The reduction to normal form of a non-normal system of differential equations (original) (raw)

This paper was edited by Sigismund Cohn, C. W. Borchardt and A. Clebsch from posthumous manuscripts of C. G. J. Jacobi. The solution of the following problem: "to transform a square table of m 2 numbers by adding minimal numbers i to each horizontal row, in such a way that it possess m transversal maxima", determines the order and the shortest normal form reduction of the system: the equations u i = 0 must be respectively differentiated i times. One also determines the number of differentiations of each equation of the given system needed to produce the differential equations necessary to reduce the proposed system to a single equation. Differential algebra • Order of a differential system • Jacobi's bound • Assignment problem • Differential resolvent • Shortest reduction in normal form Mathematics Subject Classification (2000) 12H05 • 65-03 • 65L80 • 65L08 • 90C05 • 90C27 Translator's comments The paper [Jacobi 2] is based on a single manuscript [II/23 b)], the transcription of which was considered and started by Cohn in 1859 (see his letter to Borchardt [II/13 a)] and Borchardt's abstract [II/25] that refers to his and Cohn's transcription). The first published version appeared in [VD], a volume edited by Clebsch. Nothing indicates C. G. J. Jacobi deceased on 1851.