Tropical Differential Algebraic Geometry considers difficult or even intractable problems in Differential Equations and tries to extract information on their solutions from a restricted structure of the input. The fundamental theorem of Tropical Differential Algebraic Geometry and its extensions state that the support of power series solutions of systems of ordinary differential equations (with formal power series coefficients over an uncountable algebraically closed field of characteristic zero) can be obtained either, by solving a so-called tropicalized differential system, or by testing monomial-freeness of the associated initial ideals. Tropicalized differential equations work on a completely different algebraic structure which may help in theoretical and computational questions, particularly on the existence of solutions. We show here that both of these methods can be generalized to the case of systems of partial differential equations, this is, one can go either with the solution of tropicalized systems, or test monomial-freeness of the ideal generated by the initials when looking for supports of power series solutions of systems of differential equations, regardless the (finite) number of derivatives. The key are the vertex sets of Newton polytopes, upon which relies the definition of both tropical vanishing condition and the initial of a differential polynomial.

Tropical Differential Algebraic Geometry considers difficult or even intractable problems in Differential Equations and tries to extract information on their solutions from a restricted structure of the input. The Fundamental Theorem of Tropical Differential Algebraic Geometry states that the support of solutions of systems of ordinary differential equations with formal power series coefficients over an uncountable algebraically closed field of characteristic zero can be obtained by solving a so-called tropicalized differential system. Tropicalized differential equations work on a completely different algebraic structure which may help in theoretical and computational questions. We show that the Fundamental Theorem can be extended to the case of systems of partial differential equations by introducing vertex sets of Newton polytopes.

In this paper, we provide a differential algebra algorithm for integrating fractions of differential polynomials. It is not restricted to differential fractions that are the derivatives of other differential fractions. The algorithm leads to new techniques for representing differential fractions, which may help converting differential equations to integral equations (as for example used in parameter estimation).

We propose a new algorithm for converting a characteristic set of a prime differential ideal from one ranking into another. This differential algebra algorithm computes characteristic sets by change of ranking (ordering) for prime ideals. It identifies the purely algebraic subproblems which arise during differential computations and solves them algebraically. There are two improvements w.r.t. other approaches: formerly unsolved problems could be carried out; it is conceptually simple. Different variants are implemented.

In this paper, we give three theoretical and practical contributions for solving polynomial ODE or PDE systems. The first one is practical: an algorithm which improves the purely algebraic part of Rosenfeld—Gröbner (the polynomial ODE or PDE systems simplifier which is the core of the Maple 5.5 diffalg package). It is a variant of lextriangular but does not need any Gröbner basis computation. The second one is theoretical: a characterization of the output of Rosenfeld—Gröbner and a clarification of the existing relationship between algebraic and differential characteristic sets. The third one is theoretical as well as practical: an algorithm to compute canonical representatives of differential polynomials modulo regular differential ideals without any use of Gröbner bases. This algorithm simplifies the theory (somehow a “pedagogic” contribution) but permits us also to perform easily linear algebra over the base field in the factor differential ring defined by a regular differential ideal.

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