FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2011/2012, VOLUME 17, NUMBER 1, PAGES 3-21

On singular points of solutions of linear differential systems with polynomial coefficients

S. A. Abramov
D. E. Khmelnov

Abstract

View as HTML     View as gif image

We consider systems of linear ordinary differential equations containing $m$ unknown functions of a single variable $x$. The coefficients of the systems are polynomials over a field $k$ of characteristic $0$. Each of the systems consists of $m$ equations independent over $k[x,d/dx]$. The equations are of arbitrary orders. We propose a computer algebra algorithm that, given a system $S$ of this form, constructs a polynomial $d(x)$Î k[x]\{0} such that if $S$ possesses a solution in $\$\; \backslash bar\; k((x-\backslash alpha\; ))^m\; \$$ for some $\$\; \backslash alpha\; \backslash in\; \backslash bar\; k\; \$$ and a component of this solution has a nonzero polar part, then $d($a)=0. In the case where $k$Í C and $S$ possesses an analytic solution having a singularity of an arbitrary type (not necessarily a pole) at $$a, the equality $d($a)=0 is also satisfied.

Location: http://mech.math.msu.su/~fpm/eng/k1112/k111/k11101h.htm
Last modified: January 31, 2012