Title: Effective H-fields

Topics: automatic asymptotics, differential algebra, computer algebra

Address

Research team: MAX, Algebraic modeling and symbolic computation

Contact

Context

The MAX team is searching for PhD candidates on the themes of the “ODELIX” ERC Advanced Grant. The present M2 internship proposal allows applicants to familiarize themselves with these themes. Upon successful completion of the internship, there will be an opportunity to pursue with a PhD.

Applying

Applications can be sent by email to Joris van der Hoeven; they should include a CV, a transcript of records, a letter of motivation, and optional recommendation letters. Note that we have no hard deadlines; new applications will be considered at any time of the year, as long as the Odelix project runs.

In a similar way as differential algebra was introduced in order to study solutions of differential equations in a purely formal way, the theory of -fields aims at studying the asymptotic behavior of such solutions from a formal perspective.

The language of -fields consists of the field operations , differentiation , and the ordering . The axioms for an -field comprise the usual ordered field and differential fields axioms, as well as two special axioms to express the compatability between the derivation and the ordering: setting , , , we require that (for all ) and .

The field of transseries is a typical example of an -field with a strong algebraic flavor. A transseries [4, 7, 1] is a generalized power series that is allowed to recursively involve exponentials and logarithms. For instance, and are transseries at infinity . On the analytic side, any Hardy field (e.g. a differential subfield of the ring of germs of differentiable real functions at infinity [5, 6, 3, 2]) is an -field.

The ordering on an -field naturally induces the asymptotic relations and defined by and . For this reason, -fields are a suitable framework for “asymptotic differential algebra” [1].

The aim of this internship is to develop an effective counterpart for the theory of -fields. Now there exist various algorithms for asymptotic computations in specific -fields like the field of “exp-log” functions [8]. The idea here is to develop this theory in a more abstract way. For this, we first define an -field to be effective if we can represent its elements on a computer and if we have algorithms for the operations and for the relations and . It is not hard to show that the real closure of an effective -field is again effective. The internship will start with showing that we can also effectively close under integration and exponentiation. After that, we will pursue with the study of more complicated asymptotic differential equations.