2021/10/11 by Cano, Jose, Falkensteiner, Sebastian, Robertz, Daniel +1
#12H05 Differential algebra #34A25 Analytical theory of ordinary differential equations #68W30 Symbolic computation and algebraic computation #Algebraic Geometry (math.AG) #FOS: Computer and information sciences #FOS: Mathematics #Symbolic Computation (cs.SC)
paper · doi:10.48550/arxiv.2110.05558
In this paper we study systems of autonomous algebraic ODEs in several differential indeterminates. We develop a notion of algebraic dimension of such systems by considering them as algebraic systems. Afterwards we apply differential elimination and analyze the behavior of the dimension in the resulting Thomas decomposition. For such systems of algebraic dimension one, we show that all formal Puiseux series solutions can be approximated up to an arbitrary order by convergent solutions. We show that the existence of Puiseux series and algebraic solutions can be decided algorithmically. Moreover, we present a symbolic algorithm to compute all algebraic solutions. The output can either be represented by triangular systems or by their minimal polynomials.