Part I gives algebraic basic notions necessary to generating a graded sheaf of rings from a Galois extension, i.e. essentially a specialization, called emergent, from a ring of polynomials A[x1, ..., xm] giving rise to a set of compact connected algebraic subgroups which correspond to the different sections of the sheaf of rings θ. Part II refers to the introduction of the Eisenstein homology b...

The number of linear independent algebraic relations among elementary symmetric polynomial functions over finite fields is computed. An algorithm able to find all such relations is described. The algorithm consists essentially of Gauss’ upper triangular form algorithm. It is proved that the basis of the ideal of algebraic relations found by the algorithm consists of polynomials having coefficie...

We analyze several families of two-dimensional quantum random walks. The feasible region (the region where probabilities do not decay exponentially with time) grows linearly with time, as is the case with one-dimensional QRW. The limiting shape of the feasible region is, however, quite different. The limit region turns out to be an algebraic set, which we characterize as the rational image of a...

This survey covers a decade and a half of joint work with L. Lehmann, G. M. Mbakop, and L. M. Pardo. We address the problem of finding a smooth algebraic sample point for each connected component of a real algebraic variety, being only interested in components which are generically smooth locally complete intersections. The complexity of our algorithms is essentially polynomial in the degree of...

When solving data analysis problems it is important to integrate prior knowledge and/or structural invariances. This paper contributes by a novel framework for incorporating algebraic invariance structure into kernels. In particular, we show that algebraic properties such as sign symmetries in data, phase independence, scaling etc. can be included easily by essentially performing the kernel tri...

Background. For X a smooth scheme over ~, K.t(X; 2g/{") is isomorphic to the usual rood(" topological K-theory of X as a complex manifold ([6]). In this case, Theorem 1 essentially answers a question of Fulton, [9], w 5. For many such X the mapp: Ko(X)~K~ot(X) cannot be a surjection, as non-trivial Hodge conditions on the Chern classes prevent some topological bundles from being stably algebrai...

We consider algorithms to compute the genus and rational parametric equations, for implicitly defined irreducible rational plane algebraic curves of arbitrary degree. Rational panuneterizations exist for all irreducible algebraic curves of genus O. The genus is compuled by a complete analysis of the singularities of plane algebraic curves, using affine quadratic transformations. The rational pa...

Building on the work of Davenport and Schmidt, we mainly prove two results. The first one is a version of Gel’fond’s transcendence criterion which provides a sufficient condition for a complex or p-adic number ξ to be algebraic in terms of the existence of polynomials of bounded degree taking small values at ξ together with most of their derivatives. The second one, which follows from this crit...