An efficient method for computing Liouvillian first integrals of planar polynomial vector fields

[摘要] Here we present an efficient method to compute Darboux polynomials for polynomial vector fields in the plane. This approach is restricted to polynomial vector fields presenting a Liouvillian first integral (or, equivalently, to rational first order differential equations (rational 1ODEs) presenting a Liouvillian general solution). The key to obtaining this method was to separate the procedure of solving the (non-linear) algebraic system resulting from the equation that translates the condition for the existence of a Darboux polynomial (i.e., from the equation D(p) = q p) into feasible steps (procedures that require less memory consumption). We also present a brief performance analysis of the algorithms developed. (C) 2021 Elsevier Inc. All rights reserved.

[发布日期] 2021-11-05

[关键词] Liouvillian first integrals;Planar polynomial vector fields;Darboux-Prelle-Singer methods;Darboux polynomials