Analytic reparametrization of semi-algebraic sets

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• 作者：Y. Yomdin
• 关键词：Semi-algebraic sets ; Analytic reparametrization ; Analytic complexity
• 刊名：Journal of Complexity
• 出版年：2008
• 出版时间：February 2008
• 年：2008
• 卷：24
• 期：1
• 页码：54-76
• 全文大小：303 K

文摘

In many problems in analysis, dynamics, and in their applications, it is important to subdivide objects under consideration into simple pieces, keeping control of high-order derivatives. It is known that semi-algebraic sets and mappings allow for such a controlled subdivision: this is the “me=""mml1"">style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6WHX-4NJ209N-2&_mathId=mml1&_user=10&_cdi=6862&_rdoc=8&_acct=C000050221&_version=1&_userid=10&md5=4e0ddf02e692a8cfdb2a01134849797a"" title=""Click to view the MathML source"">C<sup>ksup> reparametrization theorem” which is a high-order quantitative version of the well-known results on the existence of a triangulation of semi-algebraic sets. In a me=""mml2"">style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6WHX-4NJ209N-2&_mathId=mml2&_user=10&_cdi=6862&_rdoc=8&_acct=C000050221&_version=1&_userid=10&md5=d63c167a9d433c6f8998edcb4e89f362"" title=""Click to view the MathML source"">C<sup>ksup>-version we just require in addition that each simplex be represented as an image, under the “reparametrization mapping” me=""mml3"">style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6WHX-4NJ209N-2&_mathId=mml3&_user=10&_cdi=6862&_rdoc=8&_acct=C000050221&_version=1&_userid=10&md5=e6eaa4da66c3af93d1fbd3c5115d8d4a"" title=""Click to view the MathML source"">ψ, of the standard simplex, with all the derivatives of me=""mml4"">style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6WHX-4NJ209N-2&_mathId=mml4&_user=10&_cdi=6862&_rdoc=8&_acct=C000050221&_version=1&_userid=10&md5=8f0f7543af6d46c22bda7fa4f548d29a"" title=""Click to view the MathML source"">ψ up to order k uniformly bounded. The main result of this paper is, that if we reparametrize all the set A but its small part of a size me=""mml5"">style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6WHX-4NJ209N-2&_mathId=mml5&_user=10&_cdi=6862&_rdoc=8&_acct=C000050221&_version=1&_userid=10&md5=22fc1f20e5d12691c130c67563048a61"" title=""Click to view the MathML source"">δ, we can do much more: not only to “kill” the derivatives, but also to bound uniformly the analytic complexity of the pieces, while their number remains of order me=""mml6"">science?_ob=MathURL&_method=retrieve&_udi=B6WHX-4NJ209N-2&_mathId=mml6&_user=10&_cdi=6862&_rdoc=8&_acct=C000050221&_version=1&_userid=10&md5=a5abd758e523dcec42503b0c808651b0"">mg src=""http://www.sciencedirect.com/cache/MiamiImageURL/B6WHX-4NJ209N-2-J/0?wchp=dGLzVlz-zSkWW"" alt=""Click to view the MathML source"" align=""absbottom"" border=""0"" height=32 width=57>.