Mathematics

In this paper, we prove that if R is a local ring of dimension d, d≥2 and 1 d! ∈R then the group U m d+1 (R[X]) E d+1 (R[X]) has no k -torsion, provided k∈G L 1 (R). We also prove that if R is a regular ring of dimension d, d≥2 and 1 d! ∈R such that E d+1 (R) acts transitively on U m d+1 (R) then E d+1 (R[X]) acts transitively on U m d+1 (R[X]).

In this paper we investigate the factorization behaviour of the binomial polynomials ( x n )= x(x−1)⋯(x−n+1) n! and their powers in the ring of integer-valued polynomials Int(Z) . While it is well-known that the binomial polynomials are irreducible elements in Int(Z) , the factorization behaviour of their powers has not yet been fully understood. We fill this gap and show that the binomial polynomials are absolutely irreducible in Int(Z) , that is, ( x n ) m factors uniquely into irreducible elements in Int(Z) for all m∈N . By reformulating the problem in terms of linear algebra and number theory, we show that the question can be reduced to determining the rank of, what we call, the valuation matrix of n . A main ingredient in computing this rank is the following number-theoretical result for which we also provide a proof: If n>10 and n , n−1 , \ldots, n−(k−1) are composite integers, then there exists a prime number p>2k that divides one of these integers.

We discuss some research problems on affine monomial curves, from the perspective of computation.

Let H be a simple undirected graph and G=L(H) be its line graph. Assume that Δ(G) denotes the clique complex of G . We show that Δ(G) is sequentially Cohen-Macaulay if and only if it is shellable if and only if it is vertex decomposable. Moreover if Δ(G) is pure, we prove that these conditions are also equivalent to being strongly connected. Furthermore, we state a complete characterizations of those H for which Δ(G) is Cohen-Macaulay, sequentially Cohen-Macaulay or Gorenstein. We use these characterizations to present linear time algorithms which take a graph G , check whether G is a line graph and if yes, decide if Δ(G) is Cohen-Macaulay or sequentially Cohen-Macaulay or Gorenstein.

In this paper we review different definitions that multi-state k -out-of- n systems have received along the literature and study them in a unified way using the algebra of monomial ideals. We thus obtain formulas and algorithms to compute their reliability and bounds for it.

In this paper we study algebraic sets of pairs of matrices defined by the vanishing of either the diagonal of their commutator matrix or its anti-diagonal. We find a system of parameters for the coordinate rings of these two sets and their intersection and show that they are complete intersections. Moreover, we prove that these algebraic sets are F -pure over a field of positive prime characteristic and the algebraic set of pairs of matrices with the zero diagonal commutator is F -regular.

Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that, if the algorithm is guaranteed to terminate on every input, then there is a computable upper bound for the size of the output of the algorithm in terms of the input. We also generalize this to algorithms working with models of good enough theories (including for example, difference fields). We then apply this to differential algebraic geometry to show that there exists a computable uniform upper bound for the number of components of any variety defined by a system of polynomial PDEs. We then use this bound to show the existence of a computable uniform upper bound for the elimination problem in systems of polynomial PDEs with delays.

We propose the notion of GAS numerical semigroup which generalizes both almost symmetric and 2-AGL numerical semigroups. Moreover, we introduce the concept of almost canonical ideal which generalizes the notion of canonical ideal in the same way almost symmetric numerical semigroups generalize symmetric ones. We prove that a numerical semigroup with maximal ideal M and multiplicity e is GAS if and only if M−e is an almost canonical ideal of M−M . This generalizes a result of Barucci about almost symmetric semigroups and a theorem of Chau, Goto, Kumashiro, and Matsuoka about 2-AGL semigroups. We also study the transfer of the GAS property from a numerical semigroup to its gluing, numerical duplication and dilatation.

In this paper, we introduce a new invariant of Cohen-Macaulay local rings in terms of canonical ideals. The invariant measures how close to being Gorenstein, and preserved by localizations, dividing non-zerodivisors, and flat local homomorphisms. Furthermore it builds bridges between almost Gorenstein and nearly Gorenstein in dimension one. We also explore the invariant in numerical semigroup rings and rings arising from idealizations.

Let R be a polynomial ring over a field and M= �?n M n a finitely generated graded R -module, minimally generated by homogeneous elements of degree zero with a graded R -minimal free resolution F . A Cohen-Macaulay module M is Gorenstein when the graded resolution is symmetric. We give an upper bound for the first Hilbert coefficient, e 1 in terms of the shifts in the graded resolution of M . When M=R/I , a Gorenstein algebra, this bound agrees with the bound obtained in \cite{ES} in Gorenstein algebras with quasi-pure resolution. We conjecture a similar bound for the higher coefficients.