{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,23]],"date-time":"2026-01-23T08:34:15Z","timestamp":1769157255384,"version":"3.49.0"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>New combinatorial properties of Catalan trees are established and used to prove a number of algebraic results related to the Jacobian conjecture.  Let $F=(x_1+H_1,x_2+H_2,\\dots,x_n+H_n)$ be a system of $n$ polynomials in $C[x_1,x_2,\\dots,x_n]$, the ring of polynomials in the variables $x_1,x_2, \\dots, x_n$ over the field of complex numbers.  Let $H=(H_1,H_2,\\dots,H_n)$.   Our principal algebraic result is that if the Jacobian of $F$ is equal to 1, the polynomials $H_i$ are each homogeneous of total degree 2, and $({{\\partial H_i}\\over {\\partial x_j}})^3=0$, then $H\\circ H\\circ H=0$ and $F$ has an  inverse of the form $G=(G_1,G_2,\\dots,G_n)$, where each $G_i$ is a polynomial of total degree $\\le6$.   We prove this by showing that the sum of weights of Catalan trees over certain equivalence classes is equal to zero.  We also show that if all of the polynomials $H_i$ are homogeneous of the same total degree $d\\ge2$ and $({{\\partial H_i}\\over {\\partial x_j}})^2=0$, then $H\\circ H=0$ and the inverse of $F$ is $G=(x_1-H_1,\\dots,x_n-H_n)$.<\/jats:p>","DOI":"10.37236\/1546","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T02:09:41Z","timestamp":1578708581000},"source":"Crossref","is-referenced-by-count":3,"title":["On Catalan Trees and the Jacobian Conjecture"],"prefix":"10.37236","volume":"8","author":[{"given":"Dan","family":"Singer","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2000,11,28]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v8i1r2\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v8i1r2\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T05:20:45Z","timestamp":1579324845000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v8i1r2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,11,28]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2001,1,1]]}},"URL":"https:\/\/doi.org\/10.37236\/1546","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2000,11,28]]},"article-number":"R2"}}