min/max of degenerate critical points and Newton diagrams - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T05:46:30Z http://mathoverflow.net/feeds/question/26615 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/26615/min-max-of-degenerate-critical-points-and-newton-diagrams min/max of degenerate critical points and Newton diagrams Dmitry Kerner 2010-05-31T18:30:37Z 2010-05-31T23:02:52Z <p>Given a smooth function of several variables, whose first derivatives vanish at the origin. Suppose the matrix of second derivatives is degenerate at the origin. For example all the second derivatives vanish.</p> <p>What are the ways of classical Calculus to check whether this is a min/max/saddle? (Some non-calculus ways?)</p> <p>For example, is the origin min/max/saddle for \$f(x,y)=x^{10}+x^2y^2+y^{10}-10000xy^8\$?</p> <p>Sometimes a case can be checked by a locally analytic change of variables, i.e. in a constructive manner. In most cases the needed change of variables is a local homeomorphism, i.e. smth non-constructive.</p> <p>Singularity theory provides some invariants that sometimes help to answer this question. (The simplest such invariant: the Newton diagram.) I do not know any general method to attack these problems.</p> <p>Suggestions?</p>