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"> 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">. 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"> for any prime p.