We introduce a coloured generalization $\mathrm{NSym}_A$ of the Hopf algebra non-commutative symmetric functions described as subalgebra rooted ordered trees algebra. Its natural basis can be identified with set sentences over alphabet $A$ (the colours). present also its graded dual $\mathrm{QSym}_A$ quasi-symmetric together realization in terms power series partially commutative variables. pro...

The multigraded Poincaré-Betti series P k R(x̄; t) of a monomial ring k[x̄]/〈M〉 on a finite number of monomial generators has the form ∏ xi∈x̄ (1+xit)/bR,k(x̄; t), where bR,k(x̄; t) is a polynomial depending only on the monomial set M and the characteristic of the field k. I present a computer program designed to calculate the polynomial bR,k for a given field characteristic and a given set of monom...

We study blowups of affine n-space with center an arbitrary monomial ideal and call monomial ideals that render smooth blowups tame ideals. We give a combinatorial criterion to decide whether the blowup is smooth and apply this criterion to discuss a smoothing procedure proposed by Rosenberg, monomial building sets and permutohedra.

d-Monomial tests are statistical randomness tests based on Algebraic Normal Form representation of a Boolean function, and were first introduced by Filiol in 2002. We show that there are strong indications that the Gate Complexity of a Boolean function is related to a bias detectable in a d-Monomial test. We then discuss how to effectively apply d-Monomial tests in chosen-IV attacks against str...

Let J be a strongly stable monomial ideal in S = K[x1, . . . , xn] and let Mf(J) be the family of all homogeneous ideals I in S such that the set of all terms outside J is a K-vector basis of the quotient S/I. We show that an ideal I belongs toMf(J) if and only if it is generated by a special set of polynomials, the J-marked basis of I, that in some sense generalizes the notion of reduced Gröbn...

Hilbert schemes of zero-dimensional ideals in a polynomial ring can be covered with suitable affine open subschemes whose construction is achieved using border bases. Moreover, border bases have proved to be an excellent tool for describing zero-dimensional ideals when the coefficients are inexact. And in this situation they show a clear advantage with respect to Gröbner bases which, neverthele...

Some algorithms for singularity theory and algebraic geometry The use of Gröbner basis computations for treating systems of polynomial equations has become an important tool in many areas. This paper introduces of the concept of standard bases (a generalization of Gröbner bases) and the application to some problems from algebraic geometry. The examples are presented as Singular commands. A gene...

We show that certain two-dimensional, integrally closed monomial modules can be uniquely written as a countable product of isomorphic copies of simple integrally closed monomial ideals.