Recurrence relations for polynomial sequences via Riordan matrices

We give recurrence relations for any family of generalized Appell polynomials unifying so some known recurrences for many classical sequences of polynomials. Our main tool to get our goal is the Riordan group. We use the product of Riordan matrices to interpret some relationships between different polynomial families. Moreover using the Hadamard product of series we get a general recurrence relation for the polynomial sequences associated to the so called generalized umbral calculus.     

Linear spaces of matrices of constant rank and instanton bundles

We present a new method to study 4-dimensional linear spaces of skew-symmetric matrices of constant co-rank 2, based on rank 2 vector bundles on P³${\mathbb{P}}^3$ and derived category tools. The method allows one to prove the existence of new examples of size 10×10$10\times 10$ and 14×14$14\times 14$ via instanton bundles of charge 2 and 4 respectively, and it provides an explanation for what used to be the only known example (Westwick 1996 [25]). We also give an algorithm to construct explicitly a matrix of size 14 of this type.     

Hermitian matrices with a bounded number of eigenvalues

Conjugation covariants of matrices are applied to study the real algebraic variety consisting of complex Hermitian matrices with a bounded number of distinct eigenvalues. A minimal generating system of the vanishing ideal of degenerate three by three Hermitian matrices is given, and the structure of the corresponding coordinate ring as a module over the special unitary group is determined. The method applies also for degenerate real symmetric three by three matrices. For arbitrary n   partial information on the minimal degree component of the vanishing ideal of the variety of n×n$n\times n$ Hermitian matrices with a bounded number of eigenvalues is obtained, and some known results on sum of squares presentations of subdiscriminants of real symmetric matrices are extended to the case of complex Hermitian matrices.     

A semialgebraic closure for commutative algebra

Let A⊂R be rings containing the rationals. In R let S be a multiplicatively closed subset such that 1∈S and 0∉S, T a preorder of R (a proper subsemiring containing the squares) such that S⊂T and I an A-submodule of R. Define ρ(I) (or ρ_S,T(I)) to be

ρ(I)={a∈R|sa^2m+t∈I^2m for some m∈N,s∈S and t∈T}.     

Borel sets and sectional matrices

Following the path trodden by several authors along the border between Algebraic Geometry and Algebraic Combinatorics, we present some new results on the combinatorial structure of Borel ideals. This enables us to prove theorems on the shape of thesectional matrix of a homogeneous ideal, which is a new invariant stronger than the Hilbert function. The authors were partially supported by the Consiglio Nazionale delle Ricerche (CNR).     

Determinant identities for Laplace matrices

We show that every minor of an n×n Laplace matrix, i.e., a symmetric matrix whose row- and column sums are 0, can be written in terms of those minors that are obtained by deleting two rows and the corresponding columns. The proof is based on a classical determinant identity due to Sylvester. Furthermore, we show how our result can be applied in the context of electrical networks and spanning tree enumeration.     

Degree formulae for offset curves

In this paper, we present three different formulae for computing the degree of the offset of a real irreducible affine plane curve C given implicitly, and we see how these formulae particularize to the case of rational curves. The first formula is based on an auxiliary curve, called S, that is defined depending on a non-empty Zariski open subset of R². The second formula is based on the resultant of the defining polynomial of C, and the polynomial defining generically S. The third formula expresses the offset degree by means of the degree of C and the multiplicity of intersection of C and the hodograph H to C, at their intersection points.     

CMV matrices: Five years after

CMV matrices are the unitary analog of Jacobi matrices; we review their general theory.     

Tridiagonal matrices with nonnegative entries

In this paper, we characterize the nonnegative irreducible tridiagonal matrices and their permutations, using certain entries in their primitive idempotents. Our main result is summarized as follows. Let d denote a nonnegative integer. Let A denote a matrix in R and let denote the roots of the characteristic polynomial of A. We say A is multiplicity-free whenever these roots are mutually distinct and contained in R. In this case E_i will denote the primitive idempotent of A associated with theta_i(0?i?d). We say A is symmetrizable whenever there exists an invertible diagonal matrix Δ∈R such that ΔAΔ^-1 is symmetric. Let Γ(A) denote the directed graph with vertex set {0,1,…,d}, where i→j whenever i≠j and A_ij≠0.Theorem.Assume that each entry ofAis nonnegative. Then the following are equivalent for0≤s,t≤d.

(i)

The graphΓ(A)is a bidirected path with endpointss,t:s↔*↔*↔?↔*↔t.

(ii)

The matrixAis symmetrizable and multiplicity-free. Moreover the(s,t)-entry ofE_itimes(θ_i-θ₀)?(θ_i-θ_i-1)(θ_i-θ_i+1)?(θ_i-θ_d)is independent of i for0≤i≤d, and this common value is nonzero.

Recently Kurihara and Nozaki obtained a theorem that characterizes the Q-polynomial property for symmetric association schemes. We view the above result as a linear algebraic generalization of their theorem.     

Plethystic algebra

The notion of a Z-algebra has a non-linear analogue, whose purpose it is to control operations on commutative rings rather than linear operations on abelian groups. These plethories can also be considered non-linear generalizations of cocommutative bialgebras. We establish a number of category-theoretic facts about plethories and their actions, including a Tannaka-Krein-style reconstruction theorem. We show that the classical ring of Witt vectors, with all its concomitant structure, can be understood in a formula-free way in terms of a plethystic version of an affine blow-up applied to the plethory generated by the Frobenius map. We also discuss the linear and infinitesimal structure of plethories and explain how this gives Bloch's Frobenius operator on the de Rham-Witt complex.     

On iteration of Cox rings

We characterize all varieties with a torus action of complexity one that admit iteration of Cox rings.     

On a class of algebras associated to directed graphs

To any directed graph we associate an algebra with edges of the graph as generators and with relations defined by all pairs of directed paths with the same origin and terminus. Such algebras are related to factorizations of polynomials over noncommutative algebras. We also construct a basis for our algebras associated to layered graphs.     

Regarding an adaptive algorithm for testing multivariate linear dependence

For a finite collection of functions within some differential field of several variables, there exists an adaptive algorithm for calculating a basis of their linear relations. We study the complexity of this algorithm, noting how it compares to some other existing techniques. Also we demonstrate some modifications for improving implementation. In the course of our analysis, we define the marginal set of a Young-like set and show how the size of the former can be bounded in terms of the size of the latter.     

Quasideterminants

The determinant is a main organizing tool in commutative linear algebra. In this review we present a theory of the quasideterminants defined for matrices over a division ring.     

A note on Cartan matrices for symmetric groups

Using generating functions, a very simple explicit formula for the determinants of the p-Cartan matrices of symmetric groups is given. Our method works also when p is a composite number.Received: 5 September 2001