Title: Zero-testing for differentially algebraic transseries

Topics: automatic asymptotics, differential algebra, computer algebra

Address

Research team: MAX, Algebraic modeling and symbolic computation

Contacts

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 the above contact persons; 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.

How to compute exactly and reliably with special functions in computer algebra systems? One strategy is to adopt a local approach and systematically represent such functions using power series around some non-singular point. For instance, the function is the unique power series solution to the system of equations , , , with initial conditions , , . Assuming this kind of representation, how to check equalities like ?

Clearly, it suffices to have an algorithm for checking whether a given expression represent the zero function. When all power series are given as solutions of explicit differential equations with explicit initial conditions, several algorithms have been proposed for this zero test problem [2, 7, 6, 5]. One first challenge is to implement one or more of these algorithms and investigate possible improvements.

The aim of this internship is to generalize these algorithms to expressions that represent so-called transseries instead of ordinary power series [3, 4, 1]. A transseries is a generalized power series that is allowed to recursively involve exponentials and logarithms. For instance, and are transseries at infinity . Transseries naturally arise when studying the asymptotic behavior of solutions to ordinary differential equations.

We seek for excellent candidates with a background in mathematics and with an appetite for programming and making mathematical theory more effective.