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This is a copy of my question from math.stackexchange: https://math.stackexchange.com/questions/4517289/derivation-for-genus-degree-formula-from-algebraic-functions-field-theory. I didn't get any response there, so may be here is a more appropriate place to ask such a question.

The following is Proposition 3.11.5 from Stichtenoth's "Algebraic Function Fields and Codes".

Consider an algebraic function field (i.e. a field extension $K \subset F$ such that $F$ is a finite algebraic extension of $K(x)$ for some element $x \in F$ which is transcendental over $K$) $F = K(x,y)$ where the irreducible equation of $y$ over $K(x)$ has the form $$f(x,y) = y^n + f_1(x)y^{n−1} + \dotsc + f_{n−1}(x)y + f_n(x) = 0$$ with $f_j(x) \in K[x]$ and $\deg f_j(x) \leq j$ for $j = 1,\dotsc,n$. Then the genus $g$ of function field $F/K$ satisfies the inequality $$g \leq \frac{(n − 1)(n − 2)}{2}.$$

The key idea of the proof is to construct the Riemann-Roch space $\mathscr{L}(lA)$ for $A = (x)_\infty$ the pole divisor of $x$ and sufficiently large $l \geq n$ and estimate $\dim(\mathscr{L}(lA))$. It is then proved that elements $$ x_iy_j \text{ with } 0 \leq j \leq n−1 \text{ and } 0 \leq i \leq l−j \tag{$\ast$}$$ belong to $\mathscr{L}(lA)$ and are linear independent over $K$. Therefore $$\dim(\mathscr{L}(lA)) \geq \sum_{j=0}^{n-1} (l-j+1) = n(l + 1) − \frac{1}{2}n(n − 1).$$

From this and the Riemann-Roch theorem (we use the fact that $l$ could be taken arbitrary large): $$\dim(\mathscr{L}(lA)) = l · \deg A + 1 − g = ln + 1 − g$$ the upper bound for genus follows.

I know that in the case of nonsigular $f(x, y)$ (as polynomaial in variables $x, y$) the following bound for genus g is attained: this is precisely the reformulation of genus-degree formula for smooth curves. I wonder how to derive this formula from afforementioned proof. Basically, it boils down to showing that $(\ast)$ in the case of nonsingular $f(x,y)$ are not only linearly-independent but constitute the basis of $\mathscr{L}(lA)$ and I got stuck there. I do especially appreciate solutions that do not appeal to geometry - i.e. that do not refer to correspondense between function fields and curves and are solely based on function-field facts from Stichtenoth's book itself.

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