-
Note on the Theorem of Balog, Szemerédi, and Gowers Combinatorica (IF 1.1) Pub Date : 2024-04-23 Christian Reiher, Tomasz Schoen
We prove that every additive set A with energy \(E(A)\ge |A|^3/K\) has a subset \(A'\subseteq A\) of size \(|A'|\ge (1-\varepsilon )K^{-1/2}|A|\) such that \(|A'-A'|\le O_\varepsilon (K^{4}|A'|)\). This is, essentially, the largest structured set one can get in the Balog–Szemerédi–Gowers theorem.
-
A Lower Bound for Essential Covers of the Cube Combinatorica (IF 1.1) Pub Date : 2024-04-23 Gal Yehuda, Amir Yehudayoff
-
Turán Density of Long Tight Cycle Minus One Hyperedge Combinatorica (IF 1.1) Pub Date : 2024-04-17 József Balogh, Haoran Luo
-
Bounding the Chromatic Number of Dense Digraphs by Arc Neighborhoods Combinatorica (IF 1.1) Pub Date : 2024-04-17 Felix Klingelhoefer, Alantha Newman
The chromatic number of a directed graph is the minimum number of induced acyclic subdigraphs that cover its vertex set, and accordingly, the chromatic number of a tournament is the minimum number of transitive subtournaments that cover its vertex set. The neighborhood of an arc uv in a tournament T is the set of vertices that form a directed triangle with arc uv. We show that if the neighborhood of
-
Solution to a Problem of Grünbaum on the Edge Density of 4-Critical Planar Graphs Combinatorica (IF 1.1) Pub Date : 2024-04-17 Zdeněk Dvořák, Carl Feghali
-
On the ordinary Hecke orbit conjecture Algebra Number Theory (IF 1.3) Pub Date : 2024-04-16 Pol van Hoften
We prove the ordinary Hecke orbit conjecture for Shimura varieties of Hodge type at primes of good reduction. We make use of the global Serre–Tate coordinates of Chai as well as recent results of D’Addezio about the monodromy groups of isocrystals. The new ingredients in this paper are a general monodromy theorem for Hecke-stable subvarieties for Shimura varieties of Hodge type, and a rigidity result
-
Locally analytic vector bundles on the Fargues–Fontaine curve Algebra Number Theory (IF 1.3) Pub Date : 2024-04-16 Gal Porat
We develop a version of Sen theory for equivariant vector bundles on the Fargues–Fontaine curve. We show that every equivariant vector bundle canonically descends to a locally analytic vector bundle. A comparison with the theory of (φ,Γ)-modules in the cyclotomic case then recovers the Cherbonnier–Colmez decompletion theorem. Next, we focus on the subcategory of de Rham locally analytic vector bundles
-
Multiplicity structure of the arc space of a fat point Algebra Number Theory (IF 1.3) Pub Date : 2024-04-16 Rida Ait El Manssour, Gleb Pogudin
The equation xm = 0 defines a fat point on a line. The algebra of regular functions on the arc space of this scheme is the quotient of k[x,x′,x(2),… ] by all differential consequences of xm = 0. This infinite-dimensional algebra admits a natural filtration by finite-dimensional algebras corresponding to the truncations of arcs. We show that the generating series for their dimensions equals m∕(1 −
-
Theta correspondence and simple factors in global Arthur parameters Algebra Number Theory (IF 1.3) Pub Date : 2024-04-16 Chenyan Wu
By using results on poles of L-functions and theta correspondence, we give a bound on b for (χ,b)-factors of the global Arthur parameter of a cuspidal automorphic representation π of a classical group or a metaplectic group where χ is a conjugate self-dual automorphic character and b is an integer which is the dimension of an irreducible representation of SL 2(ℂ). We derive a more precise relation
-
Equidistribution theorems for holomorphic Siegel cusp forms of general degree: the level aspect Algebra Number Theory (IF 1.3) Pub Date : 2024-04-16 Henry H. Kim, Satoshi Wakatsuki, Takuya Yamauchi
This paper is an extension of Kim et al. (2020a), and we prove equidistribution theorems for families of holomorphic Siegel cusp forms of general degree in the level aspect. Our main contribution is to estimate unipotent contributions for general degree in the geometric side of Arthur’s invariant trace formula in terms of Shintani zeta functions in a uniform way. Several applications, including the
-
Small Subgraphs with Large Average Degree Combinatorica (IF 1.1) Pub Date : 2024-04-15 Oliver Janzer, Benny Sudakov, István Tomon
-
A Hypergraph Analog of Dirac’s Theorem for Long Cycles in 2-Connected Graphs Combinatorica (IF 1.1) Pub Date : 2024-04-15 Alexandr Kostochka, Ruth Luo, Grace McCourt
-
Globally Linked Pairs of Vertices in Generic Frameworks Combinatorica (IF 1.1) Pub Date : 2024-04-08 Tibor Jordán, Soma Villányi
-
Induced Subgraphs and Tree Decompositions VIII: Excluding a Forest in (Theta, Prism)-Free Graphs Combinatorica (IF 1.1) Pub Date : 2024-04-08 Tara Abrishami, Bogdan Alecu, Maria Chudnovsky, Sepehr Hajebi, Sophie Spirkl
-
Bounded-Diameter Tree-Decompositions Combinatorica (IF 1.1) Pub Date : 2024-04-08 Eli Berger, Paul Seymour
When does a graph admit a tree-decomposition in which every bag has small diameter? For finite graphs, this is a property of interest in algorithmic graph theory, where it is called having bounded “tree-length”. We will show that this is equivalent to being “boundedly quasi-isometric to a tree”, which for infinite graphs is a much-studied property from metric geometry. One object of this paper is to
-
The Number of Topological Types of Trees Combinatorica (IF 1.1) Pub Date : 2024-04-04
Abstract Two graphs are of the same topological type if they can be mutually embedded into each other topologically. We show that there are exactly \(\aleph _1\) distinct topological types of countable trees. In general, for any infinite cardinal \(\kappa \) there are exactly \(\kappa ^+\) distinct topological types of trees of size \(\kappa \) . This solves a problem of van der Holst from 2005.
-
Lattice Path Matroids and Quotients Combinatorica (IF 1.1) Pub Date : 2024-04-04 Carolina Benedetti-Velásquez, Kolja Knauer
We characterize the quotients among lattice path matroids (LPMs) in terms of their diagrams. This characterization allows us to show that ordering LPMs by quotients yields a graded poset, whose rank polynomial has the Narayana numbers as coefficients. Furthermore, we study full lattice path flag matroids and show that—contrary to arbitrary positroid flag matroids—they correspond to points in the nonnegative
-
Flashes and Rainbows in Tournaments Combinatorica (IF 1.1) Pub Date : 2024-04-04 António Girão, Freddie Illingworth, Lukas Michel, Michael Savery, Alex Scott
Colour the edges of the complete graph with vertex set \({\{1, 2, \dotsc , n\}}\) with an arbitrary number of colours. What is the smallest integer f(l, k) such that if \(n > f(l,k)\) then there must exist a monotone monochromatic path of length l or a monotone rainbow path of length k? Lefmann, Rödl, and Thomas conjectured in 1992 that \(f(l, k) = l^{k - 1}\) and proved this for \(l \ge (3 k)^{2 k}\)
-
Upper Tail Behavior of the Number of Triangles in Random Graphs with Constant Average Degree Combinatorica (IF 1.1) Pub Date : 2024-04-04 Shirshendu Ganguly, Ella Hiesmayr, Kyeongsik Nam
Let N be the number of triangles in an Erdős–Rényi graph \({\mathcal {G}}(n,p)\) on n vertices with edge density \(p=d/n,\) where \(d>0\) is a fixed constant. It is well known that N weakly converges to the Poisson distribution with mean \({d^3}/{6}\) as \(n\rightarrow \infty \). We address the upper tail problem for N, namely, we investigate how fast k must grow, so that \({\mathbb {P}}(N\ge k)\)
-
Isoperimetric Inequalities and Supercritical Percolation on High-Dimensional Graphs Combinatorica (IF 1.1) Pub Date : 2024-04-04
Abstract It is known that many different types of finite random subgraph models undergo quantitatively similar phase transitions around their percolation thresholds, and the proofs of these results rely on isoperimetric properties of the underlying host graph. Recently, the authors showed that such a phase transition occurs in a large class of regular high-dimensional product graphs, generalising a
-
Boolean Function Analysis on High-Dimensional Expanders Combinatorica (IF 1.1) Pub Date : 2024-03-18 Yotam Dikstein, Irit Dinur, Yuval Filmus, Prahladh Harsha
We initiate the study of Boolean function analysis on high-dimensional expanders. We give a random-walk based definition of high-dimensional expansion, which coincides with the earlier definition in terms of two-sided link expanders. Using this definition, we describe an analog of the Fourier expansion and the Fourier levels of the Boolean hypercube for simplicial complexes. Our analog is a decomposition
-
Ramsey Problems for Monotone Paths in Graphs and Hypergraphs Combinatorica (IF 1.1) Pub Date : 2024-02-28 Lior Gishboliner, Zhihan Jin, Benny Sudakov
-
Fundamental exact sequence for the pro-étale fundamental group Algebra Number Theory (IF 1.3) Pub Date : 2024-02-26 Marcin Lara
The pro-étale fundamental group of a scheme, introduced by Bhatt and Scholze, generalizes formerly known fundamental groups — the usual étale fundamental group π1 ét defined in SGA1 and the more general π1SGA3 . It controls local systems in the pro-étale topology and leads to an interesting class of “geometric coverings” of schemes, generalizing finite étale coverings. We prove exactness of the fundamental
-
Infinitesimal dilogarithm on curves over truncated polynomial rings Algebra Number Theory (IF 1.3) Pub Date : 2024-02-26 Sinan Ünver
We construct infinitesimal invariants of thickened one dimensional cycles in three dimensional space, which are the simplest cycles that are not in the Milnor range. This generalizes Park’s work on the regulators of additive cycles. The construction also allows us to prove the infinitesimal version of the strong reciprocity conjecture for thickenings of all orders. Classical analogs of our invariants
-
Wide moments of L-functions I : Twists by class group characters of imaginary quadratic fields Algebra Number Theory (IF 1.3) Pub Date : 2024-02-26 Asbjørn Christian Nordentoft
We calculate certain “wide moments” of central values of Rankin–Selberg L-functions L(π ⊗Ω, 1 2) where π is a cuspidal automorphic representation of GL 2 over ℚ and Ω is a Hecke character (of conductor 1) of an imaginary quadratic field. This moment calculation is applied to obtain “weak simultaneous” nonvanishing results, which are nonvanishing results for different Rankin–Selberg L-functions where
-
On Ozaki’s theorem realizing prescribed p-groups as p-class tower groups Algebra Number Theory (IF 1.3) Pub Date : 2024-02-26 Farshid Hajir, Christian Maire, Ravi Ramakrishna
We give a streamlined and effective proof of Ozaki’s theorem that any finite p-group Γ is the Galois group of the p-Hilbert class field tower of some number field F . Our work is inspired by Ozaki’s and applies in broader circumstances. While his theorem is in the totally complex setting, we obtain the result in any mixed signature setting for which there exists a number field k 0 with class number
-
Supersolvable descent for rational points Algebra Number Theory (IF 1.3) Pub Date : 2024-02-26 Yonatan Harpaz, Olivier Wittenberg
We construct an analogue of the classical descent theory of Colliot-Thélène and Sansuc in which algebraic tori are replaced with finite supersolvable groups. As an application, we show that rational points are dense in the Brauer–Manin set for smooth compactifications of certain quotients of homogeneous spaces by finite supersolvable groups. For suitably chosen homogeneous spaces, this implies the
-
On Kato and Kuzumaki’s properties for the Milnor K2 of function fields of p-adic curves Algebra Number Theory (IF 1.3) Pub Date : 2024-02-26 Diego Izquierdo, Giancarlo Lucchini Arteche
Let K be the function field of a curve C over a p-adic field k. We prove that, for each n,d ≥ 1 and for each hypersurface Z in ℙKn of degree d with d2 ≤ n, the second Milnor K-theory group of K is spanned by the images of the norms coming from finite extensions L of K over which Z has a rational point. When the curve C has a point in the maximal unramified extension of k, we generalize this result
-
The Ungar Games Combinatorica (IF 1.1) Pub Date : 2024-02-21 Colin Defant, Noah Kravitz, Nathan Williams
-
-
Finite size corrections for real eigenvalues of the elliptic Ginibre matrices Random Matrices Theory Appl. (IF 0.9) Pub Date : 2024-02-20 Sung-Soo Byun, Yong-Woo Lee
In this paper, we consider the elliptic Ginibre matrices in the orthogonal symmetry class that interpolates between the real Ginibre ensemble and the Gaussian orthogonal ensemble. We obtain the finite size corrections of the real eigenvalue densities in both the global and edge scaling regimes, as well as in both the strong and weak non-Hermiticity regimes. Our results extend and provide the rate of
-
Quotients of admissible formal schemes and adic spaces by finite groups Algebra Number Theory (IF 1.3) Pub Date : 2024-02-16 Bogdan Zavyalov
We give a self-contained treatment of finite group quotients of admissible (formal) schemes and adic spaces that are locally topologically finite type over a locally strongly noetherian adic space.
-
Subconvexity bound for GL(3) × GL(2) L-functions : Hybrid level aspect Algebra Number Theory (IF 1.3) Pub Date : 2024-02-16 Sumit Kumar, Ritabrata Munshi, Saurabh Kumar Singh
Let F be a GL (3) Hecke–Maass cusp form of prime level P1 and let f be a GL (2) Hecke–Maass cuspform of prime level P2. We will prove a subconvex bound for the GL (3) × GL (2) Rankin–Selberg L-function L(s,F × f) in the level aspect for certain ranges of the parameters P1 and P2.
-
A categorical Künneth formula for constructible Weil sheaves Algebra Number Theory (IF 1.3) Pub Date : 2024-02-16 Tamir Hemo, Timo Richarz, Jakob Scholbach
We prove a Künneth-type equivalence of derived categories of lisse and constructible Weil sheaves on schemes in characteristic p > 0 for various coefficients, including finite discrete rings, algebraic field extensions E ⊃ ℚℓ, ℓ≠p, and their rings of integers 𝒪E. We also consider a variant for ind-constructible sheaves which applies to the cohomology of moduli stacks of shtukas over global function
-
Generalized Igusa functions and ideal growth in nilpotent Lie rings Algebra Number Theory (IF 1.3) Pub Date : 2024-02-16 Angela Carnevale, Michael M. Schein, Christopher Voll
We introduce a new class of combinatorially defined rational functions and apply them to deduce explicit formulae for local ideal zeta functions associated to the members of a large class of nilpotent Lie rings which contains the free class-2-nilpotent Lie rings and is stable under direct products. Our results unify and generalize a substantial number of previous computations. We show that the new
-
On Tamagawa numbers of CM tori Algebra Number Theory (IF 1.3) Pub Date : 2024-02-16 Pei-Xin Liang, Yasuhiro Oki, Hsin-Yi Yang, Chia-Fu Yu
We investigate the problem of computing Tamagawa numbers of CM tori. This problem arises naturally from the problem of counting polarized abelian varieties with commutative endomorphism algebras over finite fields, and polarized CM abelian varieties and components of unitary Shimura varieties in the works of Achter, Altug, Garcia and Gordon and of Guo, Sheu and Yu, respectively. We make a systematic
-
Decidability via the tilting correspondence Algebra Number Theory (IF 1.3) Pub Date : 2024-02-06 Konstantinos Kartas
We prove a relative decidability result for perfectoid fields. This applies to show that the fields ℚp(p1∕p∞ ) and ℚp(ζp∞) are (existentially) decidable relative to the perfect hull of 𝔽p((t)) and ℚpab is (existentially) decidable relative to the perfect hull of 𝔽¯p((t)). We also prove some unconditional decidability results in mixed characteristic via reduction to characteristic p.
-
Differentially large fields Algebra Number Theory (IF 1.3) Pub Date : 2024-02-06 Omar León Sánchez, Marcus Tressl
We introduce the notion of differential largeness for fields equipped with several commuting derivations (as an analogue to largeness of fields). We lay out the foundations of this new class of “tame” differential fields. We state several characterizations and exhibit plenty of examples and applications. Our results strongly indicate that differentially large fields will play a key role in differential
-
p-groups, p-rank, and semistable reduction of coverings of curves Algebra Number Theory (IF 1.3) Pub Date : 2024-02-06 Yu Yang
We prove various explicit formulas concerning p-rank of p-coverings of pointed semistable curves over discrete valuation rings. In particular, we obtain a full generalization of Raynaud’s formula for p-rank of fibers over nonmarked smooth closed points in the case of arbitrary closed points. As an application, for abelian p-coverings, we give an affirmative answer to an open problem concerning boundedness
-
A deterministic algorithm for Harder–Narasimhan filtrations for representations of acyclic quivers Algebra Number Theory (IF 1.3) Pub Date : 2024-02-06 Chi-Yu Cheng
Let M be a representation of an acyclic quiver Q over an infinite field k. We establish a deterministic algorithm for computing the Harder–Narasimhan filtration of M. The algorithm is polynomial in the dimensions of M, the weights that induce the Harder–Narasimhan filtration of M, and the number of paths in Q. As a direct application, we also show that when k is algebraically closed and when M is unstable
-
Sur les espaces homogènes de Borovoi–Kunyavskii Algebra Number Theory (IF 1.3) Pub Date : 2024-02-06 Mạnh Linh Nguyễn
Nous établissons le principe de Hasse et l’approximation faible pour certains espaces homogènes de SL m à stabilisateur géométrique nilpotent de classe 2, construits par Borovoi et Kunyavskii. Ces espaces homogènes vérifient donc une conjecture de Colliot-Thélène concernant l’obstruction de Brauer–Manin pour les variétés géométriquement rationnellement connexes. We establish the Hasse principle and
-
Partial sums of typical multiplicative functions over short moving intervals Algebra Number Theory (IF 1.3) Pub Date : 2024-02-06 Mayank Pandey, Victor Y. Wang, Max Wenqiang Xu
We prove that the k-th positive integer moment of partial sums of Steinhaus random multiplicative functions over the interval (x,x + H] matches the corresponding Gaussian moment, as long as H ≪ x∕(log x)2k2+2+o(1) and H tends to infinity with x. We show that properly normalized partial sums of typical multiplicative functions arising from realizations of random multiplicative functions have Gaussian
-
An Upper Bound for the Height of a Tree with a Given Eigenvalue Combinatorica (IF 1.1) Pub Date : 2024-02-02 Artūras Dubickas
In this paper we prove that every totally real algebraic integer \(\lambda \) of degree \(d \ge 2\) occurs as an eigenvalue of some tree of height at most \(d(d+1)/2+3\). In order to prove this, for a given algebraic number \(\alpha \ne 0\), we investigate an additive semigroup that contains zero and is closed under the map \(x \mapsto \alpha /(1-x)\) for \(x \ne 1\). The problem of finding the smallest
-
Asymptotic cyclic-conditional freeness of random matrices Random Matrices Theory Appl. (IF 0.9) Pub Date : 2024-01-31 Guillaume Cébron, Nicolas Gilliers
Voiculescu’s freeness emerges when computing the asymptotic spectra of polynomials on N×N random matrices with eigenspaces in generic positions: they are randomly rotated with a uniform unitary random matrix UN. In this paper, we elaborate on the previous result by proposing a random matrix model, which we name the Vortex model, where UN has the law of a uniform unitary random matrix conditioned to
-
The distribution of sample mean-variance portfolio weights Random Matrices Theory Appl. (IF 0.9) Pub Date : 2024-01-31 Raymond Kan, Nathan Lassance, Xiaolu Wang
We present a simple stochastic representation for the joint distribution of sample estimates of three scalar parameters and two vectors of portfolio weights that characterize the minimum-variance frontier. This stochastic representation is useful for sampling observations efficiently, deriving moments in closed-form, and studying the distribution and performance of many portfolio strategies that are
-
Asymptotics of the determinant of the modified Bessel functions and the second Painlevé equation Random Matrices Theory Appl. (IF 0.9) Pub Date : 2024-01-31 Yu Chen, Shuai-Xia Xu, Yu-Qiu Zhao
In the paper, we consider the extended Gross–Witten–Wadia unitary matrix model by introducing a logarithmic term in the potential. The partition function of the model can be expressed equivalently in terms of the Toeplitz determinant with the (i,j)-entry being the modified Bessel functions of order i−j−ν, ν∈ℂ. When the degree n is finite, we show that the Toeplitz determinant is described by the isomonodromy
-
The Boolean quadratic forms and tangent law Random Matrices Theory Appl. (IF 0.9) Pub Date : 2024-01-29 Wiktor Ejsmont, Patrycja Hęćka
In [W. Ejsmont and F. Lehner, The free tangent law, Adv. Appl. Math. 121 (2020) 102093], we study the limit sums of free commutators and anticommutators and show that the generalized tangent function tanz1−xtanz describes the limit distribution. This is the generating function of the higher order tangent numbers of Carlitz and Scoville (see (1.6) in [L. Carlitz and R. Scoville, Tangent numbers and
-
Laplace transform of the $x-y$ symplectic transformation formula in Topological Recursion Commun. Number Theory Phys. (IF 1.9) Pub Date : 2024-01-24 Alexander Hock
The functional relation coming from the $x-y$ symplectic transformation of Topological Recursion has a lot of applications; for instance it is the higher order moment-cumulant relation in free probability or can be used to compute intersection numbers on the moduli space of complex curves. We derive the Laplace transform of this functional relation, which has a very nice and compact form as a formal
-
Cohomological Hall algebras and perverse coherent sheaves on toric Calabi–Yau $3$-folds Commun. Number Theory Phys. (IF 1.9) Pub Date : 2024-01-24 Miroslav Rapčák, Yan Soibelman, Yaping Yang, Gufang Zhao
We study the Drinfeld double of the (equivariant spherical) Cohomological Hall algebra in the sense of Kontsevich and Soibelman, associated to a smooth toric Calabi–Yau $3$-fold $X$. By general reasons, the COHA acts on the cohomology of the moduli spaces of certain perverse coherent systems on $X$ via “raising operators”. Conjecturally the COHA action extends to an action of the Drinfeld double by
-
Numerical experiments on coefficients of instanton partition functions Commun. Number Theory Phys. (IF 1.9) Pub Date : 2024-01-24 Aradhita Chattopadhyaya, Jan Manschot
We analyze the coefficients of partition functions of Vafa–Witten (VW) theory on a four-manifold. These partition functions factorize into a product of a function enumerating pointlike instantons and a function enumerating smooth instantons. For gauge groups $SU(2)$ and $SU(3)$ and four-manifold the complex projective plane $\mathbb{CP}^2$, we experimentally study the latter functions, which are examples
-
On the Generating Rank and Embedding Rank of the Hexagonic Lie Incidence Geometries Combinatorica (IF 1.1) Pub Date : 2024-01-05
Abstract Given a (thick) irreducible spherical building \(\Omega \) , we establish a bound on the difference between the generating rank and the embedding rank of its long root geometry and the dimension of the corresponding Weyl module, by showing that this difference does not grow when taking certain residues of \(\Omega \) (in particular the residue of a vertex corresponding to a point of the long
-
Euler’s Theorem for Regular CW-Complexes Combinatorica (IF 1.1) Pub Date : 2024-01-05 Richard H. Hammack, Paul C. Kainen
-
On special solutions to the Ermakov–Painlevé XXV equation Random Matrices Theory Appl. (IF 0.9) Pub Date : 2024-01-03 Alexander Chichurin, Galina Filipuk
In this paper, we study a nonlinear second-order ordinary differential equation which we call the Ermakov–Painlevé XXV equation since under certain restrictions on its coefficients it can be reduced either to the Ermakov or the Painlevé XXV equation. The Ermakov–Painlevé XXV equation arises from a generalized Riccati equation and the related third-order linear differential equation via the Schwarzian
-
A Topological Version of Hedetniemi’s Conjecture for Equivariant Spaces Combinatorica (IF 1.1) Pub Date : 2023-12-19 Vuong Bui, Hamid Reza Daneshpajouh
-
Tight Bound on Treedepth in Terms of Pathwidth and Longest Path Combinatorica (IF 1.1) Pub Date : 2023-12-19
Abstract We show that every graph with pathwidth strictly less than a that contains no path on \(2^b\) vertices as a subgraph has treedepth at most 10ab. The bound is best possible up to a constant factor.
-
Edge-Connectivity and Pairwise Disjoint Perfect Matchings in Regular Graphs Combinatorica (IF 1.1) Pub Date : 2023-12-19 Yulai Ma, Davide Mattiolo, Eckhard Steffen, Isaak H. Wolf
-
A Characterization of Graphs Whose Small Powers of Their Edge Ideals Have a Linear Free Resolution Combinatorica (IF 1.1) Pub Date : 2023-11-27 Nguyen Cong Minh, Thanh Vu
Let I(G) be the edge ideal of a simple graph G. We prove that \(I(G)^2\) has a linear free resolution if and only if G is gap-free and \({{\,\textrm{reg}\,}}I(G) \le 3\). Similarly, we show that \(I(G)^3\) has a linear free resolution if and only if G is gap-free and \({{\,\textrm{reg}\,}}I(G) \le 4\). We deduce these characterizations by establishing a general formula for the regularity of powers
-
A Group Ring Approach to Fuglede’s Conjecture in Cyclic Groups Combinatorica (IF 1.1) Pub Date : 2023-11-27 Tao Zhang
Fuglede’s conjecture states that a subset \(\Omega \subseteq \mathbb {R}^{n}\) with positive and finite Lebesgue measure is a spectral set if and only if it tiles \(\mathbb {R}^{n}\) by translation. However, this conjecture does not hold in both directions for \(\mathbb {R}^n\), \(n\ge 3\). While the conjecture remains unsolved in \(\mathbb {R}\) and \(\mathbb {R}^2\), cyclic groups are instrumental
-
Tail bounds on the spectral norm of sub-exponential random matrices Random Matrices Theory Appl. (IF 0.9) Pub Date : 2023-11-24 Guozheng Dai, Zhonggen Su, Hanchao Wang
Let X be an n×n symmetric random matrix with independent but non-identically distributed entries. The deviation inequalities of the spectral norm of X with Gaussian entries have been obtained by using the standard concentration of Gaussian measure results. This paper establishes an upper tail bound of the spectral norm of X with sub-exponential entries. Our method relies upon a crucial ingredient of
-
Universal Planar Graphs for the Topological Minor Relation Combinatorica (IF 1.1) Pub Date : 2023-11-21 Florian Lehner