A sufficient condition in order that the real Jacobian conjecture in holds

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• 作者：Francisco Braun^a ; ^{franciscobraun@dm.ufscar.br" class="auth_mail" title="E-mail the corresponding author} ; Jaume Giné ; ^b ; ^{gine@matematica.udl.cat" class="auth_mail" title="E-mail the corresponding author} ; Jaume Llibre^c ; ^{jllibre@mat.uab.cat" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 14R15 ; secondary ; 35F05 ; 35A30
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：15 March 2016
• 年：2016
• 卷：260
• 期：6
• 页码：5250-5258
• 全文大小：235 K

文摘

Let class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615006683&_mathId=si133.gif&_user=111111111&_pii=S0022039615006683&_rdoc=1&_issn=00220396&md5=4e3a1e0292eda329dc9f1450945a8a65" title="Click to view the MathML source">F=(f,g):R²→R²class="mathContainer hidden">class="mathCode">$F=(f,g):{\mathbb{R}}^2\to {\mathbb{R}}^2$ be a polynomial map such that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615006683&_mathId=si3.gif&_user=111111111&_pii=S0022039615006683&_rdoc=1&_issn=00220396&md5=97b2f025160216155a9de14166ece9f6" title="Click to view the MathML source">det⁡DF(x,y)class="mathContainer hidden">class="mathCode">$\det DF(x,y)$ is different from zero for all class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615006683&_mathId=si140.gif&_user=111111111&_pii=S0022039615006683&_rdoc=1&_issn=00220396&md5=35343c1c2fb50d946224ee5b9622a10a" title="Click to view the MathML source">(x,y)∈R²class="mathContainer hidden">class="mathCode">$(x,y)\in {\mathbb{R}}^2$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615006683&_mathId=si11.gif&_user=111111111&_pii=S0022039615006683&_rdoc=1&_issn=00220396&md5=1f9ab7748603426d0b8bb64450d8212d" title="Click to view the MathML source">F(0,0)=(0,0)class="mathContainer hidden">class="mathCode">$F(0,0)=(0,0)$. We prove that for the injectivity of F   it is sufficient to assume that the higher homogeneous terms of the polynomials class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615006683&_mathId=si12.gif&_user=111111111&_pii=S0022039615006683&_rdoc=1&_issn=00220396&md5=3e077d1d82140159318700360e411160" title="Click to view the MathML source">ff_x+gg_xclass="mathContainer hidden">class="mathCode">$f{f}_x+g{g}_x$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615006683&_mathId=si13.gif&_user=111111111&_pii=S0022039615006683&_rdoc=1&_issn=00220396&md5=579ac9160c152714a77c42731f295dd0" title="Click to view the MathML source">ff_y+gg_yclass="mathContainer hidden">class="mathCode">$f{f}_y+g{g}_y$ do not have real linear factors in common. The proofs are based on qualitative theory of dynamical systems.