\u4ee3\u6570\u5b66\u57fa\u672c\u5b9a\u7406\u5373\u5728\u57df $\\mathbb C$ \u4e2d\uff0c$n$ \u6b21\u591a\u9879\u5f0f\u65b9\u7a0b\u6709\u4e14\u4ec5\u6709 $n$ \u4e2a\u6839\u3002<\/p>\n
<\/p>\n
\u8fd8\u8bb0\u5f97\u7b2c\u4e00\u6b21\u63a5\u89e6\u4ee3\u6570\u5b66\u57fa\u672c\u5b9a\u7406\u662f\u5728\u521a\u5b66\u6570\u5b66\u5206\u6790\u505a\u4e00\u9053\u4e60\u9898\u65f6\uff1a<\/p>\n
\n\u8bc1\u660e\u4ee3\u6570\u6570\u96c6\u5408\u662f\u53ef\u6570\u96c6\u3002<\/p>\n<\/blockquote>\n
\u5f53\u65f6\u6570\u5206\u8001\u5e08\u8bf4\u8ba9\u6211\u4eec\u628a\u4ee3\u6570\u5b66\u57fa\u672c\u5b9a\u7406\u5f53\u6210\u4e00\u4e2a\u7ed3\u8bba\u4f7f\u7528\u6765\u8bc1\u660e\u8fd9\u4e2a\u9898\uff0c\u4f46\u662f\u8fd9\u5bf9\u4e8e\u6211\u6765\u8bf4\u8fd8\u662f\u975e\u5e38\u4e0d\u723d\u7684\u3002<\/p>\n
\u4e00\u65b9\u9762\uff0c\u8bc1\u660e\u8fd9\u4e2a\u9898\u5176\u5b9e\u5e76\u4e0d\u9700\u8981\u4ee3\u6570\u5b66\u57fa\u672c\u5b9a\u7406\uff1b\u53e6\u4e00\u65b9\u9762\uff0c\u6211\u6bd4\u8f83\u6297\u62d2\u4f7f\u7528\u4e00\u4e2a\u5bf9\u6211\u6765\u8bf4\u8fd8\u6ca1\u6709\u8bc1\u660e\u7684\u547d\u9898\u3002<\/p>\n
\u5728\u5b66\u4e60\u62d3\u6251\u5b66\u7684\u8fc7\u7a0b\u4e2d\uff0c\u6ca1\u60f3\u5230\u80fd\u5b66\u5230\u8fd9\u4e2a\u5b9a\u7406\u7684\u4e00\u4e2a\u7b80\u6d01\u800c\u53c8\u5de7\u5999\u7684\u8bc1\u660e\uff0c\u8fd9\u4ee4\u6211\u5341\u5206\u7684\u60ca\u559c\u3002<\/p>\n
\u672c\u6587\u5c06\u4ece\u62d3\u6251\u5b66\u7684\u89d2\u5ea6\u8bc1\u660e\u4ee3\u6570\u5b66\u57fa\u672c\u5b9a\u7406\u3002<\/p>\n
\u5b9a\u4e49 1<\/strong> $X,Y$ \u662f\u4e24\u4e2a\u62d3\u6251\u7a7a\u95f4\uff0c\u5982\u679c $f:X\\to Y,g:X\\to Y$ \u90fd\u662f\u8fde\u7eed\u6620\u5c04\u3002\u5982\u679c $\\exists H:X\\times[0,1]\\to Y $ \u662f\u8fde\u7eed\u6620\u5c04\uff0c\u6ee1\u8db3 $H(\\cdot,0)=f,H(\\cdot,1)=g$\uff0c\u5219\u79f0 $f$ \u548c $g$ \u662f\u540c\u4f26\u7684 (homotopic)<\/strong>\uff0c\u8bb0\u4f5c $f \\simeq g$. $H$ \u88ab\u79f0\u4f5c $f$ \u4e0e $g$ \u4e4b\u95f4\u7684\u540c\u4f26 (homotopy)<\/strong>\u3002<\/p>\n
\u5bb9\u6613\u9a8c\u8bc1\uff0c\u540c\u4f26\u662f\u4e00\u79cd\u7b49\u4ef7\u5173\u7cfb\u3002<\/p>\n
\u5b9a\u4e49 2<\/strong> \u5982\u679c $f$ \u548c\u4e00\u4e2a\u5e38\u503c\u51fd\u6570\u540c\u4f26\uff0c\u5219\u79f0 $f$ \u662f\u96f6\u4f26<\/strong>\u7684\uff0c\u8bb0\u4f5c $f\\simeq 0$\u3002<\/p>\n
\u5b9a\u4e49 3<\/strong> $S^1:=\\{e^{i\\theta};\\theta\\in [0,2\\pi)\\}$ \u662f\u5355\u4f4d\u5706\u5468<\/strong>\u3002\u6211\u4eec\u53ef\u4ee5\u628a $[0,2\\pi]$ \u5c06 $0$ \u548c $2\\pi$ \u9ecf\u5408\u7684\u65b9\u5f0f\u53bb\u5b9a\u4e49 $S^1$ \u4e0a\u7684\u62d3\u6251\u3002<\/p>\n
\u53ef\u4ee5\u6ce8\u610f\u5230\uff0c\u5728\u6b27\u6c0f\u7a7a\u95f4 $\\mathbb R^n\\to \\mathbb R^m$ \u4e0a\u7684\u6bcf\u4e00\u4e2a\u8fde\u7eed\u6620\u5c04\u90fd\u662f\u96f6\u4f26\u7684\uff08\u6362\u53e5\u8bdd\u8bf4\uff0c\u4efb\u610f\u4e24\u4e2a\u8fde\u7eed\u6620\u5c04\u90fd\u662f\u540c\u4f26\u7684\uff09\u3002<\/p>\n
\u4f46\u662f\uff0c\u5728 $S^1\\to S^1$ \u4e0a\u7684\u6620\u5c04\u5e76\u4e0d\u662f\u8fd9\u6837\uff0c\u6bd4\u65b9\u8bf4\u6052\u7b49\u6620\u5c04 $\\operatorname{id}:S^1\\to S^1,a\\mapsto a$ \u5c31\u4e0d\u662f\u96f6\u4f26\u7684\u3002\u867d\u7136\u76f4\u89c2\u7406\u89e3\u6bd4\u8f83\u597d\u7406\u89e3\uff0c\u4f46\u5176\u5b9e\u8bc1\u660e\u8fd9\u4e2a\u6620\u5c04\u4e0d\u662f\u96f6\u4f26\u7684\u662f\u6709\u70b9\u590d\u6742\u7684\uff0c\u5728\u672c\u6587\u4e2d\u4e0d\u4f5c\u8bc1\u660e\u3002<\/p>\n
\n\u6709\u4e86\u8fd9\u4e9b\u57fa\u7840\u7684\u6982\u5ff5\uff0c\u63a5\u4e0b\u6765\u6211\u4eec\u4fbf\u53ef\u4ee5\u7528\u4e00\u79cd\u5de7\u5999\u7684\u65b9\u5f0f\u8bc1\u660e\u4ee3\u6570\u5b66\u57fa\u672c\u5b9a\u7406\u3002<\/p>\n
\u8bc1\u660e\u5206\u4e3a\u4e24\u4e2a\u90e8\u5206\uff1a<\/p>\n
\n
- $\\mathbb C$ \u4e0a\u7684 $n$ \u6b21\u591a\u9879\u5f0f\u4e00\u5b9a\u5b58\u5728\u81f3\u5c11\u4e00\u4e2a\u6839\uff1b<\/li>\n
- \u6211\u4eec\u7528\u6570\u5b66\u5f52\u7eb3\u6cd5\u8bc1\u660e $n$ \u6b21\u591a\u9879\u5f0f\u6709\u4e14\u4ec5\u7531 $n$ \u4e2a\u6839\u3002<\/li>\n<\/ol>\n
\u5728\u7b2c\u4e00\u4e2a\u90e8\u5206\u4e2d\uff0c\u6211\u4eec\u5c06\u91c7\u7528\u53cd\u8bc1\u6cd5\u3002<\/p>\n
\u5047\u8bbe $f(x)=a_0+a_1x+\\dots+a_nx^n\\in \\mathbb C[x]$ \u662f\u5728 $\\mathbb C$ \u4e2d\u6ca1\u6709\u6839\u7684 $n$ \u6b21\u591a\u9879\u5f0f\u3002<\/p>\n
\u90a3\u4e48\u663e\u7136\u6709 $a_0\\ne 0$. \uff08\u56e0\u4e3a\u5982\u679c $a_0=0$ \u5219\u6709 $0$ \u8fd9\u4e2a\u6839\uff09<\/p>\n
\u5728\u63a5\u4e0b\u6765\u7684\u8bc1\u660e\u4e2d\uff0c\u6211\u4eec\u4e0d\u59a8\u8bbe $a_n=1$.<\/p>\n
\u5b9a\u4e49\u51fd\u6570 $f_r:S^1\\to S^1,e^{i\\theta}\\mapsto \\frac {f(re^{i\\theta})}{|f(re^{i\\theta})|},r\\in \\mathbb R_{\\ge 0}$. \u5219\u6613\u5f97 $f_r$ \u662f\u8fde\u7eed\u6620\u5c04\u3002<\/p>\n
\u5bf9\u4e8e\u4efb\u610f\u4e00\u4e2a $r\\in \\mathbb C$\uff0c\u5b9a\u4e49\u6620\u5c04 $H:S^1\\times [0,1]\\to S^1$, $H(\\cdot,t)=f_{tr}$\uff0c\u53ef\u4ee5\u5f97\u5230 $H$ \u662f\u4e00\u4e2a\u8fde\u7eed\u51fd\u6570\uff0c\u4e14 $H(\\cdot,0)$ \u662f\u4e00\u4e2a\u5e38\u503c\u51fd\u6570\u3002<\/p>\n
\u6240\u4ee5 $f_r\\simeq 0$ \u3002<\/p>\n
\u5bf9 $f_r$ \u7684\u8868\u8fbe\u5f0f\u505a\u4e00\u4e9b\u53d8\u6362\u53ef\u4ee5\u5f97\u5230\uff1a
\n$$f_r(e^{i\\theta})=\\frac{\\frac{a_0}{r^{n}}+\\frac{a_1}{r^{n-1}}e^{i\\theta}+\\dots+\\frac{a_{n-1}}{r}e^{i(n-1)\\theta}+e^{in\\theta}}{|\\frac{a_0}{r^{n}}+\\frac{a_1}{r^{n-1}}e^{i\\theta}+\\dots+\\frac{a_{n-1}}{r}e^{i(n-1)\\theta}+e^{in\\theta}|}
\n$$
\n\u7531\u6b64\u6613\u77e5\u5f53 $r\\to +\\infty$ \u65f6 $f_r$ \u4e00\u81f4\u6536\u655b\u4e8e $f_\\infty:S^1\\to S^1,e^{i\\theta}\\mapsto e^{in\\theta}$.<\/p>\n\u6240\u4ee5\u5b9a\u4e49\u6620\u5c04 $K:S^1\\times [0,1]\\to S^1$, $K(\\cdot,t)=\\begin{cases}f_{\\frac{r}{t}}, &t\\ne 0,\\\\f_\\infty,& t=0;\\end{cases}$ \u53ef\u4ee5\u5f97\u5230 $K$ \u662f\u4e00\u4e2a\u8fde\u7eed\u51fd\u6570\u3002\u6240\u4ee5 $f_\\infty\\simeq f_r\\simeq 0$.<\/p>\n
\u4f46 $f_{\\infty}$ \u4e0d\u662f\u96f6\u4f26\u7684\uff0c\u4ea7\u751f\u77db\u76fe\u3002<\/p>\n
\u7136\u540e\u6211\u4eec\u518d\u6765\u770b\u7b2c\u4e8c\u4e2a\u90e8\u5206\uff0c\u7b2c\u4e8c\u4e2a\u90e8\u5206\u5176\u5b9e\u5df2\u7ecf\u662f\u5f88\u5e73\u51e1\u7684\u4e86\u3002<\/p>\n
\n
- \n
\u5f53 $n=1$ \u65f6\uff0c\u663e\u7136\u6210\u7acb\u3002<\/p>\n<\/li>\n
- \n
\u82e5\u5f53 $n=k$ \u65f6\uff0c$k$ \u6b21\u591a\u9879\u5f0f\u6709\u4e14\u4ec5\u6709 $k$ \u4e2a\u6839\uff0c\u5219\u5f53 $n=k+1$ \u65f6\uff0c<\/p>\n
\u6211\u4eec\u7531\u8bc1\u660e\u7684\u7b2c\u4e00\u90e8\u5206\u77e5\u9053 $k+1$ \u6b21\u591a\u9879\u5f0f $f(x)$ \u4e00\u5b9a\u5b58\u5728\u4e00\u4e2a\u6839 $x_0$\uff0c\u5219\u7531\u5e26\u4f59\u9664\u6cd5\u77e5 $f(x)=(x-x_0)g(x)+f(x_0)=(x-x_0)g(x)$.<\/p>\n
\u7531\u4e8e $g(x)$ \u662f\u4e00\u4e2a $k$ \u6b21\u591a\u9879\u5f0f\uff0c\u5b83\u6709 $k$ \u4e2a\u6839\uff0c\u6240\u4ee5 $f(x)$ \u6709 $k+1$ \u4e2a\u6839\u3002<\/p>\n<\/li>\n<\/ul>\n
\u6570\u5b66\u5f52\u7eb3\u53ef\u77e5\uff0c\u4ee3\u6570\u5b66\u57fa\u672c\u5b9a\u7406\u6210\u7acb\u3002<\/p>\n
\n\u4e00\u4e2a\u4ee3\u6570\u5b66\u57fa\u672c\u5b9a\u7406\u7684\u63a8\u8bba\uff1a<\/p>\n
\u5bf9\u4e8e\u4efb\u610f\u7684 $f(x)\\in\\mathbb R[x]$, $f(x)$ \u53ef\u4ee5\u88ab\u5206\u89e3\u4e3a $\\mathbb R$ \u4e0a\u7684\u4e00\u6b21\u548c\u4e8c\u6b21\u591a\u9879\u5f0f\u7684\u4e58\u79ef\u3002<\/p>\n
\u8bbe $f(x)=a_0+a_1x+\\cdots+a_nx^n$, \u5176\u4e2d $a_j\\in \\mathbb R,\\forall j$<\/p>\n
\u7531\u4ee3\u6570\u5b66\u57fa\u672c\u5b9a\u7406\uff0c\u6211\u4eec\u77e5\u9053\u5728 $f(x)$ \u5728 $\\mathbb C$ \u4e0a\u53ef\u4ee5\u5206\u89e3\u4e3a\u4e00\u6b21\u591a\u9879\u5f0f\u7684\u4e58\u79ef\uff0c\u4f46\u662f\u5176\u4e2d\u4f1a\u6709\u4e00\u4e9b\u5f62\u5982 $x-x_0(x_0\\notin \\mathbb R)$ \u7684\u591a\u9879\u5f0f\u3002<\/p>\n
\u63a5\u4e0b\u6765\u6211\u4eec\u6765\u770b\u770b\u5982\u4f55\u628a\u8fd9\u79cd\u591a\u9879\u5f0f\u90fd\u7ec4\u5408\u4e3a\u4e8c\u6b21\u591a\u9879\u5f0f\u3002<\/p>\n
\u7531\u590d\u6570\u7684\u5171\u8f6d\u7684\u6027\u8d28\u6211\u4eec\u53ef\u4ee5\u77e5\u9053\uff1a
\n$$\\begin{align}
\na_0+a_1x+\\cdots+a_nx^n=0 &\\Leftrightarrow \\overline{a_0+a_1x+\\cdots+a_nx^n}=\\overline{0}\\\\
\n&\\Leftrightarrow a_0+a_1\\overline{x}+\\cdots+a_n+\\overline{x^n}=0\\\\
\n&\\Leftrightarrow a_0+a_1\\overline{x}+\\cdots+a_n+\\overline{x}^n=0
\n\\end{align}
\n$$
\n\u6240\u4ee5\u5982\u679c $x_0\\in \\mathbb C\\textbackslash\\mathbb R$ \u662f $f(x)$ \u7684\u865a\u6570\u6839\uff0c\u90a3\u4e48 $\\overline{x_0}$ \u4e5f\u662f $f(x)$ \u7684\u865a\u6570\u6839\u3002<\/p>\n\u6211\u4eec\u77e5\u9053 $(x-x_0)(x-\\overline{x_0})=x^2-(x_0+\\overline{x_0})x+|x_0|^2$<\/p>\n
\u800c $(x_0+\\overline{x_0})$ \u662f\u4e00\u4e2a\u5b9e\u6570\uff0c\u6240\u4ee5 $x^2-(x_0+\\overline{x_0})x+|x_0|^2\\in \\mathbb R[x]$\u3002<\/p>\n
\u7531\u4e0a\u9762\u7684\u6027\u8d28\u6613\u77e5\uff1a\u5982\u679c $x_0\\in \\mathbb C\\textbackslash \\mathbb R$ \u662f $f(x)$ \u7684\u6839\uff0c\u5219 $x^2-(x_0+\\overline{x_0})x+|x_0|^2\\mid f(x)$.<\/p>\n
\u518d\u7ed3\u5408\u6570\u5b66\u5f52\u7eb3\u6cd5\uff0c\u6211\u4eec\u4fbf\u53ef\u4ee5\u8bc1\u660e\u8fd9\u4e2a\u63a8\u8bba\u4e86\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"
\u4ee3\u6570\u5b66\u57fa\u672c\u5b9a\u7406\u5373\u5728\u57df $\\mathbb C$ \u4e2d\uff0c$n$ \u6b21\u591a\u9879\u5f0f\u65b9\u7a0b\u6709\u4e14\u4ec5\u6709 $n$ \u4e2a\u6839\u3002<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[11,8],"class_list":["post-366","post","type-post","status-publish","format-standard","hentry","category-6","tag-11","tag-8"],"_links":{"self":[{"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/posts\/366","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/qwq.cafe\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=366"}],"version-history":[{"count":3,"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/posts\/366\/revisions"}],"predecessor-version":[{"id":369,"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/posts\/366\/revisions\/369"}],"wp:attachment":[{"href":"https:\/\/qwq.cafe\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=366"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/qwq.cafe\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=366"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/qwq.cafe\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=366"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}