Continuity of critical points with respect to a parameterisation. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T09:34:31Z http://mathoverflow.net/feeds/question/116112 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/116112/continuity-of-critical-points-with-respect-to-a-parameterisation Continuity of critical points with respect to a parameterisation. Sam 2012-12-11T19:44:14Z 2012-12-11T21:21:19Z <p>Hello.</p> <p>I have a research note coming out soon, and I'm stuck showing that a weird kind of function is continuous. I need to it show a new method of bounding exponential growth factors in combinatorial classes.</p> <p>The function in question is $f: [0,1] \rightarrow (0,\infty)$, which sends a parameter $l \in [0,1]$ to the $z$ value of the unique positive critical point ($P''>0$) of a function $P_{S,l}(z) = \sum_{(i,j) \in S} z^{j\cdot l + i\cdot(1-l)}$, where $S \subset \{ 0,1,-1 \}^2$.</p> <p>For several different sets $S$, I have numerical experiments supporting the claim that this function is continuous. I've considered trying the $L^2$ norm, but I don't get very far before I'm swamped with unmanageable amounts of output. I'm looking for a shortcut that will give continuity, I'm not really concerned with how refined the bounds are. Any help or references are greatly appreciated.</p> <p>Cheers, Sam</p> http://mathoverflow.net/questions/116112/continuity-of-critical-points-with-respect-to-a-parameterisation/116113#116113 Answer by Michael Renardy for Continuity of critical points with respect to a parameterisation. Michael Renardy 2012-12-11T19:56:21Z 2012-12-11T19:56:21Z <p>If you know that P''>0, then the implicit function theorem should be applicable to give you continuity.</p>