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Value Distributions of Perfect Nonlinear Functions Combinatorica (IF 1.1) Pub Date : 2023-09-29 Lukas Kölsch, Alexandr Polujan
In this paper, we study the value distributions of perfect nonlinear functions, i.e., we investigate the sizes of image and preimage sets. Using purely combinatorial tools, we develop a framework that deals with perfect nonlinear functions in the most general setting, generalizing several results that were achieved under specific constraints. For the particularly interesting elementary abelian case
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A Generalization of the Chevalley–Warning and Ax–Katz Theorems with a View Towards Combinatorial Number Theory Combinatorica (IF 1.1) Pub Date : 2023-09-29 David J. Grynkiewicz
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A Solution to Babai’s Problems on Digraphs with Non-diagonalizable Adjacency Matrix Combinatorica (IF 1.1) Pub Date : 2023-09-29 Yuxuan Li, Binzhou Xia, Sanming Zhou, Wenying Zhu
The fact that the adjacency matrix of every finite graph is diagonalizable plays a fundamental role in spectral graph theory. Since this fact does not hold in general for digraphs, it is natural to ask whether it holds for digraphs with certain level of symmetry. Interest in this question dates back to the early 1980 s, when P. J. Cameron asked for the existence of arc-transitive digraphs with non-diagonalizable
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Induced Subgraphs of Induced Subgraphs of Large Chromatic Number Combinatorica (IF 1.1) Pub Date : 2023-09-25 António Girão, Freddie Illingworth, Emil Powierski, Michael Savery, Alex Scott, Youri Tamitegama, Jane Tan
We prove that, for every graph F with at least one edge, there is a constant \(c_F\) such that there are graphs of arbitrarily large chromatic number and the same clique number as F in which every F-free induced subgraph has chromatic number at most \(c_F\). This generalises recent theorems of Briański, Davies and Walczak, and Carbonero, Hompe, Moore and Spirkl. Our results imply that for every \(r\geqslant
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Special cycles on the basic locus of unitary Shimura varieties at ramified primes Algebra Number Theory (IF 1.3) Pub Date : 2023-09-19 Yousheng Shi
We study special cycles on the basic locus of certain unitary Shimura varieties over the ramified primes and their local analogs on the corresponding Rapoport–Zink spaces. We study the support and compute the dimension of these cycles.
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Hybrid subconvexity bounds for twists of GL(3) × GL(2) L-functions Algebra Number Theory (IF 1.3) Pub Date : 2023-09-19 Bingrong Huang, Zhao Xu
We prove hybrid subconvexity bounds for GL (3) × GL (2) L-functions twisted by a primitive Dirichlet character modulo M (prime) in the M- and t-aspects. We also improve hybrid subconvexity bounds for twists of GL (3) L-functions in the M- and t-aspects.
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Separation of periods of quartic surfaces Algebra Number Theory (IF 1.3) Pub Date : 2023-09-19 Pierre Lairez, Emre Can Sertöz
We give a computable lower bound for the distance between two distinct periods of a given quartic surface defined over the algebraic numbers. The main ingredient is the determination of height bounds on components of the Noether–Lefschetz loci. This makes it possible to study the Diophantine properties of periods of quartic surfaces and to certify a part of the numerical computation of their Picard groups
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Global dimension of real-exponent polynomial rings Algebra Number Theory (IF 1.3) Pub Date : 2023-09-19 Nathan Geist, Ezra Miller
The ring R of real-exponent polynomials in n variables over any field has global dimension n + 1 and flat dimension n. In particular, the residue field k = R∕𝔪 of R modulo its maximal graded ideal 𝔪 has flat dimension n via a Koszul-like resolution. Projective and flat resolutions of all R-modules are constructed from this resolution of k . The same results hold when R is replaced by the monoid
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Differences between perfect powers: prime power gaps Algebra Number Theory (IF 1.3) Pub Date : 2023-09-19 Michael A. Bennett, Samir Siksek
We develop machinery to explicitly determine, in many instances, when the difference x2 − yn is divisible only by powers of a given fixed prime. This combines a wide variety of techniques from Diophantine approximation (bounds for linear forms in logarithms, both archimedean and nonarchimedean, lattice basis reduction, methods for solving Thue–Mahler and S-unit equations, and the primitive divisor
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On fake linear cycles inside Fermat varieties Algebra Number Theory (IF 1.3) Pub Date : 2023-09-19 Jorge Duque Franco, Roberto Villaflor Loyola
We introduce a new class of Hodge cycles with nonreduced associated Hodge loci; we call them fake linear cycles. We characterize them for all Fermat varieties and show that they exist only for degrees d = 3,4,6, where there are infinitely many in the space of Hodge cycles. These cycles are pathological in the sense that the Zariski tangent space of their associated Hodge locus is of maximal dimension
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Effective Results on the Size and Structure of Sumsets Combinatorica (IF 1.1) Pub Date : 2023-09-18 Andrew Granville, George Shakan, Aled Walker
Let \(A \subset {\mathbb {Z}}^d\) be a finite set. It is known that NA has a particular size (\(\vert NA\vert = P_A(N)\) for some \(P_A(X) \in {\mathbb {Q}}[X]\)) and structure (all of the lattice points in a cone other than certain exceptional sets), once N is larger than some threshold. In this article we give the first effective upper bounds for this threshold for arbitrary A. Such explicit results
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Integer Multiflows in Acyclic Planar Digraphs Combinatorica (IF 1.1) Pub Date : 2023-09-19 Guyslain Naves
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Polynomial Bounds for Chromatic Number. IV: A Near-polynomial Bound for Excluding the Five-vertex Path Combinatorica (IF 1.1) Pub Date : 2023-09-15 Alex Scott, Paul Seymour, Sophie Spirkl
A graph G is H-free if it has no induced subgraph isomorphic to H. We prove that a \(P_5\)-free graph with clique number \(\omega \ge 3\) has chromatic number at most \(\omega ^{\log _2(\omega )}\). The best previous result was an exponential upper bound \((5/27)3^{\omega }\), due to Esperet, Lemoine, Maffray, and Morel. A polynomial bound would imply that the celebrated Erdős-Hajnal conjecture holds
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Unipotent ℓ-blocks for simply connected p-adic groups Algebra Number Theory (IF 1.3) Pub Date : 2023-09-09 Thomas Lanard
Let F be a nonarchimedean local field and G the F-points of a connected simply connected reductive group over F. We study the unipotent ℓ-blocks of G, for ℓ≠p. To that end, we introduce the notion of (d,1)-series for finite reductive groups. These series form a partition of the irreducible representations and are defined using Harish-Chandra theory and d-Harish-Chandra theory. The ℓ-blocks are then
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Isotriviality, integral points, and primitive primes in orbits in characteristic p Algebra Number Theory (IF 1.3) Pub Date : 2023-09-09 Alexander Carney, Wade Hindes, Thomas J. Tucker
We prove a characteristic p version of a theorem of Silverman on integral points in orbits over number fields and establish a primitive prime divisor theorem for polynomials in this setting. In characteristic p, the Thue–Siegel–Dyson–Roth theorem is false, so the proof requires new techniques from those used by Silverman. The problem is largely that isotriviality can arise in subtle ways, and we define
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Operations in connective K-theory Algebra Number Theory (IF 1.3) Pub Date : 2023-09-09 Alexander Merkurjev, Alexander Vishik
We classify additive operations in connective K-theory with various torsion-free coefficients. We discover that the answer for the integral case requires understanding of the ℤ^ case. Moreover, although integral additive operations are topologically generated by Adams operations, these are not reduced to infinite linear combinations of the latter ones. We describe a topological basis for stable operations
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The structure of Frobenius kernels for automorphism group schemes Algebra Number Theory (IF 1.3) Pub Date : 2023-09-09 Stefan Schröer, Nikolaos Tziolas
We establish structure results for Frobenius kernels of automorphism group schemes for surfaces of general type in positive characteristic. It turns out that there are surprisingly few possibilities. This relies on properties of the famous Witt algebra, which is a simple Lie algebra without finite-dimensional counterpart over the complex numbers, together with its twisted forms. The result actually
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Prague Dimension of Random Graphs Combinatorica (IF 1.1) Pub Date : 2023-09-06 He Guo, Kalen Patton, Lutz Warnke
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On the first nontrivial strand of syzygies of projective schemes and condition ND(ℓ) Algebra Number Theory (IF 1.3) Pub Date : 2023-08-29 Jeaman Ahn, Kangjin Han, Sijong Kwak
Let X ⊂ ℙn+e be any n-dimensional closed subscheme. We are mainly interested in two notions related to syzygies: one is the property Nd,p(d ≥ 2,p ≥ 1), which means that X is d-regular up to p-th step in the minimal free resolution and the other is a new notion ND (ℓ) which generalizes the classical “being nondegenerate” to the condition that requires a general finite linear section not to be contained
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Spectral reciprocity via integral representations Algebra Number Theory (IF 1.3) Pub Date : 2023-08-29 Ramon M. Nunes
We prove a spectral reciprocity formula for automorphic forms on GL (2) over a number field that is reminiscent of one found by Blomer and Khan. Our approach uses period representations of L-functions and the language of automorphic representations.
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Quadratic points on intersections of two quadrics Algebra Number Theory (IF 1.3) Pub Date : 2023-08-29 Brendan Creutz, Bianca Viray
We prove that a smooth complete intersection of two quadrics of dimension at least 2 over a number field has index dividing 2, i.e., that it possesses a rational 0-cycle of degree 2.
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A p-adic Simpson correspondence for rigid analytic varieties Algebra Number Theory (IF 1.3) Pub Date : 2023-08-29 Yupeng Wang
We establish a p-adic Simpson correspondence in the spirit of Liu and Zhu for rigid analytic varieties X over ℂp with a liftable good reduction by constructing a new period sheaf on X proét. To do so, we use the theory of cotangent complexes described by Beilinson and Bhatt. Then we give an integral decompletion theorem and complete the proof by local calculations. Our construction is compatible with
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On moment map and bigness of tangent bundles of G-varieties Algebra Number Theory (IF 1.3) Pub Date : 2023-08-29 Jie Liu
Let G be a connected algebraic group and let X be a smooth projective G-variety. We prove a sufficient criterion to determine the bigness of the tangent bundle TX using the moment map ΦXG : T∗X → 𝔤∗. As an application, the bigness of the tangent bundles of certain quasihomogeneous varieties are verified, including symmetric varieties, horospherical varieties and equivariant compactifications of commutative
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Rank 1 perturbations in random matrix theory — A review of exact results Random Matrices Theory Appl. (IF 0.9) Pub Date : 2023-08-19 Peter J. Forrester
A number of random matrix ensembles permitting exact determination of their eigenvalue and eigenvector statistics maintain this property under a rank 1 perturbation. Considered in this review are the additive rank 1 perturbation of the Hermitian Gaussian ensembles, the multiplicative rank 1 perturbation of the Wishart ensembles, and rank 1 perturbations of Hermitian and unitary matrices giving rise
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The Strong Nine Dragon Tree Conjecture is True for $$d \le k + 1$$ Combinatorica (IF 1.1) Pub Date : 2023-08-21 Sebastian Mies, Benjamin Moore
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On Bounded Degree Graphs with Large Size-Ramsey Numbers Combinatorica (IF 1.1) Pub Date : 2023-08-21 Konstantin Tikhomirov
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A Characterization of Edge-Ordered Graphs with Almost Linear Extremal Functions Combinatorica (IF 1.1) Pub Date : 2023-08-18 Gaurav Kucheriya, Gábor Tardos
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Separating Polynomial $$\chi $$ -Boundedness from $$\chi $$ -Boundedness Combinatorica (IF 1.1) Pub Date : 2023-08-09 Marcin Briański, James Davies, Bartosz Walczak
Extending the idea from the recent paper by Carbonero, Hompe, Moore, and Spirkl, for every function \(f:\mathbb {N}\rightarrow \mathbb {N}\cup \{\infty \}\) with \(f(1)=1\) and \(f(n)\geqslant \left( {\begin{array}{c}3n+1\\ 3\end{array}}\right) \), we construct a hereditary class of graphs \({\mathcal {G}}\) such that the maximum chromatic number of a graph in \({\mathcal {G}}\) with clique number
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Stability Through Non-Shadows Combinatorica (IF 1.1) Pub Date : 2023-08-04 Jun Gao, Hong Liu, Zixiang Xu
We study families \({\mathcal {F}}\subseteq 2^{[n]}\) with restricted intersections and prove a conjecture of Snevily in a stronger form for large n. We also obtain stability results for Kleitman’s isodiametric inequality and families with bounded set-wise differences. Our proofs introduce a new twist to the classical linear algebra method, harnessing the non-shadows of \({\mathcal {F}}\), which may
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Weak Saturation of Multipartite Hypergraphs Combinatorica (IF 1.1) Pub Date : 2023-07-27 Denys Bulavka, Martin Tancer, Mykhaylo Tyomkyn
Given q-uniform hypergraphs (q-graphs) F, G and H, where G is a spanning subgraph of F, G is called weakly H-saturated in F if the edges in \(E(F)\setminus E(G)\) admit an ordering \(e_1,\ldots , e_k\) so that for all \(i\in [k]\) the hypergraph \(G\cup \{e_1,\ldots ,e_i\}\) contains an isomorphic copy of H which in turn contains the edge \(e_i\). The weak saturation number of H in F is the smallest
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A Necessary and Sufficient Condition for $$(2d-2)$$ -Transversals in $$\mathbb {R}^{2d}$$ Combinatorica (IF 1.1) Pub Date : 2023-07-27 Daniel McGinnis
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A Factor Matching of Optimal Tail Between Poisson Processes Combinatorica (IF 1.1) Pub Date : 2023-07-25 Ádám Timár
Consider two independent Poisson point processes of unit intensity in the Euclidean space of dimension d at least 3. We construct a perfect matching between the two point sets that is a factor (i.e., a measurable function of the point configurations that commutes with translations), and with the property that the distance between two matched configuration points has a tail distribution that decays
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Spherical Two-Distance Sets and Eigenvalues of Signed Graphs Combinatorica (IF 1.1) Pub Date : 2023-07-21 Zilin Jiang, Jonathan Tidor, Yuan Yao, Shengtong Zhang, Yufei Zhao
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Limiting spectral distribution of stochastic block model Random Matrices Theory Appl. (IF 0.9) Pub Date : 2023-07-18 Giap Van Su, May-Ru Chen, Mei-Hui Guo, Hao-Wei Huang
The stochastic block model (SBM) is an extension of the Erdős–Rényi graph and has applications in numerous fields, such as data analysis, recovering community structure in graph data and social networks. In this paper, we consider the normal central SBM adjacency matrix with K communities of arbitrary sizes. We derive an explicit formula for the limiting empirical spectral density function when the
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Pure Pairs. V. Excluding Some Long Subdivision Combinatorica (IF 1.1) Pub Date : 2023-06-16 Alex Scott, Paul Seymour, Sophie Spirkl
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Matrix deviation inequality for ℓp-norm Random Matrices Theory Appl. (IF 0.9) Pub Date : 2023-06-15 Yuan-Chung Sheu, Te-Chun Wang
Motivated by the general matrix deviation inequality for i.i.d. ensemble Gaussian matrix [R. Vershynin, High-Dimensional Probability: An Introduction with Applications in Data Science, Cambridge Series in Statistical and Probabilistic Mathematics (Cambridge University Press, 2018), doi:10.1017/9781108231596 of Theorem 11.1.5], we show that this property holds for the ℓp-norm with 1≤p<∞ and i.i.d. ensemble
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A combinatorial proof of a sumset conjecture of Furstenberg Combinatorica (IF 1.1) Pub Date : 2023-06-14 Daniel Glasscock, Joel Moreira, Florian K. Richter
We give a new proof of a sumset conjecture of Furstenberg that was first proved by Hochman and Shmerkin in 2012: if \(\log r/\log s\) is irrational and X and Y are \(\times r\) - and \(\times s\)-invariant subsets of [0, 1], respectively, then \(\dim _{\text {H}}(X + Y ) = \min (1, \dim _{\text {H}}X + \dim _{\text {H}}Y )\). Our main result yields information on the size of the sumset \(\lambda X
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Maximal 3-Wise Intersecting Families Combinatorica (IF 1.1) Pub Date : 2023-06-13 József Balogh, Ce Chen, Kevin Hendrey, Ben Lund, Haoran Luo, Casey Tompkins, Tuan Tran
A family \({\mathcal {F}}\) on ground set \([n]:=\{1,2,\ldots , n\}\) is maximal k-wise intersecting if every collection of at most k sets in \({\mathcal {F}}\) has non-empty intersection, and no other set can be added to \({\mathcal {F}}\) while maintaining this property. In 1974, Erdős and Kleitman asked for the minimum size of a maximal k-wise intersecting family. We answer their question for \(k=3\)
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The Asymptotic Number of Score Sequences Combinatorica (IF 1.1) Pub Date : 2023-06-13 Brett Kolesnik
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A Large Family of Maximum Scattered Linear Sets of $${{\,\mathrm{{PG}}\,}}(1,q^n)$$ and Their Associated MRD Codes Combinatorica (IF 1.1) Pub Date : 2023-06-13 G. Longobardi, Giuseppe Marino, Rocco Trombetti, Yue Zhou
Linear sets in projective spaces over finite fields were introduced by Lunardon (Geom Dedic 75(3):245–261, 1999) and they play a central role in the study of blocking sets, semifields, rank-metric codes, etc. A linear set with the largest possible cardinality and rank is called maximum scattered. Despite two decades of study, there are only a limited number of maximum scattered linear sets of a line
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Enclosing Depth and Other Depth Measures Combinatorica (IF 1.1) Pub Date : 2023-06-13 Patrick Schnider
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A Recursive Theta Body for Hypergraphs Combinatorica (IF 1.1) Pub Date : 2023-06-13 Davi Castro-Silva, Fernando Mário de Oliveira Filho, Lucas Slot, Frank Vallentin
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An Improved Bound for the Linear Arboricity Conjecture Combinatorica (IF 1.1) Pub Date : 2023-06-13 Richard Lang, Luke Postle
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$$\Gamma $$ -Graphic Delta-Matroids and Their Applications Combinatorica (IF 1.1) Pub Date : 2023-06-13 Donggyu Kim, Duksang Lee, Sang-il Oum
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A Unifying Framework for the $$\nu $$ -Tamari Lattice and Principal Order Ideals in Young’s Lattice Combinatorica (IF 1.1) Pub Date : 2023-06-13 Matias von Bell, Rafael S. González D’León, Francisco A. Mayorga Cetina, Martha Yip
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The Chromatic Number of the Product of 5-Chromatic Graphs can be 4 Combinatorica (IF 1.1) Pub Date : 2023-06-12 Claude Tardif
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Central limit theorem for linear spectral statistics of block-Wigner-type matrices Random Matrices Theory Appl. (IF 0.9) Pub Date : 2023-06-07 Zhenggang Wang, Jianfeng Yao
Motivated by the stochastic block model, we investigate a class of Wigner-type matrices with certain block structures and establish a CLT for the corresponding linear spectral statistics (LSS) via the large-deviation bounds from local law and the cumulant expansion formula. We apply the results to the stochastic block model. Specifically, a class of renormalized adjacency matrices will be block-Wigner-type
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Strongly Regular Graphs Satisfying the 4-Vertex Condition Combinatorica (IF 1.1) Pub Date : 2023-06-05 A. E. Brouwer, F. Ihringer, W. M. Kantor
We survey the area of strongly regular graphs satisfying the 4-vertex condition and find several new families. We describe a switching operation on collinearity graphs of polar spaces that produces cospectral graphs. The obtained graphs satisfy the 4-vertex condition if the original graph belongs to a symplectic polar space.
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Growth in Sumsets of Higher Convex Functions Combinatorica (IF 1.1) Pub Date : 2023-06-05 Peter J. Bradshaw
The main results of this paper concern growth in sums of a k-convex function f. Firstly, we streamline the proof (from Hanson et al. (Combinatorica 42:71–85, 2020)) of a growth result for f(A) where A has small additive doubling, and improve the bound by removing logarithmic factors. The result yields an optimal bound for $$\begin{aligned} |2^k f(A) - (2^k-1)f(A)|. \end{aligned}$$ We also generalise
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Leray Numbers of Tolerance Complexes Combinatorica (IF 1.1) Pub Date : 2023-06-05 Minki Kim, Alan Lew
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The Number of Tangencies Between Two Families of Curves Combinatorica (IF 1.1) Pub Date : 2023-06-05 Balázs Keszegh, Dömötör Pálvölgyi
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A Local Version of Katona’s Intersecting Shadow Theorem Combinatorica (IF 1.1) Pub Date : 2023-06-05 Marcelo Sales, Bjarne Schülke
Katona’s intersection theorem states that every intersecting family \(\mathcal {F}\subseteq [n]^{(k)}\) satisfies \(\vert \partial \mathcal {F}\vert \geqslant \vert \mathcal {F}\vert \), where \(\partial \mathcal {F}=\{F\setminus \{x\}:x\in F\in \mathcal {F}\}\) is the shadow of \(\mathcal {F}\). Frankl conjectured that for \(n>2k\) and every intersecting family \(\mathcal {F}\subseteq [n]^{(k)}\)
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Bounded Cutoff Window for the Non-backtracking Random Walk on Ramanujan Graphs Combinatorica (IF 1.1) Pub Date : 2023-06-05 Evita Nestoridi, Peter Sarnak
We prove that the non-backtracking random walk on Ramanujan graphs with large girth exhibits the fastest possible cutoff with a bounded window.
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Counting abelian varieties over finite fields via Frobenius densities Algebra Number Theory (IF 1.3) Pub Date : 2023-05-30 Jeffrey D. Achter, S. Ali Altuğ, Luis Garcia, Julia Gordon
Let [X,λ] be a principally polarized abelian variety over a finite field with commutative endomorphism ring; further suppose that either X is ordinary or the field is prime. Motivated by an equidistribution heuristic, we introduce a factor νv([X,λ]) for each place v of ℚ, and show that the product of these factors essentially computes the size of the isogeny class of [X,λ]. The derivation of this mass
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The log product formula Algebra Number Theory (IF 1.3) Pub Date : 2023-05-30 Leo Herr
Let V,W be a pair of smooth varieties. We want to compare curve counts on V × W with those on V and W. The product formula in Gromov–Witten theory compares the virtual fundamental classes of stable maps to a product M¯g,n(V × W) to the product of stable maps M¯g,n(V ) ×M¯g,n(W). We prove the analogous theorem for log stable maps to log smooth varieties V,W. This extends results of Y.P. Lee and F. Qu
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Intersecting geodesics on the modular surface Algebra Number Theory (IF 1.3) Pub Date : 2023-05-30 Junehyuk Jung, Naser Talebizadeh Sardari
We introduce the modular intersection kernel, and we use it to study how geodesics intersect on the full modular surface 𝕏 = PSL 2(ℤ)∖ℍ. Let Cd be the union of closed geodesics with discriminant d and let β ⊂ 𝕏 be a compact geodesic segment. As an application of Duke’s theorem to the modular intersection kernel, we prove that {(p,𝜃p) : p ∈ β ∩ Cd} becomes equidistributed with respect to sin
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On Héthelyi–Külshammer’s conjecture for principal blocks Algebra Number Theory (IF 1.3) Pub Date : 2023-05-26 Ngoc Hung Nguyen, A. A. Schaeffer Fry
We prove that the number of irreducible ordinary characters in the principal p-block of a finite group G of order divisible by p is always at least 2p − 1. This confirms a conjecture of Héthelyi and Külshammer (2000) for principal blocks and provides an affirmative answer to Brauer’s problem 21 (1963) for principal blocks of bounded defect. Our proof relies on recent works of Maróti (2016) and Malle
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Shintani–Barnes cocycles and values of the zeta functions of algebraic number fields Algebra Number Theory (IF 1.3) Pub Date : 2023-05-26 Hohto Bekki
We construct a new Eisenstein cocycle, called the Shintani–Barnes cocycle, which specializes in a uniform way to the values of the zeta functions of general number fields at positive integers. Our basic strategy is to generalize the construction of the Eisenstein cocycle presented in the work of Vlasenko and Zagier by using some recent techniques developed by Bannai, Hagihara, Yamada, and Yamamoto
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On the commuting probability of p-elements in a finite group Algebra Number Theory (IF 1.3) Pub Date : 2023-05-26 Timothy C. Burness, Robert Guralnick, Alexander Moretó, Gabriel Navarro
Let G be a finite group, let p be a prime and let Pr p(G) be the probability that two random p-elements of G commute. In this paper we prove that Pr p(G) > (p2 + p − 1)∕p3 if and only if G has a normal and abelian Sylow p-subgroup, which generalizes previous results on the widely studied commuting probability of a finite group. This bound is best possible in the sense that for each prime p there
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Correction to the article Height bounds and the Siegel property Algebra Number Theory (IF 1.3) Pub Date : 2023-05-26 Martin Orr, Christian Schnell
This is a correction to the paper “Height bounds and the Siegel property” (Algebra Number Theory 12:2 (2018), 455–478). We correct an error in the proof of Theorem 4.1. Theorem 4.1 as stated in the original paper is correct, but the correction affects additional information about the theorem which is important for applications.