The Lie algebra E6 may be defined as the algebra of endomorphisms of a 27-dimensional complex vector space MC which annihilate a particular cubic polynomial. This raises a natural question: what is this polynomial? If we choose a basis for MC consisting of weight vectors {Xw} (for some Cartan subalgebra of E6), then any invariant cubic polynomial must be a linear combination of monomials XwXw′X...

ElimLin is a simple algorithm for solving polynomial systems of multivariate equations over small finite fields. It was initially proposed by Courtois to attack DES. It can reveal some hidden linear equations existing in the ideal generated by the system. We report a number of key theorems on ElimLin. Our main result is to characterize ElimLin in terms of a sequence of intersections of vector s...

For a given term order, the degree complexity of a projective scheme is defined by the maximal degree of the reduced Gröbner basis of its defining saturated ideal in generic coordinates [2]. It is wellknown that the degree complexity with respect to the graded reverse lexicographic order is equal to the Castelnuovo-Mumford regularity [3]. However, much less is known if one uses the graded lexic...

Let K be a eld; let P K n be a nite set and let =(P) Kx 1 ; : : :; x n ] be the ideal of P. A purely combinatorial algorithm to obtain a linear basis of the quotient algebra Kx 1 ; : : :; x n ]==(P) is given. Such a basis is represented by an n-dimensional Ferrers diagram of monomials which is minimal with respect to the inverse lexicographical order i:l:. It is also shown how this algorithm ca...

Nowadays, many strategies to solve polynomial systems use the computation of a Gröbner basis for the graded reverse lexicographical ordering, followed by a change of ordering algorithm to obtain a Gröbner basis for the lexicographical ordering. The change of ordering algorithm is crucial for these strategies. We study the p-adic stability of the main change of ordering algorithm, FGLM. We show ...

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...

We consider three modifications of our involutive algorithm for computing Janet bases. These modifications are related to degree compatible monomial orders and specify selection strategies for non-multiplicative prolongations. By using the standard data base of polynomial benchmarks for Gröbner bases software we compare the modifications and confront them with Magma that implements Faugère’s F4...

A Maple package for computing Gröbner bases of linear difference ideals is described. The underlying algorithm is based on Janet and Janet-like monomial divisions associated with finite difference operators. The package can be used, for example, for automatic generation of difference schemes for linear partial differential equations and for reduction of multiloop Feynman integrals. These two po...

It is well known that for the simpler problem of constructing translation invariants of grey scale images (1D, 2D or 3D) central moments can be used. There are plain closed formulae expressing them in terms of the ordinary geometrical moments. Moreover, central moments are ordinary moments of the properly normalized image. In this paper we present moment invariants for the more involved problem...

Let A be a Noetherian commutative ring with identity, let A[x] = A[x1, . . . , xn] be a polynomial ring over A, and let I ⊂ A[x] be an ideal. Geometrically, I defines a family of schemes over the base scheme Spec A; the fiber over each point p ∈ Spec A is a subscheme of the affine space Ak(p) = Spec k(p)[x], where k(p) = Ap/pp is the residue field of p. Let > be a total order on the monomials o...