The algeqtoseries(pol, x, y, order) command computes an expansion of all the branches at the origin.  The equation $\mathrm{pol}\left(x\,y\right)=0$ defines a multivalued function y(x).

If pos_slopes is specified, only those branches tending to 0 are computed. Note that this function is not designed to compute expansions to a large order. In this case, the differential equation should be used.  For more information, see algeqtodiffeq.

$\mathrm{with}\left(\mathrm{gfun}\right)\:$

$P≔y-x^2-x^3{y}^2+x^6{y}^5$

$P≔x^6{y}^5-x^3{y}^2-x^2+y$

$\mathrm{algeqtoseries}\left(P\,x\,y\,4\right)$

$\left[x^2+O\left(x^6\right)\,\frac{\mathrm{RootOf}\left({\mathrm{\_Z}}^4+1\right)}{x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}-\frac{{\mathrm{RootOf}\left({\mathrm{\_Z}}^4+1\right)}^2}{4}+\mathrm{O}\left(\sqrt{x}\right)\right]$

$\mathrm{algeqtoseries}\left(P\,x\,y\,10\,\mathrm{true}\right)$

$\left[x^2+x^7+O\left(x^{12}\right)\right]$