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1. LINEAR AND QUADRATIC UNIFORMITY OF THE MÖBIUS FUNCTION OVER

   https://doi.org/10.1112/S0025579319000032  

   Bienvenu, Pierre-Yves; Lê, Thái Hoàng (January 2019, Mathematika)

   We examine correlations of the Möbius function over $$\mathbb{F}_{q}[t]$$ with linear or quadratic phases, that is, averages of the form 1 $$\begin{eqnarray}\frac{1}{q^{n}}\mathop{\sum }_{\deg f0$$ if $$Q$$ is linear and $$O(q^{-n^{c}})$$ for some absolute constant $c>0$ if $$Q$$ is quadratic. The latter bound may be reduced to $$O(q^{-c^{\prime }n})$$ for some $$c^{\prime }>0$$ when $Q(f)$ is a linear form in the coefficients of $$f^{2}$$ , that is, a Hankel quadratic form, whereas, for general quadratic forms, it relies on a bilinear version of the additive-combinatorial Bogolyubov theorem. 

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2. Abelian varieties of prescribed order over finite fields

   https://doi.org/10.1007/s00208-024-03084-4  

   van_Bommel, Raymond; Costa, Edgar; Li, Wanlin; Poonen, Bjorn; Smith, Alexander (March 2025, Mathematische Annalen)

   Abstract Given a prime powerqand$$n \gg 1$$ $n\gg 1$, we prove that every integer in a large subinterval of the Hasse–Weil interval$$[(\sqrt{q}-1)^{2n},(\sqrt{q}+1)^{2n}]$$ $[{(\sqrt{q}-1)}^{2n},{(\sqrt{q}+1)}^{2n}]$is$$\#A({\mathbb {F}}_q)$$ $\#A\left(F_q\right)$for some ordinary geometrically simple principally polarized abelian varietyAof dimensionnover$${\mathbb {F}}_q$$ $F_q$. As a consequence, we generalize a result of Howe and Kedlaya for$${\mathbb {F}}_2$$ $F_2$to show that for each prime powerq, every sufficiently large positive integer is realizable, i.e.,$$\#A({\mathbb {F}}_q)$$ $\#A\left(F_q\right)$for some abelian varietyAover$${\mathbb {F}}_q$$ $F_q$. Our result also improves upon the best known constructions of sequences of simple abelian varieties with point counts towards the extremes of the Hasse–Weil interval. A separate argument determines, for fixedn, the largest subinterval of the Hasse–Weil interval consisting of realizable integers, asymptotically as$$q \rightarrow \infty $$ $q\to \infty$; this gives an asymptotically optimal improvement of a 1998 theorem of DiPippo and Howe. Our methods are effective: We prove that if$$q \le 5$$ $q\le 5$, then every positive integer is realizable, and for arbitraryq, every positive integer$$\ge q^{3 \sqrt{q} \log q}$$ $\ge q^{3\sqrt{q}logq}$is realizable. 

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3. AND testing and robust judgement aggregation

   https://doi.org/10.1145/3357713.3384254  

   Filmus, Y; Lifshitz, N; Minzer, D; Mossel, E. (June 2020, STOC 2020: Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing)

   A function f∶{0,1}n→ {0,1} is called an approximate AND-homomorphism if choosing x,y∈n uniformly at random, we have that f(x∧ y) = f(x)∧ f(y) with probability at least 1−ε, where x∧ y = (x1∧ y1,…,xn∧ yn). We prove that if f∶ {0,1}n → {0,1} is an approximate AND-homomorphism, then f is δ-close to either a constant function or an AND function, where δ(ε) → 0 as ε→ 0. This improves on a result of Nehama, who proved a similar statement in which δ depends on n. Our theorem implies a strong result on judgement aggregation in computational social choice. In the language of social choice, our result shows that if f is ε-close to satisfying judgement aggregation, then it is δ(ε)-close to an oligarchy (the name for the AND function in social choice theory). This improves on Nehama’s result, in which δ decays polynomially with n. Our result follows from a more general one, in which we characterize approximate solutions to the eigenvalue equation f = λ g, where is the downwards noise operator f(x) = y[f(x ∧ y)], f is [0,1]-valued, and g is {0,1}-valued. We identify all exact solutions to this equation, and show that any approximate solution in which f and λ g are close is close to an exact solution. 

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4. Determinantal Representations and the Image of the Principal Minor Map

   https://doi.org/10.1093/imrn/rnae038  

   Al_Ahmadieh, Abeer; Vinzant, Cynthia (March 2024, International Mathematics Research Notices)

   Abstract In this paper we explore determinantal representations of multiaffine polynomials and consequences for the image of various spaces of matrices under the principal minor map. We show that a real multiaffine polynomial has a definite Hermitian determinantal representation if and only if all of its so-called Rayleigh differences factor as Hermitian squares and use this characterization to conclude that the image of the space of Hermitian matrices under the principal minor map is cut out by the orbit of finitely many equations and inequalities under the action of $$(\textrm{SL}_{2}(\mathbb{R}))^{n} \rtimes S_{n}$$. We also study such representations over more general fields with quadratic extensions. Factorizations of Rayleigh differences prove an effective tool for capturing subtle behavior of the principal minor map. In contrast to the Hermitian case, we give examples to show for any field $$\mathbb{F}$$, there is no finite set of equations whose orbit under $$(\textrm{SL}_{2}(\mathbb{F}))^{n} \rtimes S_{n}$$ cuts out the image of $$n\times n$$ matrices over $${\mathbb{F}}$$ under the principal minor map for every $$n$$. 

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5. Expected Number of Real Zeros for Random Orthogonal Polynomials

   Lubinsky, Doron; Pritsker, Igor; Xie, Xiaoyou (January 2018, Mathematical proceedings of the Cambridge Philosophical Society)

   We study the expected number of real zeros for random linear combinations of orthogonal polynomials. It is well known that Kac polynomials, spanned by monomials with i.i.d. Gaussian coefficients, have only $$(2/\pi + o(1))\log{n}$$ expected real zeros in terms of the degree $$n$$. If the basis is given by the orthonormal polynomials associated with a compactly supported Borel measure on the real line, or associated with a Freud weight, then random linear combinations have $$n/\sqrt{3} + o(n)$$ expected real zeros. We prove that the same asymptotic relation holds for all random orthogonal polynomials on the real line associated with a large class of weights, and give local results on the expected number of real zeros. We also show that the counting measures of properly scaled zeros of these random polynomials converge weakly to either the Ullman distribution or the arcsine distribution. 

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