Sergei A. Abramov

Dorodnicyn Computing Centre, Russian Academy of Science, Moscow, Russia

Linear Differential Systems with Infinite Power Series Cofficients: 

Infinite power series play an important role in mathematical studies. Those series may appear as inputs for certain mathematical problems. In order to be able to discuss the corresponding algorithms, we must agree on representation of the infinite series (algorithm inputs are always objects, represented by specific finite words in some alphabet). This talk examines two possible solutions to the problem of representation of power series. First, we consider the algorithmic representation. For each series, an algorithm is specified that, given an integer i, finds the coefficient of x^i. Any deterministic algorithms are allowed (any such algorithm defines a so called constructive series). Second, we consider a representation in an approximate form, namely, in a truncated form.

A more extensive and detailed abstract can be found in this PDF file.

Lihong Zhi

Mathematics Mechanization Research Center, Academia Sinica, Beijing, China

On Simple Multiple Zeros of Polynomial Systems: 

Given a polynomial system $f$ with a multiple zero of simple singularity, we provide a lower bound on the minimal distance between the simple multiple zero and other of $f$. For an approximate simple multiple zero $x$ of multiplicity $k$, we give a numerical criterion for $f$ has $k$ roots in the ball of radius $r$ around $x$. This is ongoing joint work with Zhiwei Hao, Wenrong Jiang, Nan Li.