On the shape of possible counterexamples to the Jacobian Conjecture

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• 作者：Christian Valqui^d ; ^e ; ¹ ; ^{cvalqui@pucp.edu.pe" class="auth_mail" title="E-mail the corresponding author} ; Jorge A. Guccione^a ; ^b ; ² ; ^{vander@dm.uba.ar" class="auth_mail" title="E-mail the corresponding author} ; Juan J. Guccione^a ; ^c ; ² ; ^{jjgucci@dm.uba.ar" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Jacobian Conjecture ; Minimal counterexample
• 刊名：Journal of Algebra
• 出版年：2017
• 出版时间：1 February 2017
• 年：2017
• 卷：471
• 期：Complete
• 页码：13-74
• 全文大小：847 K

文摘

We improve the algebraic methods of Abhyankar for the Jacobian Conjecture in dimension two and describe the shape of possible counterexamples. We give an elementary proof of the result of Heitmann in [5], which states that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002186931630312X&_mathId=si1.gif&_user=111111111&_pii=S002186931630312X&_rdoc=1&_issn=00218693&md5=bbf6b6890f7fa14818d20212deb77436" title="Click to view the MathML source">gcd⁡(deg⁡(P),deg⁡(Q))≥16class="mathContainer hidden">class="mathCode">$\gcd (\mathrm{deg}(P),\mathrm{deg}(Q))\ge 16$ for any counterexample class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002186931630312X&_mathId=si2.gif&_user=111111111&_pii=S002186931630312X&_rdoc=1&_issn=00218693&md5=25b8cb397145e25cc2f362f54f350726" title="Click to view the MathML source">(P,Q)class="mathContainer hidden">class="mathCode">$(P,Q)$. We also prove that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002186931630312X&_mathId=si3.gif&_user=111111111&_pii=S002186931630312X&_rdoc=1&_issn=00218693&md5=228671777f194902168e1dc5f124c866" title="Click to view the MathML source">gcd⁡(deg⁡(P),deg⁡(Q))≠2pclass="mathContainer hidden">class="mathCode">$\gcd (\mathrm{deg}(P),\mathrm{deg}(Q))\ne 2p$ for any prime p.