User mfn - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T23:26:20Z http://mathoverflow.net/feeds/user/18459 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/128687/embdding-dimension Embdding dimension mfn 2013-04-25T06:56:37Z 2013-04-25T08:12:52Z <p>Take an open neighborhood $U$ of $p\in X$ which is a closed complex subvariety of a domain $D\subset \mathbb C^m$ with coordinates $z_1,\dots,z_m$. </p> <p>Let $f_1,\dots,f_k$ be functions on $D$ such that $O_{X,p}=O_{D,p}/(f_{1p},\dots,f_{kp})$, where $f_{ip}$ denotes the germ of $f_i$ at $p\in D$. </p> <p>We denote by $J_p(f_1,\dots,f_k)=\left(\frac{\partial f_i}{\partial z_j}(p)\right)$ the Jacobian matrix of the $f_i$ at $p$. </p> <p>Then $rank J_p(f_1,\dots,f_k)+embdim_pX=m$, where $embdim_pX$ is the embding dimension of $X$ at $p\in X$.</p> <p>What about some example to verify the above result?</p> http://mathoverflow.net/questions/77887/intersection-of-curves Intersection of curves mfn 2011-10-12T02:52:07Z 2013-04-25T07:05:22Z <p>Let $f(x,y)=0$ and $g(x,y)=0$ be curves in $\mathbb R^2$. Assume that the origin $(0,0)\in \mathbb R^2$ is a $d$-fold point of $f$ and an $e$-fold point of $g$, respectively. Let $f_d(x,y)$ be the sum of the terms of degree $d$ in $f(x,y)$, $g_e(x,y)$ be the sum of the terms of degree $e$ in $g(x,y)$. If $f_d(x,y)$ and $g_e(x,y)$ have a common factor of positive degree, then the intersection multiplicity $I_O(f,g)>de.$</p> http://mathoverflow.net/questions/118617/analytic-invariant Comment by mfn mfn 2013-01-13T13:18:39Z 2013-01-13T13:18:39Z Thank you very much for your reply.