Finally, it appears that, when $m>1$ and $h(0)\not=0$, it can be shown that there exists a local coordinate $w$ centered on $z=0$ such thatI write 'appears', because, while it's easy to show that there is an $a$ and a formal power series for $w$ that yields this normal form, I haven't gone through the details to show that the formal power series has a positive radius of convergence. In this case, the 'residue' is (up to a universal constant multiple) simply $a^l$. An alternative normal form is\frac{h(z)\ (The existence of the formal power series dz)^l}{z^{ml}} = \frac{(1 + b\ \zeta^{m-1})\ (d\zeta)^l}{\zeta^{ml}}for some constant $w$ already shows that there is no additional independent invariant that is a rational expression b$, and the nonlinear residue in this case is (a finite number universal constant multiple ofthe coefficients in the power series for ) $h$.)b^l$.
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R_{l,2l}\left(\frac{h(z)\ (dz)^l}{z^{2l}}\right) = \frac{h'(0)^l}{h(0)^{l-1}}\ h(0)\left(\frac{h'(0)}{h(0)}\right)^l,= h(0)\left(l\ frac{\bigl(l\ h(0)h''(0)-(l{-}1)\h'(0)^2\bigr)^l}{h(0)^{2l-1}}frac{h''(0)}{h(0)}-(l{-}1)\ \,= h(0)\left(l^2\ frac{\bigl(l^2\ h(0)^2h'''(0)-3l(l{-}1)h(0)h'(0)h''(0)+(2l{-}1)(l{-}1)\h'(0)^3\bigr)^l}{h(0)^{3l-1}}frac{h'''(0)}{h(0)}-3l(l{-}1)\frac{h'(0)h''(0)}{h(0)^2} +(2l{-}1)(l{-}1)\ \,
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Now, in all the cases in which $k=ml$, there is such an invariant. For example, when $k=4$ and $l=2$, k=2l$, one has the 'nonlinear residue'R_{2,4}\left(\frac{h(z)R_{l,2l}\left(\frac{h(z)\ (dz)^2}{z^4}\right) dz)^l}{z^{2l}}\right) = \frac{h'(0)^2}{h(0)}\ frac{h'(0)^l}{h(0)^{l-1}}\ ,R_{2,6}\left(\frac{h(z)R_{l,3l}\left(\frac{h(z)\ (dz)^2}{z^6}\right) frac{\bigl(l\ h(0)h''(0)-(l{-}1)\ h'(0)^2\bigr)^l}{h(0)^{2l-1}}\ ,R_{l,4l}\left(\frac{h(z)\ (dz)^l}{z^{4l}}\right) = \frac{\bigl(l^2\ h(0)^2h'''(0)-3l(l{-}1)h(0)h'(0)h''(0)+(2l{-}1)(l{-}1)\ h'(0)^3\bigr)^l}{h(0)^{3l-1}}\ ,where $\omega$ is a meromorphic $1$-form, well defined near $z=0$ up to multiplication by an $l$-th root of unity. The $l$-th power of the usual residue of $\omega$ at $z=0$ then provides an invariant that works out to be a rational expression in the coefficients of the power series of $h$, as in the cases noted above. Finally, it appears that, when $m>1$ and $h(0)\not=0$, there exists a local coordinate $w$ centered on $z=0$ such that\frac{h(z)\ (dz)^l}{z^{ml}} = \left(\frac{(1 + a\ w^{m-1})\ dw }{w^m}\right)^l\ .I write 'appears', because, while it's easy to show that there is an $a$ and a formal power series for $w$ that yields this normal form, I haven't gone through the details to show that the formal power series has a positive radius of convergence. In this case, the 'residue' is (up to a universal constant multiple) simply $a^l$. (The existence of the formal power series for $w$ already shows that there is no additional independent invariant that is a rational expression in a finite number of the coefficients in the power series for $h$.) |
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Let me point out a somewhat different answer. Rbega explicitly mentions $$ Q = \left(\frac1{z^3} + \frac1{z^2}\right)\ (dz)^2 $$ as an example of the sort of meromorphic quadratic differential that is of interest, and this is not at all covered by dakakov's dalakov's answer. In fact, consider Consider the more general problem of asking when there is a normal form (possibly with parameters) for differentials of the form $$ Q = \frac{h(z)\ (dz)^l}{z^k} dz)^l}{z^k}\ , $$ where $k$ and $l$ are fixed positive integers and $h$ is holomorphic near $z=0$ with $h(0)\not=0$. It is easy to see that Now, if when $k$ is not a multiple of $l$, then there exists a local coordinate $w$ centered on $z=0$ such that $$ Q = \frac{h(z)\ (dz)^l}{z^k} = \frac{(dw)^l}{w^k}, $$ so all of these differentials are locally equivalent. When However, when $k=ml$ for some integer $m>0$, though, the story is quite different. For example, it is clear that the 'residue' $$ R_l\left(\frac{h(z)R_{l,l}\left(\frac{h(z)\ (dz)^l}{z^l}\right) = h(0) $$ is well-defined, independent of the choice of 0-centered local coordinate $z$. In factMoreover, in this case, one can always find there exists a coordinate $w$ centered on $z=0$ such that $$ \frac{h(z)\ (dz)^l}{z^l} = \frac{h(0)\ (dw)^l}{w^l}, $$ so this the 'residue' $h(0)$ is the only invariant in the case $k=l$. In fact Now, in all the cases in which $k=ml$, there is such an invariant. For example, when $k=4$ and $l=2$, one has the 'nonlinear residue' $$ R_{2,4}\left(\frac{h(z)\ (dz)^2}{z^4}\right) = \frac{h'(0)^2}{h(0)} frac{h'(0)^2}{h(0)}\ , $$ which is independent of choice of coordinateswhen $k=4$ and $l=2$. . One also has $$ R_{2,6}\left(\frac{h(z)\ (dz)^2}{z^6}\right) = \frac{\left(h'(0)^2-2h(0)h''(0)\right)^2}{h(0)^3}, frac{\bigl(h'(0)^2-2h(0)h''(0)\bigr)^2}{h(0)^3}, $$ and so forth. The general rule is that, in these cases, one can write $$ Q = \frac{h(z)\ (dz)^l}{z^{ml}} = \left(\frac{\bigl(h(z)\bigr)^{1/l}\ dz}{z^m}\right)^l = \omega^l omega^l\ , $$ where $\omega$ is a meromorphic $1$-form, well-defined well defined near $z=0$ up to multiplication by an $l$-th root of unity. The $l$-th power of the usual residue of $\omega$ at $z=0$ then provides an invariant that works out to be a rational expression in the coefficients of the power series of $h$, as in the cases noted above. |
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