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In [[real algebraic geometry]], the ''' Łojasiewicz inequality''', named after [[Stanisław Łojasiewicz]], gives an upper bound for the distance of a point to the nearest zero of a given [[real analytic function]].  Specifically, let ƒ&nbsp;:&nbsp;''U''&nbsp;→&nbsp;'''R''' be a real-analytic function on an [[open set]] ''U'' in '''R'''<sup>''n''</sup>, and let ''Z'' be the zero [[locus (mathematics)|locus]] of ƒ. Assume that ''Z'' is not empty.  Then for any [[compact set]] ''K'' in ''U'', there exist positive constants α and ''C'' such that, for all  ''x'' in ''K''
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:<math>\operatorname{dist}(x,Z)^\alpha \le C|f(x)|. \, </math>
 
Here α can be large.
 
The following form of this inequality is often seen in more analytic contexts:  with the same assumptions on ƒ, for every ''p''&nbsp;∈&nbsp;''U'' there is a possibly smaller open neighborhood ''W'' of ''p'' and constants θ&nbsp;∈&nbsp;(0,1) and ''c''&nbsp;>&nbsp;0 such that
 
:<math>|f(x)-f(p)|^\theta\le c|\nabla f(x)|. \, </math>
 
==References==
*{{Citation | last1=Bierstone | first1=Edward | last2=Milman | first2=Pierre D. | title=Semianalytic and subanalytic sets | id={{MathSciNet | id = 972342}} | year=1988 | journal=[[Publications Mathématiques de l'IHÉS]] | issn=1618-1913 | issue=67 | pages=5–42|url=http://www.numdam.org/item?id=PMIHES_1988__67__5_0}}
*{{Citation | doi=10.2307/2153965 | last1=Ji | first1=Shanyu | last2=Kollár | first2=János | last3=Shiffman | first3=Bernard | title=A global Łojasiewicz inequality for algebraic varieties | id={{MathSciNet | id = 1046016}} | url=http://www.ams.org/journals/tran/1992-329-02/S0002-9947-1992-1046016-6/ | year=1992 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=329 | issue=2 | pages=813–818 | jstor=2153965}}
 
==External links==
*[http://www.encyclopediaofmath.org/index.php/Lojasiewicz_inequality Lojasiewicz inequality] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
 
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{{DEFAULTSORT:Lojasiewicz inequality}}
[[Category:Inequalities]]
[[Category:Mathematical analysis]]
[[Category:Real algebraic geometry]]

Latest revision as of 17:59, 8 October 2014

My name is Numbers and I am studying Social Service and Modern Languages and Classics at Fruitport / United States.

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