is to introduce and study the following graph-theoretic paradigm. Let$$ \begin{align*}T_Kf(x)=\int K(x,y) f(y) d\mu(y),\end{align*} $$where$$f: X \to {\Bbb R}$$,Xa set, finite or infinite, andKand$$\mu $$denote a suitable kernel and a measure, respectively. Given a connected ordered graphGonnvertices, consider the multi-linear form$$ \begin{align*}\Lambda_G(f_1,f_2, \dots, f_n)=\int_{x^1, \dots, x^n \in X} \ \prod_{(i,j) \in {\mathcal E}(G)} K(x^i,x^j) \prod_{l=1}^n f_l(x^l) d\mu(x^l),\end{align*} $$where$${\mathcal E}(G)$$is the edge set ofG. Define$$\Lambda _G(p_1, \ldots , p_n)$$as the smallest constant$$C>0$$such that the inequality(0.1)$$ \begin{align} \Lambda_G(f_1, \dots, f_n) \leq C \prod_{i=1}^n {||f_i||}_{L^{p_i}(X, \mu)} \end{align} $$holds for all nonnegative real-valued functions$$f_i$$,$$1\le i\le n$$, onX. The basic question is, how does the structure ofGand the mapping properties of the operator$$T_K$$influence the sharp exponents in (0.1). In this paper, this question is investigated mainly in the case$$X={\Bbb F}_q^d$$, thed-dimensional vector space over the field withqelements,$$K(x^i,x^j)$$is the indicator function of the sphere evaluated at$$x^i-x^j$$, and connected graphsGwith at most four vertices. 

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3. Khintchine-type double recurrence in abelian groups

   https://doi.org/10.1017/etds.2024.29  

   ACKELSBERG, ETHAN (April 2024, Ergodic Theory and Dynamical Systems)

   Abstract We prove a Khintchine-type recurrence theorem for pairs of endomorphisms of a countable discrete abelian group. As a special case of the main result, if$$\Gamma $$is a countable discrete abelian group,$$\varphi , \psi \in \mathrm {End}(\Gamma )$$, and$$\psi - \varphi $$is an injective endomorphism with finite index image, then for any ergodic measure-preserving$$\Gamma $$-system$$( X, {\mathcal {X}}, \mu , (T_g)_{g \in \Gamma } )$$, any measurable set$$A \in {\mathcal {X}}$$, and any$${\varepsilon }> 0$$, there is a syndetic set of$$g \in \Gamma$$such that$$\mu ( A \cap T_{\varphi(g)}^{-1} A \cap T_{\psi(g)}^{-1} A ) > \mu(A)^3 - \varepsilon$$. This generalizes the main results of Ackelsberget al[Khintchine-type recurrence for 3-point configurations.Forum Math. Sigma10(2022), Paper no. e107] and essentially answers a question left open in that paper [Question 1.12; Khintchine-type recurrence for 3-point configurations.Forum Math. Sigma10(2022), Paper no. e107]. For the group$$\Gamma = {\mathbb {Z}}^d$$, the result applies to pairs of endomorphisms given by matrices whose difference is non-singular. The key ingredients in the proof are: (1) a recent result obtained jointly with Bergelson and Shalom [Khintchine-type recurrence for 3-point configurations.Forum Math. Sigma10(2022), Paper no. e107] that says that the relevant ergodic averages are controlled by a characteristic factor closely related to thequasi-affine(orConze–Lesigne) factor; (2) an extension trick to reduce to systems with well-behaved (with respect to$$\varphi $$and$$\psi $$) discrete spectrum; and (3) a description of Mackey groups associated to quasi-affine cocycles over rotational systems with well-behaved discrete spectrum. 

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4. Extensions of tautological rings and motivic structures in the cohomology of $${\overline {\mathcal {M}}}_{g,n}$$

   https://doi.org/10.1017/fmp.2024.24  

   Canning, Samir; Larson, Hannah; Payne, Sam (January 2024, Forum of Mathematics, Pi)

   Abstract We study collections of subrings of$$H^*({\overline {\mathcal {M}}}_{g,n})$$that are closed under the tautological operations that map cohomology classes on moduli spaces of smaller dimension to those on moduli spaces of larger dimension and contain the tautological subrings. Such extensions of tautological rings are well-suited for inductive arguments and flexible enough for a wide range of applications. In particular, we confirm predictions of Chenevier and Lannes for the$$\ell $$-adic Galois representations and Hodge structures that appear in$$H^k({\overline {\mathcal {M}}}_{g,n})$$for$$k = 13$$,$$14$$and$$15$$. We also show that$$H^4({\overline {\mathcal {M}}}_{g,n})$$is generated by tautological classes for allgandn, confirming a prediction of Arbarello and Cornalba from the 1990s. In order to establish the final base cases needed for the inductive proofs of our main results, we use Mukai’s construction of canonically embedded pentagonal curves of genus 7 as linear sections of an orthogonal Grassmannian and a decomposition of the diagonal to show that the pure weight cohomology of$${\mathcal {M}}_{7,n}$$is generated by algebraic cycle classes, for$$n \leq 3$$. 

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5. On the weight zero compactly supported cohomology of H_{g,n}

   https://doi.org/10.1017/fms.2024.53  

   Brandt, Madeline; Chan, Melody; Kannan, Siddarth (January 2024, Forum of Mathematics, Sigma)

   Abstract For$$g\ge 2$$and$$n\ge 0$$, let$$\mathcal {H}_{g,n}\subset \mathcal {M}_{g,n}$$denote the complex moduli stack ofn-marked smooth hyperelliptic curves of genusg. A normal crossings compactification of this space is provided by the theory of pointed admissible$$\mathbb {Z}/2\mathbb {Z}$$-covers. We explicitly determine the resulting dual complex, and we use this to define a graph complex which computes the weight zero compactly supported cohomology of$$\mathcal {H}_{g, n}$$. Using this graph complex, we give a sum-over-graphs formula for the$$S_n$$-equivariant weight zero compactly supported Euler characteristic of$$\mathcal {H}_{g, n}$$. This formula allows for the computer-aided calculation, for each$$g\le 7$$, of the generating function$$\mathsf {h}_g$$for these equivariant Euler characteristics for alln. More generally, we determine the dual complex of the boundary in any moduli space of pointed admissibleG-covers of genus zero curves, whenGis abelian, as a symmetric$$\Delta $$-complex. We use these complexes to generalize our formula for$$\mathsf {h}_g$$to moduli spaces ofn-pointed smooth abelian covers of genus zero curves. 

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