Using tools from algebraic geometry and Gröbner basis theory we solve two problems in network coding. First we present a method to determine the smallest field size for which linear network coding is feasible. Second we derive improved estimates on the success probability of random linear network coding. These estimates take into account which monomials occur in the support of the determinant o...

While the computation of Gröbner bases is known to be an expspace-complete problem, the generic behaviour of algorithms for their computation is much better. We study generic properties of Gröbner bases and analyse precisely the best algorithm currently known, F5. 1. Gröbner Bases Gröbner bases are a fundamental tool in computational algebra. They provide a multivariate generalization of Euclid...

For a discrete distribution in Rd on a finite support D probabilities and moments are algebraically related. If there are n = |D| support points then there are n probabilities p(x), x ∈ D and n basic moments. By suitable interpolation of the probabilities using a Gröbner basis method, high order moments can be express linearly in terms of n basic moments. A main result is that high order cumula...

We focus on Gröbner bases for modules of univariate polynomial vectors over a ring. We identify a useful property, the " predictable leading monomial (PLM) property " that is shared by minimal Gröbner bases of modules in F[x] q , no matter what positional term order is used. The PLM property is useful in a range of applications and can be seen as a strengthening of the wellknown predictable deg...

We uncover the structure of the space of symmetric functions in non-commutative variables by showing that the underlined Hopf algebra is both free and co-free. We also introduce the Hopf algebra of quasi-symmetric functions in non-commutative variables and define the product and coproduct on the monomial basis of this space and show that this Hopf algebra is free and cofree. In the process of l...

We uncover the structure of the space of symmetric functions in non-commutative variables by showing that the underlined Hopf algebra is both free and co-free. We also introduce the Hopf algebra of quasi-symmetric functions in non-commutative variables and define the product and coproduct on the monomial basis of this space and show that this Hopf algebra is free and cofree. In the process of l...

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...

The Cahn–Hilliard equation plays an important role in the phase separation in a binary mixture. This is a fourth order nonlinear partial differential equation. In this paper, we study the behaviour of the solution by using orthogonal cubic spline collocation method and derive optimal order error estimates. We discuss some computational experiments by using monomial basis functions in the spatia...

We describe (reduced) Gröbner bases of the ideal of polynomials over a field, which vanish on the set of characterisic vectors of the complete unifom families ([n] d ). An interesting feature of the results is that they are largely independent of the monomial order selected. The bases depend only on the ordering of the variables. We can thus use past results related to the lex order in the pres...