A quite basic fact about abelian varieties (and complex tori) is that the only line bundle in Pic 0 with non-trivial cohomology is the structure sheaf. Mumford, in his treatment of the dual abelian variety, made the far-reaching remark that this yields the following sheaf-theoretic formulation. Let A be an abelian variety (over an algebraically closed field) of dimension g, let us denote A = Pi...

We demonstrate how several problems of algebraic geometry, i.e. Ax-Grothendieck, Hilbert’s Nullstellensatz, NoetherOstrowski, and Hilbert’s 17th problem, have simple proofs when approached from using model theory. The proofs use two general transfer principles. The first is the Lefschetz principle, which allows sentences that are true in algebraically closed fields of infinitely many prime char...

This paper presents an algebraically extended 2D image representation. Combining methods of tensor algebra, monogenic signal and quadrature filter, this novel image representation can be derived as the monogenic extension of a curvature tensor. From it, the monogenic signal and the monogenic curvature signal for modeling intrinsically one and two dimensional (i1D/i2D) structures are obtained as...

The algebraically constructible functions on a real algebraic set are the sums of signs of polynomials on this set. We prove a formula giving the minimal number of polynomials needed to write generically a given algebraically constructible function as a sum of signs. We also prove a characterization of the polynomials appearing in a generic presentation of the function with the minimal number o...