Roberto Pagaria, Giovanni Paolini
We study the representability problem for torsion-free arithmetic matroids. By using a new operation called "reduction" and a "signed Hermite normal form", we provide and implement an algorithm to compute all the representations, up to equivalence. As an application, we disprove two conjectures about the poset of layers and the independence poset of a toric arrangement.
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The theory of matroids has been generalized to oriented matroids and, recently, to arithmetic matroids. We want to give a definition of "oriented arithmetic matroid" and prove some properties like the "uniqueness of orientation".
In this note we generalize the convolution formula for the Tutte polynomial of Kook-Reiner-Stanton and Etienne-Las Vergnas to a more general setting that includes both arithmetic matroids and delta-matroids. As corollaries, we obtain new proofs of two positivity results for pseudo-arithmetic matroids and a combinatorial interpretation of the arithmetic Tutte polynomial at infinitely many points in terms of arithmetic flows and colorings. We also exhibit connections with a decomposition of Dahmen-Micchelli spaces and lattice point counting in zonotopes.
We introduce the notion of an arithmetic matroid, whose main example is given by a list of elements of a finitely generated abelian group. In particular we study the representability of its dual, providing an extension of the Gale duality to this setting. Guided by the geometry of generalized toric arrangements, we provide a combinatorial interpretation of the associated arithmetic Tutte polynomial, which can be seen as a generalization of Crapo's formula for the classical Tutte polynomial.
Convolution formulas for multivariate arithmetic Tutte polynomials
本研究针对算术拟阵的多元算术Tutte多项式，给出两个算术拟阵乘积及单个算术拟阵对应的卷积公式，可纯组合证明已有相关结论且证明更短，还得到算术拟阵特征多项式的卷积公式。
In this note we define a Stanley-Reisner ring for quasi-arithmetic matroids and more general structures. To this end, we define two types of CW complexes associated with a quasi-arithmetic matroid that generalize independence complexes of matroids. Then we use Stanley's construction of Stanley-Reisner rings for simplicial posets.
An arithmetic matroid is weakly multiplicative if the multiplicity of at least one of its bases is equal to the product of the multiplicities of its elements. We show that if such an arithmetic matroid can be represented by an integer matrix, then this matrix is uniquely determined. This implies that the integer cohomology ring of a centred toric arrangement whose arithmetic matroid is weakly multiplicative is determined by its poset of layers. This partially answers a question asked by Callegaro-Delucchi.
On the proportion of metric matroids whose Jacobians have nontrivial $\mathbf{p}$-torsion
该研究探讨雅可比矩阵含非平凡$p$-挠的度量拟阵的比例问题，通过建立其雅可比矩阵与配置超曲面$\mathbb{F}_p$-有理点的对应关系，经有限域点计数证明该比例渐近等价于$1/p$。
The first problem we investigate is the following: given $k\in \mathbb{R}_{\ge 0}$ and a vector $v$ of Plücker coordinates of a point in the real Grassmannian, is the vector obtained by taking the $k$th power of each entry of $v$ again a vector of Plücker coordinates? For $k\neq 1$, this is true if and only if the corresponding matroid is regular. Similar results hold over other fields. We also describe the subvariety of the Grassmannian that consists of all the points that define a regular matroid. The second topic is a related problem for arithmetic matroids. Let $\mathcal{A} = (E, rk, m)$ b…
In this note we provide a higher-dimensional analogue of Tutte's celebrated theorem on colorings and flows of graphs, by showing that the theory of arithmetic Tutte polynomials and quasi-polynomials encompasses invariants defined for CW complexes by Beck-Breuer-Godkin-Martin and Duval-Klivans-Martin. Furthermore, we answer a question by Bajo-Burdick-Chmutov, concerning the modified Tutte-Krushkal-Renhardy polynomials defined by these authors: to this end, we prove that the product of two arithmetic multiplicity functions on a matroid is again an arithmetic multiplicity function.
We prove an arithmetic analogue of the typical structure theorem for graph hereditary properties due to Alon, Balogh, Bollobás and Morris.
We describe the structure of the monoid of natural-valued monotone functions on an arbitrary poset. For this monoid we provide a presentation, a characterization of prime elements, and a description of its convex hull. We also study the associated monoid ring, proving that it is normal, and thus Cohen-Macaulay. We determine its Cohen-Macaulay type, characterize the Gorenstein property, and provide a Gröbner basis of the defining ideal. Then we apply these results to the monoid of quasi-arithmetic multiplicities on a uniform matroid. Finally we state some conjectures on the number of irreducibl…
We initiate the study of group actions on (possibly infinite) semimatroids and geometric semilattices. To every such action is naturally associated an orbit-counting function, a two-variable "Tutte" polynomial and a poset which, in the realizable case, coincides with the poset of connected components of intersections of the associated toric arrangement. In this structural framework we recover and strongly generalize many enumerative results about arithmetic matroids, arithmetic Tutte polynomials and toric arrangements by finding new combinatorial interpretations beyond the realizable case. In …
Elliptic arrangements of complex multiplication type
本文定义了椭圆排列，扩展到具有复乘的椭圆曲线，并证明其组合数据定义了算术模体和环End(E)上的模体，同时展示了补集的欧拉特征是算术特拉多项式的评估。
Let $X$ be a $(d\times N)$-matrix. We consider the variable polytope $Π_X(u) = \{w \ge 0 : X w = u \}$. It is known that the function $T_X$ that assigns to a parameter $u \in \mathbb{R}^d$ the volume of the polytope $Π_X(u)$ is piecewise polynomial. The Brion-Vergne formula implies that the number of lattice points in $Π_X(u)$ can be obtained by applying a certain differential operator to the function $T_X$. In this article we slightly improve the Brion-Vergne formula and we study two spaces of differential operators that arise in this context: the space of relevant differential operators (i.e…
Arithmetic Circuits and Neural Networks for Regular Matroids
本文证明了存在大小为 $O(n^3)$ 的电路和 ReLU 神经网络，用于计算正则拟阵的基生成多项式并解决加权基最大化问题，并首次展示了虚拟扩展构形比已知最佳单个扩展构形更高效。
A classical theorem of Fritz John allows one to describe a convex body, up to constants, as an ellipsoid. In this article we establish similar descriptions for generalized (i.e. multidimensional) arithmetic progressions in terms of proper (i.e. collision-free) generalized arithmetic progressions, in both torsion-free and torsion settings. We also obtain a similar characterization of iterated sumsets in arbitrary abelian groups in terms of progressions, thus strengthening and extending recent results of Szemerédi and Vu.
We introduce an arithmetic version of the multivariate Tutte polynomial, and (for representable arithmetic matroids) a quasi-polynomial that interpolates between the two. A generalized Fortuin-Kasteleyn representation with applications to arithmetic colorings and flows is obtained. We give a new and more general proof of the positivity of the coefficients of the arithmetic Tutte polynomial, and (in the representable case) a geometrical interpretation of them.
We give a quantum-inspired $O(n^4)$ algorithm computing the Tutte polynomial of a lattice path matroid, where $n$ is the size of the ground set of the matroid. Furthermore, this can be improved to $O(n^2)$ arithmetic operations if we evaluate the Tutte polynomial on a given input, fixing the values of the variables. The best existing algorithm, found in 2004, was $O(n^5)$, and the problem has only been known to be polynomial time since 2003. Conceptually, our algorithm embeds the computation in a determinant using a recently demonstrated equivalence of categories useful for counting problems s…
We introduce and study the notion of the $G$-Tutte polynomial for a list $\mathcal{A}$ of elements in a finitely generated abelian group $Γ$ and an abelian group $G$, which is defined by counting the number of homomorphisms from associated finite abelian groups to $G$. The $G$-Tutte polynomial is a common generalization of the (arithmetic) Tutte polynomial for realizable (arithmetic) matroids, the characteristic quasi-polynomial for integral arrangements, Brändén-Moci's arithmetic version of the partition function of an abelian group-valued Potts model, and the modified Tutte-Krushkal-Renhardy…
In this work, we study matroids over a domain and several classical combinatorial and algebraic invariants related. We define their Grothendieck-Tutte polynomial $T_{\mathcal{M}}(x,y)$, extending the definition given by Fink and Moci in 2016, and we show that such polynomial has the classical deletion-contraction property. Moreover, we study the set of independencies for a realizable matroid over a domain, generalizing the definition of \emph{poset of torsions} $Gr(\mathcal{M})$ given by the second author in 2017. This is a union of identical simplicial posets as for (quasi-)arithmetic matroid…
Employing two models, we show that various counting functions of a random variable defined by restriction or contraction of a ranked set with multiplicity (e.g., classical and arithmetic matroids) have expectations given by the corresponding multivariate Tutte polynomial. The first model is based on a generalization of a convolution formula of Kung (2010), extending from matroids to ranked sets with multiplicities. This model enables us to compute the expectations of many familiar polynomials, such as the chromatic, flow and Ehrhart polynomials, generalizing the classical results of Welsh (199…
Representation theory over semifields
本文研究挠群在幂等半域（尤其是 Boolean 半域）上的表示并加以分类，为后续推广到低秩拟阵表示奠定基础。
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