Consider the complex numbers and consider a function $ F left (x, y right) $ and a curve $ C $ defined by $ F left (x, y right) = 0 $,

I know that to construct the sort associated with Jacob $ C $one integrates a basis of global holomorphic differential forms over the contours of the homology group of the curve. I am looking for information that is based on concrete calculations for specific examples. Everything I've seen so far has been meaningless abstract or unspecific. Note: I'm new to this area – I'm an analyst who knows next to nothing about algebra, even less about differential geometry or topology.

In search of a reasonable answer, I turned to the wonderful (albeit densely written) text of H. F. Baker of the early 20th century. Reading the first few pages makes it easy *plentiful* It will be appreciated that there is a general method of creating a basis of holomorphic differential forms for a given curve. Ted Shifrin's comment on this math stack exchange problem only makes me more confident than ever that the answers I seek are out there somewhere.

By and large, my goals are as follows. My goal is to be able to use the answers to these questions to compute various concrete examples, either manually or with the help of a computer algebra system. Therefore I am looking for formulas, explanations and / or stepwise procedures / algorithms and / or relevant reference / reading material.

(1) In the case where $ F $ is a polynomial, what is / are the procedure (s) for determining a basis of holomorphic differential 1 forms $ F $? When the method varies depending on certain characteristics of $ F $ (say, if $ F $ is an affine curve or a projective curve or of a certain shape or a similar detail), what are these variations?

(2) In the case where $ F $ is a polynomial of $ x $-Degree $ d_ {x} $, $ y $-Degree $ d_ {y} $, and $ C $ is a curve of the genus $ g $I know that the basis of holomorphic differential 1-forms for $ C $ will be of dimension $ g $, In that case we say where $ C $ is an elliptic curve with:

$$ F left (x, y right) = 4x ^ {3} -g_ {2} x-g_ {3} -y ^ {2} $$

the classic *Jacobi inversion problem* arises from the consideration of a function $ wp left (z right) $ which parameterizes $ C $in that sense that $ F left ( wp left (z right), wp ^ { prime} left (z right) right) $ is zero. With the equation: $$ F left ( wp left (z right), wp ^ { prime} left (z right) right) = 0 $$ we can write: $$ wp ^ {- 1} left (z right) = int_ {z_ {0}} ^ {z} frac {ds} {4s ^ {3} -g_ {2} s-g_ {3 }} $$ and know that the multiplicity of the integral then reflects the structure of the Jacobian variety $ C $,

That being said, in the case where $ C $ is of the genus $ g geq2 $and where we can write $ F left (x, y right) = 0 $

as: $$ y = textrm {algebraic function of} x $$ Nothing prevents us from performing exactly the same calculation as in the case of an elliptic curve. Of course, this calculation must be wrong. my question is: *where and how is it going wrong?*? How would the parameterization function obtained in this way relate to the "true" parameterization function – the associated multivariable Abel function $ C $? In addition, how can this calculation, if any, be changed in order to generate the correct parameterization function (Abel function)?

(3) I hope that by understanding (1) and (2) I am able to see what happens when these classical techniques are applied *not algebraic* even curves defined with $ F $ now an analytic function (with exponential functions and other transcendental functions in addition to polynomials). Of particular interest to me are the transcendental curves associated with exponential Diophantine equations, such as: $$ a ^ {x} -b ^ {y} = c $$

$$ y ^ {n} = b ^ {x} -a $$

I wonder if this has already happened. If so, links and references would be very welcome.

Even so, I would like to know the answers to my previous questions, even if it's just for my personal edification.

Thank you in advance!