Weierstrass polynomials and plane pseudo-holomorphic curves

For an almost complex structure J on U ⊂ ℝ⁴ pseudo-holomorphically fibered over ℂ a J-holomorphic curve C ⊂ U can be described by a Weierstrass polynomial. The J-holomorphicity equation descends to a perturbed ∂̄-operator on the coefficients; the operator is typically (0, 2/m)-Hölder continuous if m is the local degree of C over ℂ. This sheds some light on the problem of parametrizing pseudo-holomorphic deformations of J-holomorphic curve singularities.