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Register a new userSome relational structures with polynomial growth and their associated algebras II: Finite generation
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The profile of a relational structure $R$ is the function $varphi_R$ which counts for every integer $n$ the number, possibly infinite, $varphi_R(n)$ of substructures of $R$ induced on the $n$-element subsets, isomorphic substructures being identified. If $varphi_R$ takes only finite values, this is the Hilbert function of a graded algebra associated with $R$, the age algebra $A(R)$, introduced by P.~J.~Cameron. In a previous paper, we studied the relationship between the properties of a relational structure and those of their algebra, particularly when the relational structure $R$ admits a finite monomorphic decomposition. This setting still encompasses well-studied graded commutative algebras like invariant rings of finite permutation groups, or the rings of quasi-symmetric polynomials. In this paper, we investigate how far the well know algebraic properties of those rings extend to age algebras. The main result is a combinatorial characterization of when the age algebra is finitely generated. In the special case of tournaments, we show that the age algebra is finitely generated if and only if the profile is bounded. We explore the Cohen-Macaulay property in the special case of invariants of permutation groupoids. Finally, we exhibit sufficient conditions on the relational structure that make naturally the age algebra into a Hopf algebra.
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Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term polynomial method to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of the polynomial method and show how the algorithmic theory of polynomial ideals can be used to detect k-colorability, unique Hamiltonicity, and automorphism rigidity of graphs. Our techniques are diverse and involve Nullstellensatz certificates, linear algebra over finite fields, Groebner bases, toric algebra, convex programming, and real algebraic geometry.
A way to construct and classify the three dimensional polynomially deformed algebras is given and the irreducible representations is presented. for the quadratic algebras 4 different algebras are obtained and for cubic algebras 12 different classes are constructed. Applications to quantum mechanical systems including supersymmetric quantum mechanics are discussed
We consider integer sequences that satisfy a recursion of the form $x_{n+1} = P(x_n)$ for some polynomial $P$ of degree $d > 1$. If such a sequence tends to infinity, then it satisfies an asymptotic formula of the form $x_n sim A alpha^{d^n}$, but little can be said about the constant $alpha$. In this paper, we show that $alpha$ is always irrational or an integer. In fact, we prove a stronger statement: if a sequence $G_n$ satisfies an asymptotic formula of the form $G_n = A alpha^n + B + O(alpha^{-epsilon n})$, where $A,B$ are algebraic and $alpha > 1$, and the sequence contains infinitely many integers, then $alpha$ is irrational or an integer.
The Constraint Satisfaction Problem (CSP) and its counting counterpart appears under different guises in many areas of mathematics, computer science, and elsewhere. Its structural and algorithmic properties have demonstrated to play a crucial role in many of those applications. For instance, in the decision CSPs, structural properties of the relational structures involved---like, for example, dismantlability---and their logical characterizations have been instrumental for determining the complexity and other properties of the problem. Topological properties of the solution set such as connectedness are related to the hardness of CSPs over random structures. Additionally, in approximate counting and statistical physics, where CSPs emerge in the form of spin systems, mixing properties and the uniqueness of Gibbs measures have been heavily exploited for approximating partition functions and free energy. In spite of the great diversity of those features, there are some eerie similarities between them. These were observed and made more precise in the case of graph homomorphisms by Brightwell and Winkler, who showed that dismantlability of the target graph, connectedness of the set of homomorphisms, and good mixing properties of the corresponding spin system are all equivalent. In this paper we go a step further and demonstrate similar connections for arbitrary CSPs. This requires much deeper understanding of dismantling and the structure of the solution space in the case of relational structures, and new refined concepts of mixing introduced by Brice~no. In addition, we develop properties related to the study of valid extensions of a given partially defined homomorphism, an approach that turns out to be novel even in the graph case. We also add to the mix the combinatorial property of finite duality and its logic counterpart, FO-definability, studied by Larose, Loten, and Tardif.
Following Britz, Johnsen, Mayhew and Shiromoto, we consider demi-ma-troids as a(nother) natural generalization of matroids. As they have shown, demi-ma-troids are the appropriate combinatorial objects for studying Weis duality. Our results here apport further evidence about the trueness of that observation. We define the Hamming polynomial of a demimatroid $M$, denoted by $W(x,y,t)$, as a generalization of the extended Hamming weight enumerator of a matroid. The polynomial $W(x,y,t)$ is a specialization of the Tutte polynomial of $M$, and actually is equivalent to it. Guided by work of Johnsen, Roksvold and Verdure for matroids, we prove that Betti numbers of a demimatroid and its elongations determine the Hamming polynomial. Our results may be applied to simplicial complexes since in a canonical way they can be viewed as demimatroids. Furthermore, following work of Brylawski and Gordon, we show how demimatroids may be generalized one step further, to combinatroids. A combinatroid, or Brylawski structure, is an integer valued function $rho$, defined over the power set of a finite ground set, satisfying the only condition $rho(emptyset)=0$. Even in this extreme generality, we will show that many concepts and invariants in coding theory can be carried on directly to combinatroids, say, Tutte polynomial, characteristic polynomial, MacWilliams identity, extended Hamming polynomial, and the $r$-th generalized Hamming polynomial; this last one, at least conjecturelly, guided by the work of Jurrius and Pellikaan for linear codes. All this largely extends the notions of deletion, contraction, duality and codes to non-matroidal structures.
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