Root geometry of polynomial sequences I: Type

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• 作者：Jonathan L. Gross^a ; ¹ ; ^{gross@cs.columbia.edu" class="auth_mail" title="E-mail the corresponding author} ; Toufik Mansour^b ; ^{tmansour@univ.haifa.ac.il" class="auth_mail" title="E-mail the corresponding author} ; Thomas W. Tucker^c ; ² ; ^{ttucker@colgate.edu" class="auth_mail" title="E-mail the corresponding author} ; David G.L. Wang^d ; ^e ; ³ ; ^{glw@bit.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Genus distribution ; Real-rooted polynomial ; Recurrence ; Root geometry
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 January 2016
• 年：2016
• 卷：433
• 期：2
• 页码：1261-1289
• 全文大小：568 K

文摘

This paper concerns the distribution in the complex plane of the roots of a polynomial sequence class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15007660&_mathId=si2.gif&_user=111111111&_pii=S0022247X15007660&_rdoc=1&_issn=0022247X&md5=fa0176679a94c0b6743ecdec19497d33" title="Click to view the MathML source">{W_n(x)}_n≥0class="mathContainer hidden">class="mathCode">${\{W_n(x)\}}_{n\ge 0}$ given by a recursion class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15007660&_mathId=si3.gif&_user=111111111&_pii=S0022247X15007660&_rdoc=1&_issn=0022247X&md5=2e864317e95af470daa599c34967b610" title="Click to view the MathML source">W_n(x)=aW_n−1(x)+(bx+c)W_n−2(x)class="mathContainer hidden">class="mathCode">$W_n(x)=a{W}_{n-1}(x)+(bx+c)W_{n-2}(x)$, with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15007660&_mathId=si4.gif&_user=111111111&_pii=S0022247X15007660&_rdoc=1&_issn=0022247X&md5=8f02a00b6021da13957c28d428778758" title="Click to view the MathML source">W₀(x)=1class="mathContainer hidden">class="mathCode">$W_0(x)=1$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15007660&_mathId=si5.gif&_user=111111111&_pii=S0022247X15007660&_rdoc=1&_issn=0022247X&md5=eb8033f7f36d769d19580656ce1591ef" title="Click to view the MathML source">W₁(x)=t(x−r)class="mathContainer hidden">class="mathCode">$W_1(x)=t(x-r)$, where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15007660&_mathId=si6.gif&_user=111111111&_pii=S0022247X15007660&_rdoc=1&_issn=0022247X&md5=2e740956a8b562f2443cb33718be18d3" title="Click to view the MathML source">a>0class="mathContainer hidden">class="mathCode">$a>0$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15007660&_mathId=si7.gif&_user=111111111&_pii=S0022247X15007660&_rdoc=1&_issn=0022247X&md5=c225d2df96e46a901decc4e5f0505871" title="Click to view the MathML source">b>0class="mathContainer hidden">class="mathCode">$b>0$, and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15007660&_mathId=si8.gif&_user=111111111&_pii=S0022247X15007660&_rdoc=1&_issn=0022247X&md5=95c2838e7e84b0f2a412fa6e85630ea9" title="Click to view the MathML source">c,t,r∈Rclass="mathContainer hidden">class="mathCode">$c,t,r\in \mathbb{R}$. Our results include proof of the distinct-real-rootedness of every such polynomial class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15007660&_mathId=si20.gif&_user=111111111&_pii=S0022247X15007660&_rdoc=1&_issn=0022247X&md5=5bde8b13a3985d54a28c1d1b89690884" title="Click to view the MathML source">W_n(x)class="mathContainer hidden">class="mathCode">$W_n(x)$, derivation of the best bound for the zero-set class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15007660&_mathId=si10.gif&_user=111111111&_pii=S0022247X15007660&_rdoc=1&_issn=0022247X&md5=95cbe28bad0f756fc51a99dc03594ee1">class="imgLazyJSB inlineImage" height="17" width="224" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X15007660-si10.gif">class="mathContainer hidden">class="mathCode">$\{x|W_n(x)=0\text{for some}n\ge 1\}$, and determination of three precise limit points of this zero-set. Also, we give several applications from combinatorics and topological graph theory.