相关论文: The differential rank of a differential-valued fie…
We introduce the notion of the definable rank of an ordered field, ordered abelian group and ordered set, respectively. We study the relation between the definable rank of an ordered field and the definable rank of the value group of its…
There are several equivalent characterizations of the valuation rank of an ordered or valued field. In this paper, we extend the theory to the case of an ordered or valued {\it difference} field (that is, ordered or valued field endowed…
We introduce the notion of differential largeness for fields equipped with several commuting derivations (as an analogue to largeness of fields). We lay out the foundations of this new class of "tame" differential fields. We state several…
We study the class of differentially henselian fields, which are henselian valued fields equipped with generic derivations in the sense of Cubides Kovacics and Point, and are special cases of differentially large fields in the sense of…
We give a first-order definition of key polynomials, we show the links with previous definitions, that it is relevant to study key degrees, and to use a kind of valuations that we call partially multiplicative. We also prove or reprove…
Given a differential or $q$-difference equation $P$ of order $n$, we prove that the set of exponents of a generalized power series solution has its rational rank bounded by the rational rank of the support of $P$ plus $n$. We also prove…
We generalize the differential dimension polynomial from prime differential ideals to characterizable differential ideals. Its computation is algorithmic, its degree and leading coefficient remain differential birational invariants, and it…
This paper deals with valuations of fields of formal meromorphic functions and their residue fields. We explicitly describe the residue fields of the monomial valuations. We also classify all the discrete rank one valuations of fields of…
Let $T$ be a polynomially bounded o-minimal theory extending the theory of real closed ordered fields. Let $K$ be a model of $T$ equipped with a $T$-convex valuation ring and a $T$-derivation. If this derivation is continuous with respect…
We study completeness in partial differential varieties. We generalize many results from ordinary differential fields to the partial differential setting. In particular, we establish a valuative criterion for differential completeness and…
We determine the most general group of equivalence transformations for a family of differential equations defined by an arbitrary vector field on a manifold. We also find all invariants and differential invariants for this group up to the…
For a simple, normal and finite extension of a valued field, we prove that we can related the order of the ramification group of the field extension and the set of key polynomials associated to the extension of the valuation. More…
A characteristic-dependent linear rank inequality is a linear inequality that holds by ranks of subspaces of a vector space over a finite field of determined characteristic, and does not in general hold over other characteristics. In this…
We show that for several notions of rank including tensor rank, Waring rank, and generalized rank with respect to a projective variety, the maximum value of rank is at most twice the generic rank. We show that over the real numbers, the…
In this paper we introduce a method of characteristic sets with respect to several term orderings for difference-differential polynomials. Using this technique, we obtain a method of computation of multivariate dimension polynomials of…
In this paper we present a characterization for the defect of a simple algebraic extension of rank one valued fields using the key polynomials that define the valuation. As a particular example, this gives the classification of defect…
We investigate fields of characteristic 0 and dp-rank 2. While we do not obtain a classification, we prove that any unstable field of characteristic 0 and dp-rank 2 admits a unique definable V-topology. If this statement could be…
In this paper, we study several definitions of generalized rank weights for arbitrary finite extensions of fields. We prove that all these definitions coincide, generalizing known results for extensions of finite fields.
We define the rank of elements of general unital rings, discuss its properties and give several examples to support the definition. In semiprime rings we give a characterization of rank in terms of invertible elements. As an application we…
In this paper, we study properties of nodal orders defined over arbitrary base fields. In particular we give a classification of complete real nodal orders.