16 research outputs found
Topics in algebra, geometry and differential equations
The study of differential equations and the study of algebraic geometry are two disciplines within mathematics that seem to be mostly disjoint from each other. Looking deeper, however, one finds that connections do exist. This thesis gives in four chapters four examples of interesting mathematical insights that can be gained from combining the concepts and techniques from both of these fields. The first project shows how the behaviour of solutions of certain differential equations can be better understood by considering algebraic curves with a differential form. The second project proves the existence of certain higher differential operators in algebraic settings where these were not known to occur before. The third project shows that the existence or non-existence of power series solutions of partial differential equations can be interpreted from the perspective of tropical geometry. And the last project relates the old theorem of Siegel about integral points on elliptic curves to the monodromy of linear differential equations on this elliptic curve
The level of pairs of polynomials
Given a polynomial with coefficients in a field of prime characteristic
, it is known that there exists a differential operator that raises to
its th power. We first discuss a relation between the `level' of this
differential operator and the notion of `stratification' in the case of
hyperelliptic curves. Next we extend the notion of level to that of a pair of
polynomials. We prove some basic properties and we compute this level in
certain special cases. In particular we present examples of polynomials and
such that there is no differential operator raising to its th
power.Comment: 14 pages, comments are welcom
Tropical initial degeneration for systems of algebraic differential equations
We study the notion of degeneration for affine schemes associated to systems
of algebraic differential equations with coefficients in the fraction field of
a multivariate formal power series ring. In order to do this, we use an
integral structure of this field that arises as the unit ball associated to the
tropical valuation, first introduced in the context of tropical differential
algebra. This unit ball turns out to be a particular type of integral domain,
known as B\'ezout domain. By applying to these systems a translation map along
a vector of weights that emulates the one used in classical tropical algebraic
geometry, the resulting translated systems will have coefficients in this unit
ball. When the resulting quotient module over the unit ball is torsion-free,
then it gives rise to integral models of the original system in which every
prime ideal of the unit ball defines an initial degeneration, and they can be
found as a base-change to the residue field of the prime ideal.
In particular, the closed fibres of our integral models can be rightfully
called initial degenerations, since we show that the maximal ideals of this
unit ball naturally correspond to monomial orders. We use this correspondence
to define initial forms of differential polynomials and initial ideals of
differential ideals, and we show that they share many features of their
classical analogues.Comment: 21 page
Autonomous first order differential equations
The problem of algebraic dependence of solutions to (non-linear) first order autonomous equations over an algebraically closed field of characteristic zero is given a ‘complete’ answer, obtained independently of model theoretic results on differentially closed fields. Instead, the geometry of curves and generalized Jacobians provides the key ingredient. Classification and formal solutions of autonomous equations are treated. The results are applied to answer a question on Dπ -finiteness of solutions of first order differential equations
On The Relationship Between Differential Algebra and Tropical Differential Algebraic Geometry
International audienceThis paper presents the relationship between differential algebra and tropical differential algebraic geometry, mostly focusing on the existence problem of formal power series solutions for systems of polynomial ODE and PDE. Moreover, it improves an approximation theorem involved in the proof of the Fundamental Theorem of tropical differential algebraic geometry which permits to improve this latter by dropping the base field uncountability hypothesis used in the original version
On The Relationship Between Differential Algebra and Tropical Differential Algebraic Geometry
International audienceThis paper presents the relationship between differential algebra and tropical differential algebraic geometry, mostly focusing on the existence problem of formal power series solutions for systems of polynomial ODE and PDE. Moreover, it improves an approximation theorem involved in the proof of the Fundamental Theorem of tropical differential algebraic geometry which permits to improve this latter by dropping the base field uncountability hypothesis used in the original version