Existence and uniqueness theorems for formal power series solutions of analytic differential systems

We present Existence ant1 Uniqueness Theorems for formal pO\Wr series solutions Of ilnd~t.i(’ s\lSteIlls Of PDF. in il cmtain form. ‘This form can be obtained by it finil.c number of differediations and cliulinat.ic)us of the original systen~~ and allows its formal power series solut,ious t.0 lx coniput~ed in a11 alg0rithmic fashion. The result.ing reduced involutiw form (rif’ form) produced by our rif’ algorit,liui is a generalizitt.ion of the ClassiCal fornl of Riquier and .Janet; and that of CauchKOV~l.lC?\?h~iL I;(: waken the assumpt.ion of linearity iu the highest dermdves iu t~hosc approaches t.O allow for systcrns which are n0nlineiK in their highest deriva.t.ives. A new fornml developn~cx~t. of Riqnicr’s theory is given: with proofs. n~otleled after t,how in Griilmcr Basis Theory. For the uonlincar tllcoryz the concept of rclatiw Riquiel Bases is introduced. This allows for t.he easy esteusion of ideas from the linear t0 tlw nonlinear t,hrory. Tile essent.ial idea is that an arbitrary noulincar system can Ix writ.teu (aft.cr tliffcrcutiatiou if necessary), as il syst.cmi which is liw ear in its highw, dcrivat.ivcs, and a constraint syst,em. which is n0nlinear in its highclst, derivatives. Our t,heorems iwe applied t,o S6T~rill eximplcs.