Blockchain Papers

David Gutiérrez Cambra

The objective of this memory is to give a purely algebraic proof of the Sauer- Shelah-Perles Lemma (inspired by the elegant proof in [FrPa,1983]), based only in duality in the Q−algebra Q[Vn] of polynomial functions de_ned on the zero-dimensional algebraic variety of subsets of the set [n] := {1, 2, . . . , n}. In fact, two di_erent proofs of this lemma will be given. Furthermore, we prove how several other classical results from Combinatorics are particular examples of a Trace (Inversion) Formula in _nite Q−algebras. For instance, one of this results is the general form of the Inclusion-Exclusion Principle (both with direct and reverse order associated to subsets inclusion). This approach also allows us to show a basis of the space of null t−designs, which di_ers from the one described in Theorem 4 of [DeFr,1982]. All results are still true if we replace Q[Vn] by K[Vn], where K is a perfect _eld of characteristic di_erent from 2. This memory has then the underlying purpose of connecting two _elds of mathematical knowledge that are not usually connected, at least not in this form.

Héctor Pastén, Thanases Pheidas, Xavier Vidaux

No abstract is available for this record.

Gennady Lyubeznik

Let k be a field, let R be a ring of polynomials in a finite number of variables over k, let D be the ring of k-linear differential operators of R and let f be a non-zero element of R. It is well-known that R_f, with its natural D-module structure, has finite length in the category of D-modules. We give a characteristic-free proof of this fact. To the best of our knowledge this is the first characteristic-free proof.