2. Derivatives 10 2.1. Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2. Hasse-Schmidt derivatives ∂ z . . . . . . . . . . . . . . . . . . . . . 11 2.3. Hasse-Schmidt derivations in general . . . . . . . . . . . . . . . . . 18 2.4. Hasse-Schmidt derivations from A to B as algebra maps A→ B [[t]] 22 2.5. Extending Hasse-Schmidt derivations to localizations . ....

The behavior of the Hasse–Schmidt algebra under étale extension is used to show that the Hasse–Schmidt algebra of a smooth algebra of finite type over a field equals the ring of differential operators. These techniques show that the formation of Hasse–Schmidt derivations does not commute with localization, providing a counterexample to a question of Brown and Kuan; their conjecture is reformula...

1. Linear series on curves 1.1. Terminology and notation 1.2. Morphisms from linear series; Castelnuovo’s genus bound 1.3. Linear series from morphisms 1.4. Relation between linear series and morphisms 1.5. Hermitian invariants; Weierstrass semigroups I 2. Weierstrass point theory 2.1. Hasse derivatives 2.2. Order sequence; Ramification divisor 2.3. D-Weierstrass points 2.4. D-osculating spaces...

We upper bound the number of common zeros over a finite grid of multivariate polynomials and an arbitrary finite collection of their consecutive Hasse derivatives (in a coordinate-wise sense). To that end, we make use of the tool from Gröbner basis theory known as footprint. Then we establish and prove extensions in this context of a family of well-known results in algebra and combinatorics. Th...

Building on the abstract notion of prolongation developed in [10], the theory of iterative Hasse-Schmidt rings and schemes is introduced, simultaneously generalising difference and (Hasse-Schmidt) differential rings and schemes. This work provides a unified formalism for studying difference and differential algebraic geometry, as well as other related geometries. As an application, Hasse-Schmid...

Abstract. Building on the abstract notion of prolongation developed in [7], the theory of iterative Hasse rings and schemes is introduced, simultaneously generalising difference and (Hasse-)differential rings and schemes. This work provides a unified formalism for studying difference and differential algebraic geometry, as well as other related geometries. As an application, Hasse jet spaces ar...

In [6] Messmer and Wood proved quantifier elimination for separably closed fields of finite Ershov invariant e equipped with a (certain) Hasse derivation. We propose a variant of their theory, using a sequence of e commuting Hasse derivations. In contrast to [6] our Hasse derivations are iterative.

Model theory, a branch of mathematical logic which involves the study of sets definable in a mathematical structure using an appropriate first order language, has recently shown its capacity to produce applications to other domains of mathematics, principally to algebra, algebraic geometry and number theory. The main example is the proof by Ehud Hrushovski of a conjecture of algebraic geometry,...

Building on the abstract notion of prolongation developed in [7], the theory of iterative Hasse rings and schemes is introduced, simultaneously generalising difference and (Hasse-)differential rings and schemes. This work provides a unified formalism for studying difference and differential algebraic geometry, as well as other related geometries. As an application, Hasse jet spaces are construc...