{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,28]],"date-time":"2022-03-28T22:50:21Z","timestamp":1648507821329},"reference-count":23,"publisher":"World Scientific Pub Co Pte Lt","issue":"02","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2007,3]]},"abstract":"<jats:p> Let z = (z<jats:sub>1<\/jats:sub>, z<jats:sub>2<\/jats:sub>,\u2026, z<jats:sub>n<\/jats:sub>) be noncommutative free variables and t a formal parameter which commutes with z. Let k be any unital integral domain of any characteristic and F<jats:sub>t<\/jats:sub>(z) = z - H<jats:sub>t<\/jats:sub>(z) with H<jats:sub>t<\/jats:sub>(z) \u2208 k[[t]]\u3008\u3008z\u3009\u3009<jats:sup>\u00d7n<\/jats:sup> and the order o(H<jats:sub>t<\/jats:sub>(z))\u2265 2. Note that F<jats:sub>t<\/jats:sub>(z) can be viewed as a deformation of the formal map F(z):= z - H<jats:sub>t=1<\/jats:sub>(z) when it makes sense (for example, when H<jats:sub>t<\/jats:sub>(z) \u2208 k[t]\u3008\u3008z\u3009\u3009<jats:sup>\u00d7n<\/jats:sup>). The inverse map G<jats:sub>t<\/jats:sub>(z) of F<jats:sub>t<\/jats:sub>(z) can always be written as G<jats:sub>t<\/jats:sub>(z) = z+M<jats:sub>t<\/jats:sub>(z) with M<jats:sub>t<\/jats:sub>(z) \u2208 k[[t]]\u3008\u3008z\u3009\u3009<jats:sup>\u00d7n<\/jats:sup> and o(M<jats:sub>t<\/jats:sub>(z)) \u2265 2. In this paper, we first derive the PDEs satisfied by M<jats:sub>t<\/jats:sub>(z) and u(F<jats:sub>t<\/jats:sub>), u(G<jats:sub>t<\/jats:sub>) \u2208 k[[t]]\u3008\u3008z\u3009\u3009 with u(z) \u2208 k\u3008\u3008z\u3009\u3009 in the general case as well as in the special case when H<jats:sub>t<\/jats:sub>(z) = tH(z) for some H(z) \u2208 k\u3008\u3008z\u3009\u3009<jats:sup>\u00d7n<\/jats:sup>. We also show that the elements above are actually characterized by certain Cauchy problems of these PDEs. Secondly, we apply the derived PDEs to prove a recurrent inversion formula for formal maps in noncommutative variables. Finally, for the case char. k = 0, we derive an expansion inversion formula by the planar binary rooted trees. <\/jats:p>","DOI":"10.1142\/s0218196707003676","type":"journal-article","created":{"date-parts":[[2007,3,23]],"date-time":"2007-03-23T09:11:58Z","timestamp":1174641118000},"page":"261-288","source":"Crossref","is-referenced-by-count":1,"title":["DEFORMATIONS AND INVERSION FORMULAS FOR FORMAL AUTOMORPHISMS IN NONCOMMUTATIVE VARIABLES"],"prefix":"10.1142","volume":"17","author":[{"given":"WENHUA","family":"ZHAO","sequence":"first","affiliation":[{"name":"Department of Mathematics, Illinois State University, Normal, IL 61790-4520, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","series-title":"Lectures in algebraic geometry","author":"Abhyankar S. S.","year":"1974"},{"key":"rf2","first-page":"240","volume":"53","author":"Andrews G. E.","journal-title":"Proc. Amer. Math. 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